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Fractint Version 18.2 Page 1 New Features in Version 18.2...........................4 Introduction...........................................9 1. Fractint Commands.....................................11 1.1 Getting Started.....................................11 1.2 Plotting Commands...................................11 1.3 Zoom box Commands...................................14 1.4 Color Cycling Commands..............................16 1.5 Palette Editing Commands............................17 1.6 Image Save/Restore Commands.........................21 1.7 Print Command.......................................22 1.8 Parameter Save/Restore Commands.....................22 1.9 "3D" Commands.......................................24 1.10 Interrupting and Resuming...........................25 1.11 Orbits Window.......................................25 1.12 View Window.........................................26 1.13 Video Mode Function Keys............................27 1.14 Hints...............................................28 2. Fractal Types.........................................29 2.1 The Mandelbrot Set..................................29 2.2 Julia Sets..........................................30 2.3 Julia Toggle Spacebar Commands......................31 2.4 Inverse Julias......................................32 2.5 Newton domains of attraction........................33 2.6 Newton..............................................34 2.7 Complex Newton......................................34 2.8 Lambda Sets.........................................35 2.9 Mandellambda Sets...................................35 2.10 Circle..............................................36 2.11 Plasma Clouds.......................................36 2.12 Lambdafn............................................37 2.13 Mandelfn............................................38 2.14 Barnsley Mandelbrot/Julia Sets......................38 2.15 Barnsley IFS Fractals...............................39 2.16 Sierpinski Gasket...................................40 2.17 Quartic Mandelbrot/Julia............................41 2.18 Distance Estimator..................................41 2.19 Pickover Mandelbrot/Julia Types.....................41 2.20 Pickover Popcorn....................................42 2.21 Peterson Variations.................................42 2.22 Unity...............................................43 2.23 Scott Taylor / Lee Skinner Variations...............43 2.24 Kam Torus...........................................44 2.25 Bifurcation.........................................44 2.26 Orbit Fractals......................................46 2.27 Lorenz Attractors...................................47 2.28 Rossler Attractors..................................48 2.29 Henon Attractors....................................48 2.30 Pickover Attractors.................................49 2.31 Gingerbreadman......................................49 2.32 Martin Attractors...................................49 2.33 Icon................................................50 2.34 Test................................................50 2.35 Formula.............................................51 Fractint Version 18.2 Page 2 2.36 Julibrots...........................................52 2.37 Diffusion Limited Aggregation.......................54 2.38 Magnetic Fractals...................................54 2.39 L-Systems...........................................55 2.40 Lyapunov Fractals...................................57 2.41 fn||fn Fractals.....................................58 2.42 Halley..............................................59 2.43 Dynamic System......................................59 2.44 Mandelcloud.........................................60 2.45 Quaternion..........................................60 2.46 HyperComplex........................................61 2.47 Cellular Automata...................................61 2.48 Phoenix.............................................62 2.49 Frothy Basins.......................................63 3. Doodads, Bells, and Whistles..........................65 3.1 Drawing Method......................................65 3.2 Palette Maps........................................65 3.3 Autokey Mode........................................66 3.4 Distance Estimator Method...........................68 3.5 Inversion...........................................69 3.6 Decomposition.......................................70 3.7 Logarithmic Palettes and Color Ranges...............71 3.8 Biomorphs...........................................72 3.9 Continuous Potential................................72 3.10 Starfields..........................................74 4. "3D" Images...........................................76 4.1 3D Mode Selection...................................76 4.2 Select Fill Type Screen.............................79 4.3 Stereo 3D Viewing...................................80 4.4 Rectangular Coordinate Transformation...............81 4.5 3D Color Parameters.................................82 4.6 Light Source Parameters.............................83 4.7 Spherical Projection................................84 4.8 3D Overlay Mode.....................................84 4.9 Special Note for CGA or Hercules Users..............85 4.10 Making Terrains.....................................85 4.11 Making 3D Slides....................................87 4.12 Interfacing with Ray Tracing Programs...............87 5. Command Line Parameters, Parameter Files, Batch Mode..90 5.1 Using the DOS Command Line..........................90 5.2 Setting Defaults (SSTOOLS.INI File).................90 5.3 Parameter Files and the <@> Command.................91 5.4 General Parameter Syntax............................92 5.5 Startup Parameters..................................92 5.6 Calculation Mode Parameters.........................93 5.7 Fractal Type Parameters.............................94 5.8 Image Calculation Parameters........................94 5.9 Color Parameters....................................96 5.10 Doodad Parameters...................................99 5.11 File Parameters....................................100 5.12 Video Parameters...................................101 5.13 Sound Parameters...................................103 5.14 Printer Parameters.................................104 Fractint Version 18.2 Page 3 5.15 PostScript Parameters..............................105 5.16 PaintJet Parameters................................107 5.17 Plotter Parameters.................................108 5.18 3D Parameters......................................108 5.19 Batch Mode.........................................110 6. Hardware Support.....................................112 6.1 Notes on Video Modes, "Standard" and Otherwise.....112 6.2 "Disk-Video" Modes.................................114 6.3 Customized Video Modes, FRACTINT.CFG...............115 7. Common Problems......................................118 8. Fractals and the PC..................................121 8.1 A Little History...................................121 8.1.1 Before Mandelbrot................................121 8.1.2 Who Is This Guy, Anyway?.........................122 8.2 A Little Code......................................123 8.2.1 Periodicity Logic................................123 8.2.2 Limitations of Integer Math (And How We Cope)....123 8.2.3 The Fractint "Fractal Engine" Architecture.......124 Appendix A Mathematics of the Fractal Types................127 Appendix B Stone Soup With Pixels: The Authors.............143 Appendix C GIF Save File Format............................150 Appendix D Other Fractal Products..........................151 Appendix E Bibliography....................................152 Appendix F Other Programs..................................153 Appendix G Revision History................................154 Appendix H Version13 to Version 14 Type Mapping............168 Fractint Version 18.2 Page 4 New Features in Version 18.2 Versions 18.1 and 18.2 are bug-fix releases for version 18.0. Changes from 18.1 to 18.2 include: The command now causes filenames only to be written in PAR files. Fractint will now search directories in the PATH for files not found in the requested the requested directory or the current directory. If you place .MAP, .FRM, etc. in directories in your PATH, then Fractint will find them. Fixed bug that caused fractals using PI symmetry to fail at high resolution. Fractals interrupted with <3> or <r> can now resume. The palette editor's (undo) now works. The <s> command in orbit/Julia window mode is no longer case sensitive. Added warnings that the POV-Ray output is obsolete (but has been left in). Use POV-Ray's height field facility instead or create and convert RAW files. Fixed several IFS bugs. Changes from 18.0 to 18.1 include: Overlay tuning - the Mandelbrot/Julia Set fractals are now back up to 17.x speeds Disk Video modes now work correctly with VESA video adapters (they used to use the same array for different purposes, confusing each other) 1024x768x256 and 2048x2048x256 disk video modes work again Parameter-file processing no longer crashes Fractint if it attempts to run a formula requiring access to a non-existent FRM file IFS arrays no longer overrun their array space type=cellular fixes "autologmap=2" now correctly picks up the minimum color The use of disk-video mode with random-access fractal types is now legal (it generates a warning message but lets you proceed if you really want to) The Lsystems "spinning-wheel" now spins slower (removing needless overhead) Changes to contributors' addresses in the Help screens Fractint Version 18.2 Page 5 (The remainder of this "new features" section is from version 18.0) New fractal types: 19 new fractal types, including: New fractal types - 'lambda(fn||fn)', 'julia(fn||fn)', 'manlam(fn||fn)', 'mandel(fn||fn)', 'halley', 'phoenix', 'mandphoenix', 'cellular', generalized bifurcation, and 'bifmay' - from Jonathan Osuch. New Mandelcloud, Quaternion, Dynamic System, Cellular Automata fractal types from Ken Shirriff. New HyperComplex fractal types from Timothy Wegner New ICON type from Dan Farmer, including a PAR file of examples. New Frothy Basin fractal types (and PAR entries) by Wesley Loewer MIIM (Modified Inverse Iteration Method) implementation of Inverse Julia from Michael Snyder. New Inverse Julia fractal type from Juan Buhler. New floating-point versions of Markslambda, Marksmandel, Mandel4, and Julia4 types (chosen automatically if the floating-point option is enabled). New options/features: New assembler-based parser logic from Chuck Ebbert - significantly faster than the C-based code it replaces! New assembler-based Lyapunov logic from Nicholas Wilt and Wes Loewer. Roughly six times faster than the old version! New Orbits-on-a-window / Julia-in-a-window options: 1) The old Overlay option is now '#' (Shift-3). 2) During generation, 'O' brings up orbits (as before) - after generation, 'O' brings up new orbits Windows mode. 3) Control-O brings up new orbits Windows mode at any time. 4) Spacebar toggles between Inverse Julia mode and the Julia set and back to the Mandelbrot set. These new "in-a-window" modes are really neat! See Orbits Window (p. 25) and Julia Toggle Spacebar Commands (p. 31) for details. New multi-image GIF support in the command. You can now generate 65535x65535x256 fractal images using Fractint (if you have the disk space, of course). This option builds special PAR entries and a MAKEMIG.BAT file that you later use to invoke Fractint multiple times to generate individual sections of the image and (in a final step) stitch them all together. If your other software can't handle Multiple-image GIFs, a SIMPLGIF program is also supplied that converts MIGS into simgle-image GIFs. Press F1 at the prompts screen for details. Fractint Version 18.2 Page 6 Fractint's decoder now handles Multi-Image Gifs. New SuperVGA/VESA Autodetect logic from the latest version of VGAKIT. Sure hope we didn't break anything. New register-compatible 8514/A code from Jonathan Osuch. By default, Fractint now looks first for the presence of an 8514/A register- compatible adapter and then (and only if it doesn't find one) the presence of the 8514/A API (IE, HDILOAD is no longer necessary for register-compatible "8514/a" adapters). Fractint can be forced to use the 8514/A API by using a new command-line option, "afi=yes". Jonathan also added ATI's "8514/a-style" 800x600x256 and 1280x1024x16 modes. New XGA-detection logic for ISA-based XGA-2 systems. The palette editor now has a "freestyle" editing option. See Palette Editing Commands (p. 17) for details. Fractint is now more "batch file" friendly. When running Fractint from a batch file, pressing any key will cause Fractint to exit with an errorlevel = 2. Any error that interrupts an image save to disk will cause an exit with errorlevel = 2. Any error that prevents an image from being generated will cause an exit with errorlevel = 1. New Control-X, Control-Y, and Control-Z options flip a fractal image along the X-axis, Y-axis, and Origin, respectively. New area calculation mode in TAB screen from Ken Shirriff (for accuracy use inside=0). The TAB screen now indicates when the Integer Math algorithms are in use. The palette must now be explicitly changed, it will not reset to the default unexpectedly when doing things like switching video modes. The Julibrot type has been generalized. Julibrot fractals can now be generated from PAR files. Added command support for viewwindows. Added room for two additional PAR comments in the command New coloring method for IFS shows which parts of fractal came from which transform. Added attractor basin phase plotting for Julia sets from Ken Shirriff. Improved finite attractor code to find more attractors from Ken Shirriff. New zero function, to be used in PAR files to replace old integer tan, tanh Fractint Version 18.2 Page 7 Debugflag=10000 now reports video chipset in use as well as CPU/FPU type and available memory Added 6 additional parameters for params= for those fractal types that need them. New 'matherr()' logic lets Fractint get more aggressive when these errors happen. New autologmap option (log=+-2) from Robin Bussell that ensures that all palette values are used by searching the screen border for the lowest value and then setting log= to +- that color. Two new diffusion options - falling and square cavity. Three new Editpal commands: '!', '@' and '#' commands (that's <shift- 1>, <shift-2>, and <shift-3>) to swap R<->G, G<->B, R<->B. Parameter files now use a slightly shorter maximum line length, making them a bit more readable when stuffed into messages on Compuserve. Plasma now has 16-bit .POT output for use with Ray tracers. The "old" algorithm has been modified so that the plasma effect is independent of resolution. Slight modification to the Raytrace code to make it compatible with Rayshade 4.0 patch level 6. Improved boundary-tracing logic from Wesley Loewer. Command-line parameters can now be entered on-the-fly using the <g> key thanks to Ken Shirriff. Dithered gif images can now be loaded onto a b/w display. Thanks to Ken Shirriff. Pictures can now be output as compressed PostScript. Thanks to Ken Shirriff. Periodicity is a new inside coloring option. Thanks to Ken Shirriff. Fixes: symmetry values for the SQR functions, bailout for the floating- pt versions of 'lambdafn' and 'mandelfn' fractals from Jonathan Osuch. "Flip", "conj" operators are now selectable in the parser New DXF Raytracing option from Dennis Bragg. Improved boundary-tracing logic from Wesley Loewer. New MSC7-style overlay structure is used if MAKEFRAC.BAT specifies MSC7. (with new FRACTINT.DEF and FRACTINT.LNK files for MSC7 users). Several modules have been re-organized to take advantage of this new overlay capability if compiled under MSC7. Fractint Version 18.2 Page 8 Fractint now looks first any embedded help inside FRACTINT.EXE, and then for an external FRACTINT.HLP file before giving up. Previous releases required that the help text be embedded inside FRACTINT.EXE. Bug fixes: Corrected formulas displayed for Marksmandel, Cmplxmarksmandel, and associated julia types. BTM and precision fixes. Symmetry logic changed for various "outside=" options Symmetry value for EXP function in lambdafn and lambda(fn||fn) fixed. Fixed bug where math errors prevented save in batch mode. The <3> and <r> commands no longer destroy image -- user can back out with ESC and image is still there. Fixed display of correct number of Julibrot parameters, and Julibrot relaxes and doesn't constantly force ALTERN.MAP. Fixed tesseral type for condition when border is all one color but center contains image. Fixed integer mandel and julia when used with parameters > +1.99 and < -1.99 Eliminated recalculation when generating a julia type from a mandelbrot type when the 'z' screen is viewed for the first time. Minor logic change to prevent double-clutching into and out of graphics mode when pressing, say, the 'x' key from a menu screen. Changed non-US phone number for the Houston Public (Software) Library The "Y" screen is now "Extended Options" instead of "Extended Doodads" ...and probably a lot more bux-fixes that we've since forgotten that we've implemented. Fractint Version 18.2 Page 9 Introduction FRACTINT plots and manipulates images of "objects" -- actually, sets of mathematical points -- that have fractal dimension. See "Fractals and the PC" (p. 121) for some historical and mathematical background on fractal geometry, a discipline named and popularized by mathematician Benoit Mandelbrot. For now, these sets of points have three important properties: 1) They are generated by relatively simple calculations repeated over and over, feeding the results of each step back into the next -- something computers can do very rapidly. 2) They are, quite literally, infinitely complex: they reveal more and more detail without limit as you plot smaller and smaller areas. Fractint lets you "zoom in" by positioning a small box and hitting <Enter> to redraw the boxed area at full-screen size; its maximum linear "magnification" is over a trillionfold. 3) They can be astonishingly beautiful, especially using PC color displays' ability to assign colors to selected points, and (with VGA displays or EGA in 640x350x16 mode) to "animate" the images by quickly shifting those color assignments. For a demonstration of some of Fractint's features, run the demonstration file included with this release (DEMO.BAT) by typing "demo" at the DOS prompt. You can stop the demonstration at any time by pressing <Esc>. The name FRACTINT was chosen because the program generates many of its images using INTeger math, rather than the floating point calculations used by most such programs. That means that you don't need a math co- processor chip (aka floating point unit or FPU), although for a few fractal types where floating point math is faster, the program recognizes and automatically uses an 80x87 chip if it's present. It's even faster on systems using Intel's 80386 and 80486 microprocessors, where the integer math can be executed in their native 32-bit mode. Fractint works with many adapters and graphics modes from CGA to the 1024x768, 256-color XGA mode. Even "larger" images, up to 2048x2048x256, can be plotted to expanded memory, extended memory, or disk: this bypasses the screen and allows you to create images with higher resolution than your current display can handle, and to run in "background" under multi-tasking control programs such as DESQview and Windows 3. Fractint is an experiment in collaboration. Many volunteers have joined Bert Tyler, the program's first author, in improving successive versions. Through electronic mail messages, first on CompuServe's PICS forum and now on GRAPHDEV, new versions are hacked out and debugged a little at a time. Fractint was born fast, and none of us has seen any other fractal plotter close to the present version for speed, versatility, and all-around wonderfulness. (If you have, tell us so we can steal somebody else's ideas instead of each other's.) See The Stone Soup Story (p. 143) and A Word About the Authors (p. 144) for information about the authors, and see Contacting the Authors (p. 146) Fractint Version 18.2 Page 10 for how to contribute your own ideas and code. Fractint is freeware. The copyright is retained by the Stone Soup Group. Fractint may be freely copied and distributed in unmodified form but may not be sold. (A nominal distribution fee may be charged for media and handling by freeware and shareware distributors.) Fractint may be used personally or in a business - if you can do your job better by using Fractint, or using images from it, that's great! It may not be given away with commercial products without explicit permission from the Stone Soup Group. There is no warranty of Fractint's suitability for any purpose, nor any acceptance of liability, express or implied. ********************************************************************** * Contribution policy: Don't want money. Got money. Want admiration. * ********************************************************************** Source code for Fractint is also freely available - see Distribution of Fractint (p. 145). See the FRACTSRC.DOC file included with the source for conditions on use. (In most cases we just want credit.) Fractint Version 18.2 Page 11 1. Fractint Commands 1.1 Getting Started To start the program, enter FRACTINT at the DOS prompt. The program displays an initial "credits" screen. If Fractint doesn't start properly, please see Common Problems (p. 118). Hitting <Enter> gets you from the initial screen to the main menu. You can select options from the menu by moving the highlight with the cursor arrow keys and pressing <Enter>, or you can enter commands directly. As soon as you select a video mode, Fractint begins drawing an image - the "full" Mandelbrot set if you haven't selected another fractal type. For a quick start, after starting Fractint try one of the following: If you have MCGA, VGA, or better: <F3> If you have EGA: <F9> If you have CGA: <F5> Otherwise, monochrome: <F6> After the initial Mandelbrot image has been displayed, try zooming into it (see Zoom Box Commands (p. 14)) and color cycling (see Color Cycling Commands (p. 16)). Once you're comfortable with these basics, start exploring other functions from the main menu. Help is available from the menu and at most other points in Fractint by pressing the <F1> key. AT ANY TIME, you can hit a command key to select a function. You do not need to wait for a calculation to finish, nor do you have to return to the main menu. When entering commands, note that for the "typewriter" keys, upper and lower case are equivalent, e.g. and have the same result. Many commands and parameters can be passed to FRACTINT as command-line arguments or read from a configuration file; see "Command Line Parameters, Parameter Files, Batch Mode" for details. 1.2 Plotting Commands Function keys & various combinations are used to select a video mode and redraw the screen. For a quick start try one of the following: If you have MCGA, VGA, or better: <F3> If you have EGA: <F9> If you have CGA: <F5> Otherwise, monochrome: <F6> <F1> Display a help screen. The function keys available in help mode are displayed at the bottom of the help screen. Fractint Version 18.2 Page 12 <M> or <Esc> Return from a displayed image to the main menu. <Esc> From the main menu, <Esc> is used to exit from Fractint. <Delete> Same as choosing "select video mode" from the main menu. Goes to the "select video mode" screen. See Video Mode Function Keys (p. 27). <\> (previously <Home>) Redraw the previous image. The program tracks 25 sets of previous coordinates and fractal types, but does not remember other options which were different for those past images. <Tab> Display the current fractal type, parameters, video mode, screen or (if displayed) zoom-box coordinates, maximum iteration count, and other information useful in keeping track of where you are. The Tab function is non-destructive - if you press it while in the midst of generating an image, you will continue generating it when you return. The Tab function tells you if your image is still being generated or has finished - a handy feature for those overnight, 1024x768 resolution fractal images. If the image is incomplete, it also tells you whether it can be interrupted and resumed. (Any function other than <Tab> and <F1> counts as an "interrupt".) The Tab screen also includes a pixel-counting function, which will count the number of pixels colored in the inside color. This gives an estimate of the area of the fractal. Note that the inside color must be different from the outside color(s) for this to work; inside=0 is a good choice. <T> Select a fractal type. Move the cursor to your choice (or type the first few letters of its name) and hit <Enter>. Next you will be prompted for any parameters used by the selected type - hit <Enter> for the defaults. See Fractal Types (p. 29) for a list of supported types. <X> Select a number of eXtended options. Brings up a full-screen menu of options, any of which you can change at will. These options are: "passes=" - see Drawing Method (p. 65) Floating point toggle - see <F> key description below "maxiter=" - see Image Calculation Parameters (p. 94) "inside=" and "outside=" - see Color Parameters (p. 96) "savename=" filename - see File Parameters (p. 100) "overwrite=" option - see File Parameters (p. 100) "sound=" option - see Sound Parameters (p. 103) "logmap=" - see Logarithmic Palettes and Color Ranges (p. 71) "biomorph=" - see Biomorphs (p. 72) "decomp=" - see Decomposition (p. 70) "fillcolor=" - see Drawing Method (p. 65) Fractint Version 18.2 Page 13 <F> Toggles the use of floating-point algorithms (see "Limitations of Integer Math (And How We Cope)" (p. 123)). Whether floating point is in use is shown on the <Tab> status screen. The floating point option can also be turned on and off using the "X" options screen. If you have a non-Intel floating point chip which supports the full 387 instruction set, see the "FPU=" command in Startup Parameters (p. 92) to get the most out of your chip. <Y> More options which we couldn't fit under the <X> command: "finattract=" - see Finite Attractors (p. 139) "potential=" parameters - see Continuous Potential (p. 72) "invert=" parameters - see Inversion (p. 69) "distest=" parameters - see Distance Estimator Method (p. 68) "cyclerange=" - see Color Cycling Commands (p. 16) <Z> Modify the parameters specific to the currently selected fractal type. This command lets you modify the parameters which are requested when you select a new fractal type with the <T> command, without having to repeat that selection. You can enter "e" or "p" in column one of the input fields to get the numbers e and pi (2.71828... and 3.14159...). From the fractal parameters screen, you can press <F6> to bring up a sub parameter screen for the coordinates of the image's corners. <+> or <-> Switch to color-cycling mode and begin cycling the palette by shifting each color to the next "contour." See Color Cycling Commands (p. 16). <C> Switch to color-cycling mode but do not start cycling. The normally black "overscan" border of the screen changes to white. See Color Cycling Commands (p. 16). <E> Enter Palette-Editing Mode. See Palette Editing Commands (p. 17). <Spacebar> Toggle between Mandelbrot set images and their corresponding Julia-set images. Read the notes in Fractal Types, Julia Sets (p. 30) before trying this option if you want to see anything interesting. <J> Toggle between Julia escape time fractal and the Inverse Julia orbit fractal. See Inverse Julias (p. 32) <Enter> Enter is used to resume calculation after a pause. It is only necessary to do this when there is a message on the screen waiting to be acknowledged, such as the message shown after you save an image to disk. Modify 3D transformation parameters used with 3D fractal types such as "Lorenz3D" and 3D "IFS" definitions, including the selection of "funny glasses" (p. 80) red/blue 3D. Fractint Version 18.2 Page 14 <A> Convert the current image into a fractal 'starfield'. See Starfields (p. 74). <O> (the letter, not the number) If pressed while an image is being generated, toggles the display of intermediate results -- the "orbits" Fractint uses as it calculates values for each point. Slows the display a bit, but shows you how clever the program is behind the scenes. (See "A Little Code" in "Fractals and the PC" (p. 121).) <D> Shell to DOS. Return to Fractint by entering "exit" at a DOS prompt. <Insert> Restart at the "credits" screen and reset most variables to their initial state. Variables which are not reset are: savename, lightname, video, startup filename. 1.3 Zoom box Commands Zoom Box functions can be invoked while an image is being generated or when it has been completely drawn. Zooming is supported for most fractal types, but not all. The general approach to using the zoom box is: Frame an area using the keys described below, then <Enter> to expand what's in the frame to fill the whole screen (zoom in); or <Ctrl><Enter> to shrink the current image into the framed area (zoom out). With a mouse, double-click the left button to zoom in, double click the right button to zoom out. <Page Up>, <Page Down> Use <Page Up> to initially bring up the zoom box. It starts at full screen size. Subsequent use of these keys makes the zoom box smaller or larger. Using <Page Down> to enlarge the zoom box when it is already at maximum size removes the zoom box from the display. Moving the mouse away from you or toward you while holding the left button down performs the same functions as these keys. Using the cursor "arrow" keys or moving the mouse without holding any buttons down, moves the zoom box. Holding <Ctrl> while pressing cursor "arrow" keys moves the box 5 times faster. (This only works with enhanced keyboards.) Panning: If you move a fullsize zoombox and don't change anything else before performing the zoom, Fractint just moves what's already on the screen and then fills in the new edges, to reduce drawing time. This feature applies to most fractal types but not all. A side effect is that while an image is incomplete, a full size zoom box moves in steps larger than one pixel. Fractint keeps the box on multiple pixel boundaries, to make panning possible. As a multi-pass (e.g. solid guessing) image approaches completion, the zoom box can move in smaller increments. Fractint Version 18.2 Page 15 In addition to resizing the zoom box and moving it around, you can do some rather warped things with it. If you're a new Fractint user, we recommend skipping the rest of the zoom box functions for now and coming back to them when you're comfortable with the basic zoom box functions. <Ctrl><Keypad->, <Ctrl><Keypad+> Holding <Ctrl> and pressing the numeric keypad's + or - keys rotates the zoom box. Moving the mouse left or right while holding the right button down performs the same function. <Ctrl><Page Up>, <Ctrl><Page Down> These commands change the zoom box's "aspect ratio", stretching or shrinking it vertically. Moving the mouse away from you or toward you while holding both buttons (or the middle button on a 3-button mouse) down performs the same function. There are no commands to directly stretch or shrink the zoom box horizontally - the same effect can be achieved by combining vertical stretching and resizing. <Ctrl><Home>, <Ctrl><End> These commands "skew" the zoom box, moving the top and bottom edges in opposite directions. Moving the mouse left or right while holding both buttons (or the middle button on a 3-button mouse) down performs the same function. There are no commands to directly skew the left and right edges - the same effect can be achieved by using these functions combined with rotation. <Ctrl><Insert>, <Ctrl><Delete> These commands change the zoom box color. This is useful when you're having trouble seeing the zoom box against the colors around it. Moving the mouse away from you or toward you while holding the right button down performs the same function. You may find it difficult to figure out what combination of size, position rotation, stretch, and skew to use to get a particular result. (We do.) A good way to get a feel for all these functions is to play with the Gingerbreadman fractal type. Gingerbreadman's shape makes it easy to see what you're doing to him. A warning though: Gingerbreadman will run forever, he's never quite done! So, pre-empt with your next zoom when he's baked enough. If you accidentally change your zoom box shape or rotate and forget which way is up, just use <PageDown> to make it bigger until it disappears, then <PageUp> to get a fresh one. With a mouse, after removing the old zoom box from the display release and re-press the left button for a fresh one. If your screen does not have a 4:3 "aspect ratio" (i.e. if the visible display area on it is not 1.333 times as wide as it is high), rotating and zooming will have some odd effects - angles will change, including the zoom box's shape itself, circles (if you are so lucky as to see any with a non-standard aspect ratio) become non-circular, and so on. The vast majority of PC screens *do* have a 4:3 aspect ratio. Fractint Version 18.2 Page 16 Zooming is not implemented for the plasma and diffusion fractal types, nor for overlayed and 3D images. A few fractal types support zooming but do not support rotation and skewing - nothing happens when you try it. 1.4 Color Cycling Commands Color-cycling mode is entered with the 'c', '+', or '-' keys from an image, or with the 'c' key from Palette-Editing mode. The color-cycling commands are available ONLY for VGA adapters and EGA adapters in 640x350x16 mode. You can also enter color-cycling while using a disk-video mode, to load or save a palette - other functions are not supported in disk-video. Note that the colors available on an EGA adapter (16 colors at a time out of a palette of 64) are limited compared to those of VGA, super-VGA, and MCGA (16 or 256 colors at a time out of a palette of 262,144). So color-cycling in general looks a LOT better in the latter modes. Also, because of the EGA palette restrictions, some commands are not available with EGA adapters. Color cycling applies to the color numbers selected by the "cyclerange=" command line parameter (also changeable via the <Y> options screen and via the palette editor). By default, color numbers 1 to 255 inclusive are cycled. On some images you might want to set "inside=0" (<X> options or command line parameter) to exclude the "lake" from color cycling. When you are in color-cycling mode, you will either see the screen colors cycling, or will see a white "overscan" border when paused, as a reminder that you are still in this mode. The keyboard commands available once you've entered color-cycling. are described below. <F1> Bring up a HELP screen with commands specific to color cycling mode. <Esc> Leave color-cycling mode. <+> or <-> Begin cycling the palette by shifting each color to the next "contour." <+> cycles the colors in one direction, <-> in the other. '<' or '>' Force a color-cycling pause, disable random colorizing, and single-step through a one color-cycle. For "fine-tuning" your image colors. Cursor up/down Increase/decrease the cycling speed. High speeds may cause a harmless flicker at the top of the screen. <F2> through <F10> Switches from simple rotation to color selection using randomly generated color bands of short (F2) to long (F10) duration. Fractint Version 18.2 Page 17 <1> through <9> Causes the screen to be updated every 'n' color cycles (the default is 1). Handy for slower computers. <Enter> Randomly selects a function key (F2 through F10) and then updates ALL the screen colors prior to displaying them for instant, random colors. Hit this over and over again (we do). <Spacebar> Pause cycling with white overscan area. Cycling restarts with any command key (including another spacebar). <Shift><F1>-<F10> Pause cycling and reset the palette to a preset two color "straight" assignment, such as a spread from black to white. (Not for EGA) <Ctrl><F1>-<F10> Pause & set a 2-color cyclical assignment, e.g. red->yellow->red (not EGA). <Alt><F1>-<F10> Pause & set a 3-color cyclical assignment, e.g. green->white->blue (not EGA). <R>, <G>, Pause and increase the red, green, or blue component of all colors by a small amount (not for EGA). Note the case distinction of this vs: <r>, <g>, Pause and decrease the red, green, or blue component of all colors by a small amount (not for EGA). <D> or <A> Pause and load an external color map from the files DEFAULT.MAP or ALTERN.MAP, supplied with the program. <L> Pause and load an external color map (.MAP file). Several .MAP files are supplied with Fractint. See Palette Maps (p. 65). <S> Pause, prompt for a filename, and save the current palette to the named file (.MAP assumed). See Palette Maps (p. 65). 1.5 Palette Editing Commands Palette-editing mode provides a number of tools for modifying the colors in an image. It can be used only with MCGA or higher adapters, and only with 16 or 256 color video modes. Many thanks to Ethan Nagel for creating the palette editor. Use the <E> key to enter palette-editing mode from a displayed image or from the main menu. Fractint Version 18.2 Page 18 When this mode is entered, an empty palette frame is displayed. You can use the cursor keys to position the frame outline, and <Pageup> and <Pagedn> to change its size. (The upper and lower limits on the size depend on the current video mode.) When the frame is positioned where you want it, hit Enter to display the current palette in the frame. Note that the palette frame shows R(ed) G(reen) and B(lue) values for two color registers at the top. The active color register has a solid frame, the inactive register's frame is dotted. Within the active register, the active color component is framed. Using the commands described below, you can assign particular colors to the registers and manipulate them. Note that at any given time there are two colors "X"d - these are pre-empted by the editor to display the palette frame. They can be edited but the results won't be visible. You can change which two colors are borrowed ("X"d out) by using the <v> command. Once the palette frame is displayed and filled in, the following commands are available: <F1> Bring up a HELP screen with commands specific to palette-editing mode. <Esc> Leave palette-editing mode <H> Hide the palette frame to see full image; the cross-hair remains visible and all functions remain enabled; hit <H> again to restore the palette display. Cursor keys Move the cross-hair cursor around. In 'auto' mode (the default) the color under the center of the cross-hair is automatically assigned to the active color register. Control-Cursor keys move the cross-hair faster. A mouse can also be used to move around. <R> <G> Select the Red, Green, or Blue component of the active color register for subsequent commands <Insert> <Delete> Select previous or next color component in active register <+> <-> Increase or decrease the active color component value by 1 Numeric keypad (gray) + and - keys do the same. <Pageup> <Pagedn> Increase or decrease the active color component value by 5; Moving the mouse up/down with left button held is the same <0> <1> <2> <3> <4> <5> Set the active color component's value to 0 10 20 ... 60 Fractint Version 18.2 Page 19 <Space> Select the other color register as the active one. In the default 'auto' mode this results in the now-inactive register being set to remember the color under the cursor, and the now-active register changing from whatever it had previously remembered to now follow the color. <,> <.> Rotate the palette one step. By default colors 1 through 255 inclusive are rotated. This range can be over-ridden with the "cyclerange" parameter, the <Y> options screen, or the <O> command described below. "<" ">" Rotate the palette continuously (until next keystroke) <O> Set the color cycling range to the range of colors currently defined by the color registers. <C> Enter Color-Cycling Mode. When you invoke color-cycling from here, it will subsequently return to palette-editing when you <Esc> from it. See Color Cycling Commands (p. 16). <=> Create a smoothly shaded range of colors between the colors selected by the two color registers. <M> Specify a gamma value for the shading created by <=>. <D> Duplicate the inactive color register's values to the active color register. <T> Stripe-shade - create a smoothly shaded range of colors between the two color registers, setting only every Nth register. After hitting <T>, hit a numeric key from 2 to 9 to specify N. For example, if you press <T> <3>, smooth shading is done between the two color registers, affecting only every 3rd color between them. The other colors between them remain unchanged. <W> Convert current palette to gray-scale. (If the <X> or <Y> exclude ranges described later are in force, only the active range of colors is converted to gray-scale.) <Shift-F2> ... <Shift-F9> Store the current palette in a temporary save area associated with the function key. The temporary save palettes are useful for quickly comparing different palettes or the effect of some changes - see next command. The temporary palettes are only remembered until you exit from palette-editing mode. Fractint Version 18.2 Page 20 <F2> ... <F9> Restore the palette from a temporary save area. If you haven't previously saved a palette for the function key, you'll get a simple grey scale. <L> Pause and load an external color map (.MAP file). See Palette Maps (p. 65). <S> Pause, prompt for a filename, and save the current palette to the named file (.MAP assumed). See Palette Maps (p. 65). Invert frame colors. With some colors the palette is easier to see when the frame colors are interchanged. <\> Move or resize the palette frame. The frame outline is drawn - it can then be repositioned and sized with the cursor keys, <Pageup> and <Pagedn>, just as was done when first entering palette-editing mode. Hit Enter when done moving/sizing. <V> Use the colors currently selected by the two color registers for the palette editor's frame. When palette editing mode is entered, the last two colors are "X"d out for use by the palette editor; this command can be used to replace the default with two other color numbers. <A> Toggle 'auto' mode on or off. When on (the default), the active color register follows the cursor; when off, <Enter> must be pressed to set the active register to the color under the cursor. <Enter> Only useful when 'auto' is off, as described above; double clicking the left mouse button is the same as Enter. <X> Toggle 'exclude' mode on or off - when toggled on, only those image pixels which match the active color are displayed. <Y> Toggle 'exclude' range on or off - similar to <X>, but all pixels matching colors in the range of the two color registers are displayed. <N> Make a negative color palette - will convert only current color if in 'x' mode or range between editors in 'y' mode or entire palette if in "normal" mode. <!> <@> <#> Swap R<->G, G<->B, and R<->B columns. These keys are shifted 1, 2, and 3, which you may find easier to remember. Fractint Version 18.2 Page 21 Undoes the last palette editor command. Will undo all the way to the beginning of the current session. <E> Redoes the undone palette editor commands. <F> Toggles "Freestyle mode" on and off (Freestyle mode changes a range of palette values smoothly from a center value outward). With your cursor inside the palette box, press the <F> key to enter Freestyle mode. A default range of colors will be selected for you centered at the cursor (the ends of the color range are noted by putting dashed lines around the corresponding palette values). While in Freestyle mode: Moving the mouse changes the location of the range of colors that are affected. Control-Insert/Delete or the shifted-right-mouse-button changes the size of the affected palette range. The normal color editing keys (R,G,B,1-6, etc) set the central color of the affected palette range. Pressing ENTER or double-clicking the left mouse button makes the palette changes permanent (if you don't perform this step, any palette changes disappear when you press the <F> key again to exit freestyle mode). 1.6 Image Save/Restore Commands <S> saves the current image to disk. All parameters required to recreate the image are saved with it. Progress is marked by colored lines moving down the screen's edges. The default filename for the first image saved after starting Fractint is FRACT001.GIF; subsequent saves in the same session are automatically incremented 002, 003... Use the "savename=" parameter or <X> options screen to change the name. By default, files left over from previous sessions are not overwritten - the first unused FRACTnnn name is used. Use the "overwrite=yes" parameter or <X> options screen) to overwrite existing files. A save operation can be interrupted by pressing any key. If you interrupt, you'll be asked whether to keep or discard the partial file. <R> restores an image previously saved with <S>, or an ordinary GIF file. After pressing <R> you are shown the file names in the current directory which match the current file mask. To select a file to restore, move the cursor to it (or type the first few letters of its name) and press <Enter>. Directories are shown in the file list with a "\" at the end of the name. When you select a directory, the contents of that directory are shown. Or, you can type the name of a different directory (and optionally a different drive) and press <Enter> for a new display. You can also type a mask such as "*.XYZ" and press <Enter> to display files whose name ends with the matching suffix (XYZ). Fractint Version 18.2 Page 22 You can use <F6> to switch directories to the default fractint directory or to your own directory which is specified through the DOS environment variable "FRACTDIR". Once you have selected a file to restore, a summary description of the file is shown, with a video mode selection list. Usually you can just press <Enter> to go past this screen and load the image. Other choices available at this point are: Cursor keys: select a different video mode <Tab>: display more information about the fractal <F1>: for help about the "err" column in displayed video modes If you restore a file into a video mode which does not have the same pixel dimensions as the file, Fractint will make some adjustments: The view window parameters (see <V> command) will automatically be set to an appropriate size, and if the image is larger than the screen dimensions, it will be reduced by using only every Nth pixel during the restore. 1.7 Print Command Print the current fractal image on your (Laserjet, Paintjet, Epson- compatible, PostScript, or HP-GL) printer. See "Setting Defaults (SSTOOLS.INI File)" (p. 90) and "Printer Parameters" (p. 104) for how to let Fractint know about your printer setup. "Disk-Video" Modes (p. 114) can be used to generate images for printing at higher resolutions than your screen supports. 1.8 Parameter Save/Restore Commands Parameter files can be used to save/restore all options and settings required to recreate particular images. The parameters required to describe an image require very little disk space, especially compared with saving the image itself. <@> The <@> command loads a set of parameters describing an image. (Actually, it can also be used to set non-image parameters such as SOUND, but at this point we're interested in images. Other uses of parameter files are discussed in "Parameter Files and the <@> Command" (p. 91).) When you hit <@>, Fractint displays the names of the entries in the currently selected parameter file. The default parameter file, FRACTINT.PAR, is included with the Fractint release and contains parameters for some sample images. After pressing <@>, highlight an entry and press <Enter> to load it, or press <F6> to change to another parameter file. Fractint Version 18.2 Page 23 Note that parameter file entries specify all calculation related parameters, but do not specify things like the video mode - the image will be plotted in your currently selected mode. The command saves the parameters required to describe the currently displayed image, which can subsequently be used with the <@> command to recreate it. After you press , Fractint prompts for: Parameter file: The name of the file to store the parameters in. You should use some name like "myimages" instead of fractint.par, so that your images are kept separate from the ones released with new versions of Fractint. You can use the PARMFILE= command in SSTOOLS.INI to set the default parameter file name to "myimages" or whatever. (See "Setting Defaults (SSTOOLS.INI File)" (p. 90) and "parmfile=" in "File Parameters" (p. 100).) Name: The name you want to assign to the entry, to be displayed when the <@> command is used. Main comment: A comment to be shown beside the entry in the <@> command display. Second, Third, and Fourth comment: Additional comments to store in the file with the entry. These comments go in the file only, and are not displayed by the <@> command. Record colors?: Whether color information should be included in the entry. Usually the default value displayed by Fractint is what you want. Allowed values are: "no" - Don't record colors. This is the default if the image is using your video adapter's default colors. "@mapfilename" - When these parameters are used, load colors from the named color map file. This is the default if you are currently using colors from a color map file. "yes" - Record the colors in detail. This is the default when you've changed the display colors by using the palette editor or by color cycling. The only reason that this isn't what Fractint always does for the command is that color information can be bulky - up to nearly 1K of disk space. That may not sound like much, but can add up when you consider the thousands of wonderful images you may find you just *have* to record... Smooth-shaded ranges of colors are compressed, so if that's used a lot in an image the color information won't be as bulky. # of colors: This only matters if "Record colors?" is set to "yes". It specifies the number of colors to record. Recording less colors will take less space. Usually the default value displayed by Fractint is what you want. You might want to increase it in some cases, e.g. if you are using a 256 color mode with maxiter 150, and have used the palette editor to set all 256 possible colors for use with color cycling, then you'll want to set the "# of colors" to 256. Fractint Version 18.2 Page 24 At the bottom of the input screen are inputs for Fractint's "pieces" divide-and-conquer feature. You can create multiple PAR entries that break an image up into pieces so that you can generate the image pieces one by one. There are two reasons for doing this. The first is in case the fractal is very slow, and you want to generate parts of the image at the same time on several computers. The second is that you might want to make an image greater than 2048 x 2048. The parameters for this feature are: X Multiples - How many divisions of final image in the x direction Y Multiples - How many divisions of final image in the y direction Video mode - Fractint video mode for each piece (e.g. "F3") The last item defaults to the current video mode. If either X Multiples or Y Multiples are greater than 1, then multiple numbered PAR entries for the pieces are added to the PAR file, and a MAKEMIG.BAT file is created that builds all of the component pieces and then stitches them together into a "multi-image" GIF. The current limitations of the "divide and conquer" algorithm are 36 or fewer X and Y multiples (so you are limited to "only" 36x36=1296 component images), and a final resolution limit in both the X and Y directions of 65,535 (a limitation of "only" four billion pixels or so). The final image generated by MAKEMIG is a "multi-image" GIF file called FRACTMIG.GIF. In case you have other software that can't handle multi-image GIF files, MAKEMIG includes a final (but commented out) call to SIMPLGIF, a companion program that reads a GIF file that may contain little tricks like multiple images and creates a simple GIF from it. Fair warning: SIMPLGIF needs room to build a composite image while it works, and it does that using a temporary disk file equal to the size of the final image - and a 64Kx64K GIF image requires a 4GB temporary disk file! <G> The <G> command lets you give a startup parameter interactively. 1.9 "3D" Commands See "3D" Images (p. 76) for details of these commands. <3> Restore a saved image as a 3D "landscape", translating its color information into "height". You will be prompted for all KINDS of options. <#> Restore in 3D and overlay the result on the current screen. Fractint Version 18.2 Page 25 1.10 Interrupting and Resuming Fractint command keys can be loosely grouped as: o Keys which suspend calculation of the current image (if one is being calculated) and automatically resume after the function. <Tab> (display status information) and <F1> (display help), are the only keys in this group. o Keys which automatically trigger calculation of a new image. Examples: selecting a video mode (e.g. <F3>); selecting a fractal type using <T>; using the <X> screen to change an option such as maximum iterations. o Keys which do something, then wait for you to indicate what to do next. Examples: <M> to go to main menu; <C> to enter color cycling mode; <PageUp> to bring up a zoom box. After using a command in this group, calculation automatically resumes when you return from the function (e.g. <Esc> from color cycling, <PageDn> to clear zoom box). There are a few fractal types which cannot resume calculation, they are noted below. Note that after saving an image with <S>, you must press <Enter> to clear the "saved" message from the screen and resume. An image which is <S>aved before it completes can later be <R>estored and continued. The calculation is automatically resumed when you restore such an image. When a slow fractal type resumes after an interruption in the third category above, there may be a lag while nothing visible happens. This is because most cases of resume restart at the beginning of a screen line. If unsure, you can check whether calculation has resumed with the <Tab> key. The following fractal types cannot (currently) be resumed: plasma, 3d transformations, julibrot, and 3d orbital types like lorenz3d. To check whether resuming an image is possible, use the <Tab> key while it is calculating. It is resumable unless there is a note under the fractal type saying it is not. The Batch Mode (p. 110) section discusses how to resume in batch mode. To <R>estore and resume a "formula", "lsystem", or "ifs" type fractal your "formulafile", "lfile", or "ifsfile" must contain the required name. 1.11 Orbits Window The <O> key turns on the Orbit mode. In this mode a cursor appears over the fractal. A window appears showing the orbit used in the calculation of the color at the point where the cursor is. Move the cursor around the fractal using the arrow keys or the mouse and watch the orbits change. Try entering the Orbits mode with View Windows (<V>) turned on. The following keys take effect in Orbits mode. <c> Circle toggle - makes little circles with radii inversely Fractint Version 18.2 Page 26 proportional to the iteration. Press <c> again to toggle back to point-by-point display of orbits. <l> Line toggle - connects orbits with lines (can use with <c>) <n> Numbers toggle - shows coordinates of the cursor on the screen. Press <n> again to turn off numbers. Enter pixel coordinates directly <h> Hide fractal toggle. Works only if View Windows is turned on and set for a small window (such as the default size.) Hides the fractal, allowing the orbit to take up the whole screen. Press <h> again to uncover the fractal. <s> Saves the fractal, cursor, orbits, and numbers as they appear on the screen. <<> or <,> Zoom orbits image smaller <>> or <.> Zoom orbits image larger <z> Restore default zoom. 1.12 View Window The <V> command is used to set the view window parameters described below. These parameters can be used to: o Define a small window on the screen which is to contain the generated images. Using a small window speeds up calculation time (there are fewer pixels to generate). You can use a small window to explore quickly, then turn the view window off to recalculate the image at full screen size. o Generate an image with a different "aspect ratio"; e.g. in a square window or in a tall skinny rectangle. o View saved GIF images which have pixel dimensions different from any mode supported by your hardware. This use of view windows occurs automatically when you restore such an image. "Preview display" Set this to "yes" to turn on view window, "no" for full screen display. While this is "no", the only view parameter which has any affect is "final media aspect ratio". When a view window is being used, all other Fractint functions continue to operate normally - you can zoom, color- cycle, and all the rest. "Reduction factor" When an explicit size is not given, this determines the view window size, as a factor of the screen size. E.g. a reduction factor of 2 makes the window 1/2 as big as the screen in both dimensions. "Final media aspect ratio" This is the height of the final image you want, divided by the width. The default is 0.75 because standard PC monitors have a height:width ratio of 3:4. E.g. set this to 2.0 for an image twice as high as it is wide. The effect of this parameter is visible only when "preview display" is enabled. "Crop starting coordinates" This parameter affects what happens when you change the aspect ratio. If set to "no", then when you change aspect ratio, the prior image will be squeezed or stretched to fit into the new shape. If set to "yes", the Fractint Version 18.2 Page 27 prior image is "cropped" to avoid squeezing or stretching. "Explicit size" Setting these to non-zero values over-rides the "reduction factor" with explicit sizes in pixels. If only the "x pixels" size is specified, the "y pixels" size is calculated automatically based on x and the aspect ratio. More about final aspect ratio: If you want to produce a high quality hard-copy image which is say 8" high by 5" down, based on a vertical "slice" of an existing image, you could use a procedure like the following. You'll need some method of converting a GIF image to your final media (slide or whatever) - Fractint can only do the whole job with a PostScript printer, it does not preserve aspect ratio with other printers. o restore the existing image o set view parameters: preview to yes, reduction to anything (say 2), aspect ratio to 1.6, and crop to yes o zoom, rotate, whatever, till you get the desired final image o set preview display back to no o trigger final calculation in some high res disk video mode, using the appropriate video mode function key o print directly to a PostScript printer, or save the result as a GIF file and use external utilities to convert to hard copy. 1.13 Video Mode Function Keys Fractint supports *so* many video modes that we've given up trying to reserve a keyboard combination for each of them. Any supported video mode can be selected by going to the "Select Video Mode" screen (from main menu or by using <Delete>), then using the cursor up and down arrow keys and/or <PageUp> and <PageDown> keys to highlight the desired mode, then pressing <Enter>. Up to 39 modes can be assigned to the keys F2-F10, SF1-SF10 <Shift>+<Fn>), CF1-CF10 (<Ctrl>+<Fn>), and AF1-AF10 (<Alt>+<Fn>). The modes assigned to function keys can be invoked directly by pressing the assigned key, without going to the video mode selection screen. 30 key combinations can be reassigned: <F1> to <F10> combined with any of <Shift>, <Ctrl>, or <Alt>. The video modes assigned to <F2> through <F10> can not be changed - these are assigned to the most common video modes, which might be used in demonstration files or batches. To reassign a function key to a mode you often use, go to the "select video mode" screen, highlight the video mode, press the keypad (gray) <+> key, then press the desired function key or key combination. The new key assignment will be remembered for future runs. To unassign a key (so that it doesn't invoke any video mode), highlight the mode currently selected by the key and press the keypad (gray) <-> key. Fractint Version 18.2 Page 28 A note about the "select video modes" screen: the video modes which are displayed with a 'B' suffix in the number of colors are modes which have no custom programming - they use the BIOS and are S-L-O-W ones. See "Video Adapter Notes" (p. 112) for comments about particular adapters. See "Disk-Video" Modes (p. 114) for a description of these non-display modes. See "Customized Video Modes, FRACTINT.CFG" (p. 115) for information about adding your own video modes. 1.14 Hints Remember, you do NOT have to wait for the program to finish a full screen display before entering a command. If you see an interesting spot you want to zoom in on while the screen is half-done, don't wait -- do it! If you think after seeing the first few lines that another video mode would look better, go ahead -- Fractint will shift modes and start the redraw at once. When it finishes a display, it beeps and waits for your next command. In general, the most interesting areas are the "border" areas where the colors are changing rapidly. Zoom in on them for the best results. The first Mandelbrot-set (default) fractal image has a large, solid-colored interior that is the slowest to display; there's nothing to be seen by zooming there. Plotting time is directly proportional to the number of pixels in a screen, and hence increases with the resolution of the video mode. You may want to start in a low-resolution mode for quick progress while zooming in, and switch to a higher-resolution mode when things get interesting. Or use the solid guessing mode and pre-empt with a zoom before it finishes. Plotting time also varies with the maximum iteration setting, the fractal type, and your choice of drawing mode. Solid- guessing (the default) is fastest, but it can be wrong: perfectionists will want to use dual-pass mode (its first-pass preview is handy if you might zoom pre-emptively) or single-pass mode. When you start systematically exploring, you can save time (and hey, every little bit helps -- these "objects" are INFINITE, remember!) by <S>aving your last screen in a session to a file, and then going straight to it the next time by using the command FRACTINT FRACTxxx (the .GIF extension is assumed), or by starting Fractint normally and then using the <R> command to reload the saved file. Or you could hit to create a parameter file entry with the "recipe" for a given image, and next time use the <@> command to re-plot it. Fractint Version 18.2 Page 29 2. Fractal Types A list of the fractal types and their mathematics can be found in the Summary of Fractal Types (p. 127). Some notes about how Fractint calculates them are in "A Little Code" in "Fractals and the PC" (p. 121) . Fractint starts by default with the Mandelbrot set. You can change that by using the command-line argument "TYPE=" followed by one of the fractal type names, or by using the <T> command and selecting the type - if parameters are needed, you will be prompted for them. In the text that follows, due to the limitations of the ASCII character set, "a*b" means "a times b", and "a^b" means "a to the power b". 2.1 The Mandelbrot Set (type=mandel) This set is the classic: the only one implemented in many plotting programs, and the source of most of the printed fractal images published in recent years. Like most of the other types in Fractint, it is simply a graph: the x (horizontal) and y (vertical) coordinate axes represent ranges of two independent quantities, with various colors used to symbolize levels of a third quantity which depends on the first two. So far, so good: basic analytic geometry. Now things get a bit hairier. The x axis is ordinary, vanilla real numbers. The y axis is an imaginary number, i.e. a real number times i, where i is the square root of -1. Every point on the plane -- in this case, your PC's display screen -- represents a complex number of the form: x-coordinate + i * y-coordinate If your math training stopped before you got to imaginary and complex numbers, this is not the place to catch up. Suffice it to say that they are just as "real" as the numbers you count fingers with (they're used every day by electrical engineers) and they can undergo the same kinds of algebraic operations. OK, now pick any complex number -- any point on the complex plane -- and call it C, a constant. Pick another, this time one which can vary, and call it Z. Starting with Z=0 (i.e., at the origin, where the real and imaginary axes cross), calculate the value of the expression Z^2 + C Take the result, make it the new value of the variable Z, and calculate again. Take that result, make it Z, and do it again, and so on: in mathematical terms, iterate the function Z(n+1) = Z(n)^2 + C. For certain values of C, the result "levels off" after a while. For all others, it grows without limit. The Mandelbrot set you see at the start -- the solid-colored lake (blue by default), the blue circles sprouting Fractint Version 18.2 Page 30 from it, and indeed every point of that color -- is the set of all points C for which the value of Z is less than 2 after 150 iterations (150 is the default setting, changeable via the <X> options screen or "maxiter=" parameter). All the surrounding "contours" of other colors represent points for which Z exceeds 2 after 149 iterations (the contour closest to the M-set itself), 148 iterations, (the next one out), and so on. We actually don't test for Z exceeding 2 - we test Z squared against 4 instead because it is easier. This value (FOUR usually) is known as the "bailout" value for the calculation, because we stop iterating for the point when it is reached. The bailout value can be changed on the <Z> options screen but the default is usually best. Some features of interest: 1. Use the <X> options screen to increase the maximum number of iterations. Notice that the boundary of the M-set becomes more and more convoluted (the technical terms are "wiggly," "squiggly," and "utterly bizarre") as the Z-values for points that were still within the set after 150 iterations turn out to exceed 2 after 200, 500, or 1200. In fact, it can be proven that the true boundary is infinitely long: detail without limit. 2. Although there appear to be isolated "islands" of blue, zoom in -- that is, plot for a smaller range of coordinates to show more detail -- and you'll see that there are fine "causeways" of blue connecting them to the main set. As you zoomed, smaller islands became visible; the same is true for them. In fact, there are no isolated points in the M-set: it is "connected" in a strict mathematical sense. 3. The upper and lower halves of the first image are symmetric (a fact that Fractint makes use of here and in some other fractal types to speed plotting). But notice that the same general features -- lobed discs, spirals, starbursts -- tend to repeat themselves (although never exactly) at smaller and smaller scales, so that it can be impossible to judge by eye the scale of a given image. 4. In a sense, the contour colors are window-dressing: mathematically, it is the properties of the M-set itself that are interesting, and no information about it would be lost if all points outside the set were assigned the same color. If you're a serious, no-nonsense type, you may want to cycle the colors just once to see the kind of silliness that other people enjoy, and then never do it again. Go ahead. Just once, now. We trust you. 2.2 Julia Sets (type=julia) These sets were named for mathematician Gaston Julia, and can be generated by a simple change in the iteration process described for the Mandelbrot Set (p. 29). Start with a specified value of C, "C-real + i * C-imaginary"; use as the initial value of Z "x-coordinate + i * y- coordinate"; and repeat the same iteration, Z(n+1) = Z(n)^2 + C. Fractint Version 18.2 Page 31 There is a Julia set corresponding to every point on the complex plane -- an infinite number of Julia sets. But the most visually interesting tend to be found for the same C values where the M-set image is busiest, i.e. points just outside the boundary. Go too far inside, and the corresponding Julia set is a circle; go too far outside, and it breaks up into scattered points. In fact, all Julia sets for C within the M-set share the "connected" property of the M-set, and all those for C outside lack it. Fractint's spacebar toggle lets you "flip" between any view of the M-set and the Julia set for the point C at the center of that screen. You can then toggle back, or zoom your way into the Julia set for a while and then return to the M-set. So if the infinite complexity of the M-set palls, remember: each of its infinite points opens up a whole new Julia set. Historically, the Julia sets came first: it was while looking at the M- set as an "index" of all the Julia sets' origins that Mandelbrot noticed its properties. The relationship between the Mandelbrot (p. 29) set and Julia set can hold between other sets as well. Many of Fractint's types are "Mandelbrot/Julia" pairs (sometimes called "M-sets" or "J-sets". All these are generated by equations that are of the form z(k+1) = f(z(k),c), where the function orbit is the sequence z(0), z(1), ..., and the variable c is a complex parameter of the equation. The value c is fixed for "Julia" sets and is equal to the first two parameters entered with the "params=Creal/Cimag" command. The initial orbit value z(0) is the complex number corresponding to the screen pixel. For Mandelbrot sets, the parameter c is the complex number corresponding to the screen pixel. The value z(0) is c plus a perturbation equal to the values of the first two parameters. See the discussion of Mandellambda Sets (p. 35). This approach may or may not be the "standard" way to create "Mandelbrot" sets out of "Julia" sets. Some equations have additional parameters. These values are entered as the third for fourth params= value for both Julia and Mandelbrot sets. The variables x and y refer to the real and imaginary parts of z; similarly, cx and cy are the real and imaginary parts of the parameter c and fx(z) and fy(z) are the real and imaginary parts of f(z). The variable c is sometimes called lambda for historical reasons. NOTE: if you use the "PARAMS=" argument to warp the M-set by starting with an initial value of Z other than 0, the M-set/J-sets correspondence breaks down and the spacebar toggle no longer works. 2.3 Julia Toggle Spacebar Commands The spacebar toggle has been enhanced for the classic Mandelbrot and Julia types. When viewing the Mandelbrot, the spacebar turns on a window mode that displays the Inverse Julia corresponding to the cursor position in a window. Pressing the spacebar then causes the regular Julia escape time fractal corresponding to the cursor position to be generated. The following keys take effect in Inverse Julia mode. Fractint Version 18.2 Page 32 <Space> Generate the escape-time Julia Set corresponding to the cursor position. Only works if fractal is a "Mandelbrot" type. <n> Numbers toggle - shows coordinates of the cursor on the screen. Press <n> again to turn off numbers. Enter new pixel coordinates directly <h> Hide fractal toggle. Works only if View Windows is turned on and set for a small window (such as the default size.) Hides the fractal, allowing the orbit to take up the whole screen. Press <h> again to uncover the fractal. <s> Saves the fractal, cursor, orbits, and numbers. <<> or <,> Zoom inverse julia image smaller. <>> or <.> Zoom inverse julia image larger. <z> Restore default zoom. The Julia Inverse window is only implemented for the classic Mandelbrot (type=mandel). For other "Mandelbrot" types <space> turns on the cursor without the Julia window, and allows you to select coordinates of the matching Julia set in a way similar to the use of the zoom box with the Mandelbrot/Julia toggle in previous Fractint versions. 2.4 Inverse Julias (type=julia_inverse) Pick a function, such as the familiar Z(n) = Z(n-1) squared plus C (the defining function of the Mandelbrot Set). If you pick a point Z(0) at random from the complex plane, and repeatedly apply the function to it, you get a sequence of new points called an orbit, which usually either zips out toward infinity or zooms in toward one or more "attractor" points near the middle of the plane. The set of all points that are "attracted" to infinity is called the "Basin of Attraction" of infinity. Each of the other attractors also has its own Basin of Attraction. Why is it called a Basin? Imagine a lake, and all the water in it "draining" into the attractor. The boundary between these basins is called the Julia Set of the function. The boundary between the basins of attraction is sort of like a repeller; all orbits move away from it, toward one of the attractors. But if we define a new function as the inverse of the old one, as for instance Z(n) = sqrt(Z(n-1) minus C), then the old attractors become repellers, and the former boundary itself becomes the attractor! Now, starting from any point, all orbits are drawn irresistibly to the Julia Set! In fact, once an orbit reaches the boundary, it will continue to hop about until it traces the entire Julia Set! This method for drawing Julia Sets is called the Inverse Iteration Method, or IIM for short. Unfortunately, some parts of each Julia Set boundary are far more attractive to inverse orbits than others are, so that as an orbit traces out the set, it keeps coming back to these attractive parts again and again, only occasionally visiting the less attractive parts. Thus it may take an infinite length of time to draw the entire set. To hasten the process, we can keep track of how many times each pixel on our computer screen is visited by an orbit, and whenever an orbit reaches a pixel that has already been visited more than a certain number of times, Fractint Version 18.2 Page 33 we can consider that orbit finished and move on to another one. This "hit limit" thus becomes similar to the iteration limit used in the traditional escape-time fractal algorithm. This is called the Modified Inverse Iteration Method, or MIIM, and is much faster than the IIM. Now, the inverse of Mandelbrot's classic function is a square root, and the square root actually has two solutions; one positive, one negative. Therefore at each step of each orbit of the inverse function there is a decision; whether to use the positive or the negative square root. Each one gives rise to a new point on the Julia Set, so each is a good choice. This series of choices defines a binary decision tree, each point on the Julia Set giving rise to two potential child points. There are many interesting ways to traverse a binary tree, among them Breadth first, Depth first (left or negative first), Depth first (right or positive first), and completely at random. It turns out that most traversal methods lead to the same or similar pictures, but that how the image evolves as the orbits trace it out differs wildly depending on the traversal method chosen. As far as I know, this fact is an original discovery, and this version of FRACTINT is its first publication. Pick a Julia constant such as Z(0) = (-.74543, .11301), the popular Seahorse Julia, and try drawing it first Breadth first, then Depth first (right first), Depth first (left first), and finally with Random Walk. Caveats: the video memory is used in the algorithm, to keep track of how many times each pixel has been visited (by changing it's color). Therefore the algorithm will not work well if you zoom in far enough that part of the Julia Set is off the screen. Bugs: Not working with Disk Video. Not resumeable. The <J> key toggles between the Inverse Julia orbit and the corresponding Julia escape time fractal. 2.5 Newton domains of attraction (type=newtbasin) The Newton formula is an algorithm used to find the roots of polynomial equations by successive "guesses" that converge on the correct value as you feed the results of each approximation back into the formula. It works very well -- unless you are unlucky enough to pick a value that is on a line BETWEEN two actual roots. In that case, the sequence explodes into chaos, with results that diverge more and more wildly as you continue the iteration. This fractal type shows the results for the polynomial Z^n - 1, which has n roots in the complex plane. Use the <T>ype command and enter "newtbasin" in response to the prompt. You will be asked for a parameter, the "order" of the equation (an integer from 3 through 10 -- 3 for x^3-1, 7 for x^7-1, etc.). A second parameter is a flag to turn on alternating shades showing changes in the number of iterations needed to attract an orbit. Some people like stripes and some don't, as always, Fractint gives you a choice! Fractint Version 18.2 Page 34 The coloring of the plot shows the "basins of attraction" for each root of the polynomial -- i.e., an initial guess within any area of a given color would lead you to one of the roots. As you can see, things get a bit weird along certain radial lines or "spokes," those being the lines between actual roots. By "weird," we mean infinitely complex in the good old fractal sense. Zoom in and see for yourself. This fractal type is symmetric about the origin, with the number of "spokes" depending on the order you select. It uses floating-point math if you have an FPU, or a somewhat slower integer algorithm if you don't have one. 2.6 Newton (type=newton) The generating formula here is identical to that for newtbasin (p. 33), but the coloring scheme is different. Pixels are colored not according to the root that would be "converged on" if you started using Newton's formula from that point, but according to the iteration when the value is close to a root. For example, if the calculations for a particular pixel converge to the 7th root on the 23rd iteration, NEWTBASIN will color that pixel using color #7, but NEWTON will color it using color #23. If you have a 256-color mode, use it: the effects can be much livelier than those you get with type=newtbasin, and color cycling becomes, like, downright cosmic. If your "corners" choice is symmetrical, Fractint exploits the symmetry for faster display. The applicable "params=" values are the same as newtbasin. Try "params=4." Other values are 3 through 10. 8 has twice the symmetry and is faster. As with newtbasin, an FPU helps. 2.7 Complex Newton (type=complexnewton/complexbasin) Well, hey, "Z^n - 1" is so boring when you can use "Z^a - b" where "a" and "b" are complex numbers! The new "complexnewton" and "complexbasin" fractal types are just the old "newton" (p. 34) and "newtbasin" (p. 33) fractal types with this little added twist. When you select these fractal types, you are prompted for four values (the real and imaginary portions of "a" and "b"). If "a" has a complex portion, the fractal has a discontinuity along the negative axis - relax, we finally figured out that it's *supposed* to be there! Fractint Version 18.2 Page 35 2.8 Lambda Sets (type=lambda) This type calculates the Julia set of the formula lambda*Z*(1-Z). That is, the value Z[0] is initialized with the value corresponding to each pixel position, and the formula iterated. The pixel is colored according to the iteration when the sum of the squares of the real and imaginary parts exceeds 4. Two parameters, the real and imaginary parts of lambda, are required. Try 0 and 1 to see the classical fractal "dragon". Then try 0.2 and 1 for a lot more detail to zoom in on. It turns out that all quadratic Julia-type sets can be calculated using just the formula z^2+c (the "classic" Julia"), so that this type is redundant, but we include it for reason of it's prominence in the history of fractals. 2.9 Mandellambda Sets (type=mandellambda) This type is the "Mandelbrot equivalent" of the lambda (p. 35) set. A comment is in order here. Almost all the Fractint "Mandelbrot" sets are created from orbits generated using formulas like z(n+1) = f(z(n),C), with z(0) and C initialized to the complex value corresponding to the current pixel. Our reasoning was that "Mandelbrots" are maps of the corresponding "Julias". Using this scheme each pixel of a "Mandelbrot" is colored the same as the Julia set corresponding to that pixel. However, Kevin Allen informs us that the MANDELLAMBDA set appears in the literature with z(0) initialized to a critical point (a point where the derivative of the formula is zero), which in this case happens to be the point (.5,0). Since Kevin knows more about Dr. Mandelbrot than we do, and Dr. Mandelbrot knows more about fractals than we do, we defer! Starting with version 14 Fractint calculates MANDELAMBDA Dr. Mandelbrot's way instead of our way. But ALL THE OTHER "Mandelbrot" sets in Fractint are still calculated OUR way! (Fortunately for us, for the classic Mandelbrot Set these two methods are the same!) Well now, folks, apart from questions of faithfulness to fractals named in the literature (which we DO take seriously!), if a formula makes a beautiful fractal, it is not wrong. In fact some of the best fractals in Fractint are the results of mistakes! Nevertheless, thanks to Kevin for keeping us accurate! (See description of "initorbit=" command in Image Calculation Parameters (p. 94) for a way to experiment with different orbit intializations). Fractint Version 18.2 Page 36 2.10 Circle (type=circle) This fractal types is from A. K. Dewdney's "Computer Recreations" column in "Scientific American". It is attributed to John Connett of the University of Minnesota. (Don't tell anyone, but this fractal type is not really a fractal!) Fascinating Moire patterns can be formed by calculating x^2 + y^2 for each pixel in a piece of the complex plane. After multiplication by a magnification factor (the parameter), the number is truncated to an integer and mapped to a color via color = value modulo (number of colors). That is, the integer is divided by the number of colors, and the remainder is the color index value used. The resulting image is not a fractal because all detail is lost after zooming in too far. Try it with different resolution video modes - the results may surprise you! 2.11 Plasma Clouds (type=plasma) Plasma clouds ARE real live fractals, even though we didn't know it at first. They are generated by a recursive algorithm that randomly picks colors of the corner of a rectangle, and then continues recursively quartering previous rectangles. Random colors are averaged with those of the outer rectangles so that small neighborhoods do not show much change, for a smoothed-out, cloud-like effect. The more colors your video mode supports, the better. The result, believe it or not, is a fractal landscape viewed as a contour map, with colors indicating constant elevation. To see this, save and view with the <3> command (see "3D" Images (p. 76)) and your "cloud" will be converted to a mountain! You've GOT to try color cycling (p. 16) on these (hit "+" or "-"). If you haven't been hypnotized by the drawing process, the writhing colors will do it for sure. We have now implemented subliminal messages to exploit the user's vulnerable state; their content varies with your bank balance, politics, gender, accessibility to a Fractint programmer, and so on. A free copy of Microsoft C to the first person who spots them. This type accepts four parameters. The first determines how abruptly the colors change. A value of .5 yields bland clouds, while 50 yields very grainy ones. The default value is 2. The second determines whether to use the original algorithm (0) or a modified one (1). The new one gives the same type of images but draws the dots in a different order. It will let you see what the final image will look like much sooner than the old one. Fractint Version 18.2 Page 37 The third determines whether to use a new seed for generating the next plasma cloud (0) or to use the previous seed (1). The fourth parameter turns on 16-bit .POT output which provides much smoother height gradations. This is especially useful for creating mountain landscapes when using the plasma output with a ray tracer such as POV-Ray. With parameter three set to 1, the next plasma cloud generated will be identical to the previous but at whatever new resolution is desired. Zooming is ignored, as each plasma-cloud screen is generated randomly. The random number seed used for each plasma image is displayed on the <tab> information screen, and can be entered with the command line parameter "rseed=" to recreate a particular image. The algorithm is based on the Pascal program distributed by Bret Mulvey as PLASMA.ARC. We have ported it to C and integrated it with Fractint's graphics and animation facilities. This implementation does not use floating-point math. The algorithm was modified starting with version 18 so that the plasma effect is independent of screen resolution. Saved plasma-cloud screens are EXCELLENT starting images for fractal "landscapes" created with the "3D" commands (p. 24). 2.12 Lambdafn (type=lambdafn) Function=[sin|cos|sinh|cosh|exp|log|sqr|...]) is specified with this type. Prior to version 14, these types were lambdasine, lambdacos, lambdasinh, lambdacos, and lambdaexp. Where we say "lambdasine" or some such below, the good reader knows we mean "lambdafn with function=sin".) These types calculate the Julia set of the formula lambda*fn(Z), for various values of the function "fn", where lambda and Z are both complex. Two values, the real and imaginary parts of lambda, should be given in the "params=" option. For the feathery, nested spirals of LambdaSines and the frost-on-glass patterns of LambdaCosines, make the real part = 1, and try values for the imaginary part ranging from 0.1 to 0.4 (hint: values near 0.4 have the best patterns). In these ranges the Julia set "explodes". For the tongues and blobs of LambdaExponents, try a real part of 0.379 and an imaginary part of 0.479. A coprocessor used to be almost mandatory: each LambdaSine/Cosine iteration calculates a hyperbolic sine, hyperbolic cosine, a sine, and a cosine (the LambdaExponent iteration "only" requires an exponent, sine, and cosine operation)! However, Fractint now computes these transcendental functions with fast integer math. In a few cases the fast math is less accurate, so we have kept the old slow floating point code. To use the old code, invoke with the float=yes option, and, if you DON'T have a coprocessor, go on a LONG vacation! Fractint Version 18.2 Page 38 2.13 Mandelfn (type=mandelfn) Function=[sin|cos|sinh|cosh|exp|log|sqr|...]) is specified with this type. Prior to version 14, these types were mandelsine, mandelcos, mandelsinh, mandelcos, and mandelexp. Same comment about our lapses into the old terminology as above! These are "pseudo-Mandelbrot" mappings for the LambdaFn (p. 37) Julia functions. They map to their corresponding Julia sets via the spacebar command in exactly the same fashion as the original M/J sets. In general, they are interesting mainly because of that property (the function=exp set in particular is rather boring). Generate the appropriate "Mandelfn" set, zoom on a likely spot where the colors are changing rapidly, and hit the spacebar key to plot the Julia set for that particular point. Try "FRACTINT TYPE=MANDELFN CORNERS=4.68/4.76/-.03/.03 FUNCTION=COS" for a graphic demonstration that we're not taking Mandelbrot's name in vain here. We didn't even know these little buggers were here until Mark Peterson found this a few hours before the version incorporating Mandelfns was released. Note: If you created images using the lambda or mandel "fn" types prior to version 14, and you wish to update the fractal information in the "*.fra" file, simply read the files and save again. You can do this in batch mode via a command line such as: "fractint oldfile.fra savename=newfile.gif batch=yes" For example, this procedure can convert a version 13 "type=lambdasine" image to a version 14 "type=lambdafn function=sin" GIF89a image. We do not promise to keep this "backward compatibility" past version 14 - if you want to keep the fractal information in your *.fra files accurate, we recommend conversion. See GIF Save File Format (p. 150). 2.14 Barnsley Mandelbrot/Julia Sets (type=barnsleym1/.../j3) Michael Barnsley has written a fascinating college-level text, "Fractals Everywhere," on fractal geometry and its graphic applications. (See Bibliography (p. 152).) In it, he applies the principle of the M and J sets to more general functions of two complex variables. We have incorporated three of Barnsley's examples in Fractint. Their appearance suggests polarized-light microphotographs of minerals, with patterns that are less organic and more crystalline than those of the M/J sets. Each example has both a "Mandelbrot" and a "Julia" type. Toggle between them using the spacebar. The parameters have the same meaning as they do for the "regular" Mandelbrot and Julia. For types M1, M2, and M3, they are used to "warp" the image by setting the initial value of Z. For the types J1 through Fractint Version 18.2 Page 39 J3, they are the values of C in the generating formulas. Be sure to try the <O>rbit function while plotting these types. 2.15 Barnsley IFS Fractals (type=ifs) One of the most remarkable spin-offs of fractal geometry is the ability to "encode" realistic images in very small sets of numbers -- parameters for a set of functions that map a region of two-dimensional space onto itself. In principle (and increasingly in practice), a scene of any level of complexity and detail can be stored as a handful of numbers, achieving amazing "compression" ratios... how about a super-VGA image of a forest, more than 300,000 pixels at eight bits apiece, from a 1-KB "seed" file? Again, Michael Barnsley and his co-workers at the Georgia Institute of Technology are to be thanked for pushing the development of these iterated function systems (IFS). When you select this fractal type, Fractint scans the current IFS file (default is FRACTINT.IFS, a set of definitions supplied with Fractint) for IFS definitions, then prompts you for the IFS name you wish to run. Fern and 3dfern are good ones to start with. You can press <F6> at the selection screen if you want to select a different .IFS file you've written. Note that some Barnsley IFS values generate images quite a bit smaller than the initial (default) screen. Just bring up the zoom box, center it on the small image, and hit <Enter> to get a full-screen image. To change the number of dots Fractint generates for an IFS image before stopping, you can change the "maximum iterations" parameter on the <X> options screen. Fractint supports two types of IFS images: 2D and 3D. In order to fully appreciate 3D IFS images, since your monitor is presumably 2D, we have added rotation, translation, and perspective capabilities. These share values with the same variables used in Fractint's other 3D facilities; for their meaning see "Rectangular Coordinate Transformation" (p. 81). You can enter these values from the command line using: rotation=xrot/yrot/zrot (try 30/30/30) shift=xshift/yshift (shifts BEFORE applying perspective!) perspective=viewerposition (try 200) Alternatively, entering from main screen will allow you to modify these values. The defaults are the same as for regular 3D, and are not always optimum for 3D IFS. With the 3dfern IFS type, try rotation=30/30/30. Note that applying shift when using perspective changes the picture -- your "point of view" is moved. Fractint Version 18.2 Page 40 A truly wild variation of 3D may be seen by entering "2" for the stereo mode (see "Stereo 3D Viewing" (p. 80)), putting on red/blue "funny glasses", and watching the fern develop with full depth perception right there before your eyes! This feature USED to be dedicated to Bruce Goren, as a bribe to get him to send us MORE knockout stereo slides of 3D ferns, now that we have made it so easy! Bruce, what have you done for us *LATELY* ?? (Just kidding, really!) Each line in an IFS definition (look at FRACTINT.IFS with your editor for examples) contains the parameters for one of the generating functions, e.g. in FERN: a b c d e f p ___________________________________ 0 0 0 .16 0 0 .01 .85 .04 -.04 .85 0 1.6 .85 .2 -.26 .23 .22 0 1.6 .07 -.15 .28 .26 .24 0 .44 .07 The values on each line define a matrix, vector, and probability: matrix vector prob |a b| |e| p |c d| |f| The "p" values are the probabilities assigned to each function (how often it is used), which add up to one. Fractint supports up to 32 functions, although usually three or four are enough. 3D IFS definitions are a bit different. The name is followed by (3D) in the definition file, and each line of the definition contains 13 numbers: a b c d e f g h i j k l p, defining: matrix vector prob |a b c| |j| p |d e f| |k| |g h i| |l| The program FDESIGN can be used to design IFS fractals - see FDESIGN (p. 153). You can save the points in your IFS fractal in the file ORBITS.RAW which is overwritten each time a fractal is generated. The program Acrospin can read this file and will let you view the fractal from any angle using the cursor keys. See Acrospin (p. 153). 2.16 Sierpinski Gasket (type=sierpinski) Another pre-Mandelbrot classic, this one found by W. Sierpinski around World War I. It is generated by dividing a triangle into four congruent smaller triangles, doing the same to each of them, and so on, yea, even unto infinity. (Notice how hard we try to avoid reiterating "iterating"?) Fractint Version 18.2 Page 41 If you think of the interior triangles as "holes", they occupy more and more of the total area, while the "solid" portion becomes as hopelessly fragile as that gasket you HAD to remove without damaging it -- you remember, that Sunday afternoon when all the parts stores were closed? There's a three-dimensional equivalent using nested tetrahedrons instead of triangles, but it generates too much pyramid power to be safely unleashed yet. There are no parameters for this type. We were able to implement it with integer math routines, so it runs fairly quickly even without an FPU. 2.17 Quartic Mandelbrot/Julia (type=mandel4/julia4) These fractal types are the moral equivalent of the original M and J sets, except that they use the formula Z(n+1) = Z(n)^4 + C, which adds additional pseudo-symmetries to the plots. The "Mandel4" set maps to the "Julia4" set via -- surprise! -- the spacebar toggle. The M4 set is kind of boring at first (the area between the "inside" and the "outside" of the set is pretty thin, and it tends to take a few zooms to get to any interesting sections), but it looks nice once you get there. The Julia sets look nice right from the start. Other powers, like Z(n)^3 or Z(n)^7, work in exactly the same fashion. We used this one only because we're lazy, and Z(n)^4 = (Z(n)^2)^2. 2.18 Distance Estimator (distest=nnn/nnn) This used to be type=demm and type=demj. These types have not died, but are only hiding! They are equivalent to the mandel and julia types with the "distest=" option selected with a predetermined value. The Distance Estimator Method (p. 68) can be used to produce higher quality images of M and J sets, especially suitable for printing in black and white. If you have some *.fra files made with the old types demm/demj, you may want to convert them to the new form. See the Mandelfn (p. 38) section for directions to carry out the conversion. 2.19 Pickover Mandelbrot/Julia Types (type=manfn+zsqrd/julfn+zsqrd, manzpowr/julzpowr, manzzpwr/julzzpwr, manfn+exp/julfn+exp - formerly included man/julsinzsqrd and man/julsinexp which have now been generalized) These types have been explored by Clifford A. Pickover, of the IBM Thomas J. Watson Research center. As implemented in Fractint, they are regular Mandelbrot/Julia set pairs that may be plotted with or without the "biomorph" (p. 72) option Pickover used to create organic-looking Fractint Version 18.2 Page 42 beasties (see below). These types are produced with formulas built from the functions z^z, z^n, sin(z), and e^z for complex z. Types with "power" or "pwr" in their name have an exponent value as a third parameter. For example, type=manzpower params=0/0/2 is our old friend the classical Mandelbrot, and type=manzpower params=0/0/4 is the Quartic Mandelbrot. Other values of the exponent give still other fractals. Since these WERE the original "biomorph" types, we should give an example. Try: FRACTINT type=manfn+zsqrd biomorph=0 corners=-8/8/-6/6 function=sin to see a big biomorph digesting little biomorphs! 2.20 Pickover Popcorn (type=popcorn/popcornjul) Here is another Pickover idea. This one computes and plots the orbits of the dynamic system defined by: x(n+1) = x(n) - h*sin(y(n)+tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n)+tan(3*x(n)) with the initializers x(0) and y(0) equal to ALL the complex values within the "corners" values, and h=.01. ALL these orbits are superimposed, resulting in "popcorn" effect. You may want to use a maxiter value less than normal - Pickover recommends a value of 50. As a bonus, type=popcornjul shows the Julia set generated by these same equations with the usual escape-time coloring. Turn on orbit viewing with the "O" command, and as you watch the orbit pattern you may get some insight as to where the popcorn comes from. Although you can zoom and rotate popcorn, the results may not be what you'd expect, due to the superimposing of orbits and arbitrary use of color. Just for fun we added type popcornjul, which is the plain old Julia set calculated from the same formula. 2.21 Peterson Variations (type=marksmandel, marksjulia, cmplxmarksmand, cmplxmarksjul, marksmandelpwr, tim's_error) These fractal types are contributions of Mark Peterson. MarksMandel and MarksJulia are two families of fractal types that are linked in the same manner as the classic Mandelbrot/Julia sets: each MarksMandel set can be considered as a mapping into the MarksJulia sets, and is linked with the spacebar toggle. The basic equation for these sets is: Z(n+1) = ((lambda^exp) * Z(n)^2) + lambda where Z(0) = 0.0 and lambda is (x + iy) for MarksMandel. For MarksJulia, Z(0) = (x + iy) and lambda is a constant (taken from the MarksMandel spacebar toggle, if that method is used). The exponent is a positive integer or a complex number. We call these "families" because each value of the exponent yields a different MarksMandel set, which turns out to be a kinda- polygon with (exponent+1) sides. The exponent value is the third parameter, after the "initialization warping" values. Typically one Fractint Version 18.2 Page 43 would use null warping values, and specify the exponent with something like "PARAMS=0/0/4", which creates an unwarped, pentagonal MarksMandel set. In the process of coding MarksMandelPwr formula type, Tim Wegner created the type "tim's_error" after making an interesting coding mistake. 2.22 Unity (type=unity) This Peterson variation began with curiosity about other "Newton-style" approximation processes. A simple one, One = (x * x) + (y * y); y = (2 - One) * x; x = (2 - One) * y; produces the fractal called Unity. One of its interesting features is the "ghost lines." The iteration loop bails out when it reaches the number 1 to within the resolution of a screen pixel. When you zoom a section of the image, the bailout criterion is adjusted, causing some lines to become thinner and others thicker. Only one line in Unity that forms a perfect circle: the one at a radius of 1 from the origin. This line is actually infinitely thin. Zooming on it reveals only a thinner line, up (down?) to the limit of accuracy for the algorithm. The same thing happens with other lines in the fractal, such as those around |x| = |y| = (1/2)^(1/2) = .7071 Try some other tortuous approximations using the TEST stub (p. 50) and let us know what you come up with! 2.23 Scott Taylor / Lee Skinner Variations (type=fn(z*z), fn*fn, fn*z+z, fn+fn, sqr(1/fn), sqr(fn), spider, tetrate, manowar) Two of Fractint's faithful users went bonkers when we introduced the "formula" type, and came up with all kinds of variations on escape-time fractals using trig functions. We decided to put them in as regular types, but there were just too many! So we defined the types with variable functions and let you, the, overwhelmed user, specify what the functions should be! Thus Scott Taylor's "z = sin(z) + z^2" formula type is now the "fn+fn" regular type, and EITHER function can be one of sin, cos, tan, cotan, sinh, cosh, tanh, cotanh, exp, log, sqr, recip, ident, conj, flip, or cosxx. Plus we give you 4 parameters to set, the complex coefficients of the two functions! Thus the innocent-looking "fn+fn" type is really 256 different types in disguise, not counting the damage done by the parameters! Some functions that require further explanation: Fractint Version 18.2 Page 44 conj() - returns the complex conjugate of the argument. That is, changes sign of the imaginary component of argument: (x,y) becomes (x,-y) ident() - identity function. Leaves the value of the argument unchanged, acting like a "z" term in a formula. flip() - Swap the real and imaginary components of the complex number. e.g. (4,5) would become (5,4) Lee informs us that you should not judge fractals by their "outer" appearance. For example, the images produced by z = sin(z) + z^2 and z = sin(z) - z^2 look very similar, but are different when you zoom in. 2.24 Kam Torus (type=kamtorus, kamtorus3d) This type is created by superimposing orbits generated by a set of equations, with a variable incremented each time. x(0) = y(0) = orbit/3; x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a) y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a) After each orbit, 'orbit' is incremented by a step size. The parameters are angle "a", step size for incrementing 'orbit', stop value for 'orbit', and points per orbit. Try this with a stop value of 5 with sound=x for some weird fractal music (ok, ok, fractal noise)! You will also see the KAM Torus head into some chaotic territory that Scott Taylor wanted to hide from you by setting the defaults the way he did, but now we have revealed all! The 3D variant is created by treating 'orbit' as the z coordinate. With both variants, you can adjust the "maxiter" value (<X> options screen or parameter maxiter=) to change the number of orbits plotted. 2.25 Bifurcation (type=bifxxx) The wonder of fractal geometry is that such complex forms can arise from such simple generating processes. A parallel surprise has emerged in the study of dynamical systems: that simple, deterministic equations can yield chaotic behavior, in which the system never settles down to a steady state or even a periodic loop. Often such systems behave normally up to a certain level of some controlling parameter, then go through a transition in which there are two possible solutions, then four, and finally a chaotic array of possibilities. This emerged many years ago in biological models of population growth. Consider a (highly over-simplified) model in which the rate of growth is partly a function of the size of the current population: Fractint Version 18.2 Page 45 New Population = Growth Rate * Old Population * (1 - Old Population) where population is normalized to be between 0 and 1. At growth rates less than 200 percent, this model is stable: for any starting value, after several generations the population settles down to a stable level. But for rates over 200 percent, the equation's curve splits or "bifurcates" into two discrete solutions, then four, and soon becomes chaotic. Type=bifurcation illustrates this model. (Although it's now considered a poor one for real populations, it helped get people thinking about chaotic systems.) The horizontal axis represents growth rates, from 190 percent (far left) to 400 percent; the vertical axis normalized population values, from 0 to 4/3. Notice that within the chaotic region, there are narrow bands where there is a small, odd number of stable values. It turns out that the geometry of this branching is fractal; zoom in where changing pixel colors look suspicious, and see for yourself. Three parameters apply to bifurcations: Filter Cycles, Seed Population, and Function or Beta. Filter Cycles (default 1000) is the number of iterations to be done before plotting maxiter population values. This gives the iteration time to settle into the characteristic patterns that constitute the bifurcation diagram, and results in a clean-looking plot. However, using lower values produces interesting results too. Set Filter Cycles to 1 for an unfiltered map. Seed Population (default 0.66) is the initial population value from which all others are calculated. For filtered maps the final image is independent of Seed Population value in the valid range (0.0 < Seed Population < 1.0). Seed Population becomes effective in unfiltered maps - try setting Filter Cycles to 1 (unfiltered) and Seed Population to 0.001 ("PARAMS=1/.001" on the command line). This results in a map overlaid with nice curves. Each Seed Population value results in a different set of curves. Function (default "ident") is the function applied to the old population before the new population is determined. The "ident" function calculates the same bifurcation fractal that was generated before these formulae were generalized. Beta is used in the bifmay bifurcations and is the power to which the denominator is raised. Note that fractint normally uses periodicity checking to speed up bifurcation computation. However, in some cases a better quality image will be obtained if you turn off periodicity checking with "periodicity=no"; for instance, if you use a high number of iterations and a smooth colormap. Many formulae can be used to produce bifurcations. Mitchel Feigenbaum studied lots of bifurcations in the mid-70's, using a HP-65 calculator (IBM PCs, Fractals, and Fractint, were all Sci-Fi then !). He studied Fractint Version 18.2 Page 46 where bifurcations occurred, for the formula r*p*(1-p), the one described above. He found that the ratios of lengths of adjacent areas of bifurcation were four and a bit. These ratios vary, but, as the growth rate increases, they tend to a limit of 4.669+. This helped him guess where bifurcation points would be, and saved lots of time. When he studied bifurcations of r*sin(PI*p) he found a similar pattern, which is not surprising in itself. However, 4.669+ popped out, again. Different formulae, same number ? Now, THAT's surprising ! He tried many other formulae and ALWAYS got 4.669+ - Hot Damn !!! So hot, in fact, that he phoned home and told his Mom it would make him Famous ! He also went on to tell other scientists. The rest is History... (It has been conjectured that if Feigenbaum had a copy of Fractint, and used it to study bifurcations, he may never have found his Number, as it only became obvious from long perusal of hand-written lists of values, without the distraction of wild color-cycling effects !). We now know that this number is as universal as PI or E. It appears in situations ranging from fluid-flow turbulence, electronic oscillators, chemical reactions, and even the Mandelbrot Set - yup, fraid so: "budding" of the Mandelbrot Set along the negative real axis occurs at intervals determined by Feigenbaum's Number, 4.669201660910..... Fractint does not make direct use of the Feigenbaum Number (YET !). However, it does now reflect the fact that there is a whole sub-species of Bifurcation-type fractals. Those implemented to date, and the related formulae, (writing P for pop[n+1] and p for pop[n]) are : bifurcation P = p + r*fn(p)*(1-fn(p)) Verhulst Bifurcations. biflambda P = r*fn(p)*(1-fn(p)) Real equivalent of Lambda Sets. bif+sinpi P = p + r*fn(PI*p) Population scenario based on... bif=sinpi P = r*fn(PI*p) ...Feigenbaum's second formula. bifstewart P = r*fn(p)*fn(p) - 1 Stewart Map. bifmay P = r*p / ((1+p)^b) May Map. It took a while for bifurcations to appear here, despite them being over a century old, and intimately related to chaotic systems. However, they are now truly alive and well in Fractint! 2.26 Orbit Fractals Orbit Fractals are generated by plotting an orbit path in two or three dimensional space. See Lorenz Attractors (p. 47), Rossler Attractors (p. 48), Henon Attractors (p. 48), Pickover Attractors (p. 49), Gingerbreadman (p. 49), and Martin Attractors (p. 49). The orbit trajectory for these types can be saved in the file ORBITS.RAW by invoking Fractint with the "orbitsave=yes" command-line option. This file will be overwritten each time you generate a new fractal, so rename it if you want to save it. A nifty program called Acrospin can read Fractint Version 18.2 Page 47 these files and rapidly rotate them in 3-D - see Acrospin (p. 153). 2.27 Lorenz Attractors (type=lorenz/lorenz3d) The "Lorenz Attractor" is a "simple" set of three deterministic equations developed by Edward Lorenz while studying the non- repeatability of weather patterns. The weather forecaster's basic problem is that even very tiny changes in initial patterns ("the beating of a butterfly's wings" - the official term is "sensitive dependence on initial conditions") eventually reduces the best weather forecast to rubble. The lorenz attractor is the plot of the orbit of a dynamic system consisting of three first order non-linear differential equations. The solution to the differential equation is vector-valued function of one variable. If you think of the variable as time, the solution traces an orbit. The orbit is made up of two spirals at an angle to each other in three dimensions. We change the orbit color as time goes on to add a little dazzle to the image. The equations are: dx/dt = -a*x + a*y dy/dt = b*x - y -z*x dz/dt = -c*z + x*y We solve these differential equations approximately using a method known as the first order taylor series. Calculus teachers everywhere will kill us for saying this, but you treat the notation for the derivative dx/dt as though it really is a fraction, with "dx" the small change in x that happens when the time changes "dt". So multiply through the above equations by dt, and you will have the change in the orbit for a small time step. We add these changes to the old vector to get the new vector after one step. This gives us: xnew = x + (-a*x*dt) + (a*y*dt) ynew = y + (b*x*dt) - (y*dt) - (z*x*dt) znew = z + (-c*z*dt) + (x*y*dt) (default values: dt = .02, a = 5, b = 15, c = 1) We connect the successive points with a line, project the resulting 3D orbit onto the screen, and voila! The Lorenz Attractor! We have added two versions of the Lorenz Attractor. "Type=lorenz" is the Lorenz attractor as seen in everyday 2D. "Type=lorenz3d" is the same set of equations with the added twist that the results are run through our perspective 3D routines, so that you get to view it from different angles (you can modify your perspective "on the fly" by using the command.) If you set the "stereo" option to "2", and have red/blue funny glasses on, you will see the attractor orbit with depth perception. Fractint Version 18.2 Page 48 Hint: the default perspective values (x = 60, y = 30, z = 0) aren't the best ones to use for fun Lorenz Attractor viewing. Experiment a bit - start with rotation values of 0/0/0 and then change to 20/0/0 and 40/0/0 to see the attractor from different angles.- and while you're at it, use a non-zero perspective point Try 100 and see what happens when you get *inside* the Lorenz orbits. Here comes one - Duck! While you are at it, turn on the sound with the "X". This way you'll at least hear it coming! Different Lorenz attractors can be created using different parameters. Four parameters are used. The first is the time-step (dt). The default value is .02. A smaller value makes the plotting go slower; a larger value is faster but rougher. A line is drawn to connect successive orbit values. The 2nd, third, and fourth parameters are coefficients used in the differential equation (a, b, and c). The default values are 5, 15, and 1. Try changing these a little at a time to see the result. 2.28 Rossler Attractors (type=rossler3D) This fractal is named after the German Otto Rossler, a non-practicing medical doctor who approached chaos with a bemusedly philosophical attitude. He would see strange attractors as philosophical objects. His fractal namesake looks like a band of ribbon with a fold in it. All we can say is we used the same calculus-teacher-defeating trick of multiplying the equations by "dt" to solve the differential equation and generate the orbit. This time we will skip straight to the orbit generator - if you followed what we did above with type Lorenz (p. 47) you can easily reverse engineer the differential equations. xnew = x - y*dt - z*dt ynew = y + x*dt + a*y*dt znew = z + b*dt + x*z*dt - c*z*dt Default parameters are dt = .04, a = .2, b = .2, c = 5.7 2.29 Henon Attractors (type=henon) Michel Henon was an astronomer at Nice observatory in southern France. He came to the subject of fractals via investigations of the orbits of astronomical objects. The strange attractor most often linked with Henon's name comes not from a differential equation, but from the world of discrete mathematics - difference equations. The Henon map is an example of a very simple dynamic system that exhibits strange behavior. The orbit traces out a characteristic banana shape, but on close inspection, the shape is made up of thicker and thinner parts. Upon magnification, the thicker bands resolve to still other thick and thin components. And so it goes forever! The equations that generate this strange pattern perform the mathematical equivalent of repeated stretching and folding, over and over again. Fractint Version 18.2 Page 49 xnew = 1 + y - a*x*x ynew = b*x The default parameters are a=1.4 and b=.3. 2.30 Pickover Attractors (type=pickover) Clifford A. Pickover of the IBM Thomas J. Watson Research center is such a creative source for fractals that we attach his name to this one only with great trepidation. Probably tomorrow he'll come up with another one and we'll be back to square one trying to figure out a name! This one is the three dimensional orbit defined by: xnew = sin(a*y) - z*cos(b*x) ynew = z*sin(c*x) - cos(d*y) znew = sin(x) Default parameters are: a = 2.24, b = .43, c = -.65, d = -2.43 2.31 Gingerbreadman (type=gingerbreadman) This simple fractal is a charming example stolen from "Science of Fractal Images", p. 149. xnew = 1 - y + |x| ynew = x The initial x and y values are set by parameters, defaults x=-.1, y = 0. 2.32 Martin Attractors (type=hopalong/martin) These fractal types are from A. K. Dewdney's "Computer Recreations" column in "Scientific American". They are attributed to Barry Martin of Aston University in Birmingham, England. Hopalong is an "orbit" type fractal like lorenz. The image is obtained by iterating this formula after setting z(0) = y(0) = 0: x(n+1) = y(n) - sign(x(n))*sqrt(abs(b*x(n)-c)) y(n+1) = a - x(n) Parameters are a, b, and c. The function "sign()" returns 1 if the argument is positive, -1 if argument is negative. This fractal continues to develop in surprising ways after many iterations. Fractint Version 18.2 Page 50 Another Martin fractal is simpler. The iterated formula is: x(n+1) = y(n) - sin(x(n)) y(n+1) = a - x(n) The parameter is "a". Try values near the number pi. 2.33 Icon (type=icon/icon3d) This fractal type was inspired by the book "Symmetry in Chaos" by Michael Field and Martin Golubitsky (ISBN 0-19-853689-5, Oxford Press) To quote from the book's jacket, "Field and Golubitsky describe how a chaotic process eventually can lead to symmetric patterns (in a river, for instance, photographs of the turbulent movement of eddies, taken over time, often reveal patterns on the average." The Icon type implemented here maps the classic population logistic map of bifurcation fractals onto the complex plane in Dn symmetry. The initial points plotted are the more chaotic initial orbits, but as you wait, delicate webs will begin to form as the orbits settle into a more periodic pattern. Since pixels are colored by the number of times they are hit, the more periodic paths will become clarified with time. These fractals run continuously. There are 6 parameters: Lambda, Alpha, Beta, Gamma, Omega, and Degree Omega 0 = Dn, or dihedral (rotation + reflectional) symmetry !0 = Zn, or cyclic (rotational) symmetry Degree = n, or Degree of symmetry 2.34 Test (type=test) This is a stub that we (and you!) use for trying out new fractal types. "Type=test" fractals make use of Fractint's structure and features for whatever code is in the routine 'testpt()' (located in the small source file TESTPT.C) to determine the color of a particular pixel. If you have a favorite fractal type that you believe would fit nicely into Fractint, just rewrite the C function in TESTPT.C (or use the prototype function there, which is a simple M-set implementation) with an algorithm that computes a color based on a point in the complex plane. After you get it working, send your code to one of the authors and we might just add it to the next release of Fractint, with full credit to you. Our criteria are: 1) an interesting image and 2) a formula significantly different from types already supported. (Bribery may also work. THIS author is completely honest, but I don't trust those other guys.) Be sure to include an explanation of your algorithm and the Fractint Version 18.2 Page 51 parameters supported, preferably formatted as you see here to simplify folding it into the documentation. 2.35 Formula (type=formula) This is a "roll-your-own" fractal interpreter - you don't even need a compiler! To run a "type=formula" fractal, you first need a text file containing formulas (there's a sample file - FRACTINT.FRM - included with this distribution). When you select the "formula" fractal type, Fractint scans the current formula file (default is FRACTINT.FRM) for formulas, then prompts you for the formula name you wish to run. After prompting for any parameters, the formula is parsed for syntax errors and then the fractal is generated. If you want to use a different formula file, press <F6> when you are prompted to select a formula name. There are two command-line options that work with type=formula ("formulafile=" and "formulaname="), useful when you are using this fractal type in batch mode. The following documentation is supplied by Mark Peterson, who wrote the formula interpreter: Formula fractals allow you to create your own fractal formulas. The general format is: Mandelbrot(XAXIS) { z = Pixel: z = sqr(z) + pixel, |z| <= 4 } | | | | | Name Symmetry Initial Iteration Bailout Condition Criteria Initial conditions are set, then the iterations performed until the bailout criteria is true or 'z' turns into a periodic loop. All variables are created automatically by their usage and treated as complex. If you declare 'v = 2' then the variable 'v' is treated as a complex with an imaginary value of zero. Predefined Variables (x, y) -------------------------------------------- z used for periodicity checking p1 parameters 1 and 2 p2 parameters 3 and 4 pixel screen coordinates LastSqr Modulus from the last sqr() function rand Complex random number Precedence -------------------------------------------- 1 sin(), cos(), sinh(), cosh(), cosxx(), tan(), cotan(), tanh(), cotanh(), sqr, log(), exp(), abs(), conj(), real(), imag(), flip(), fn1(), fn2(), fn3(), fn4(), Fractint Version 18.2 Page 52 srand() 2 - (negation), ^ (power) 3 * (multiplication), / (division) 4 + (addition), - (subtraction) 5 = (assignment) 6 < (less than), <= (less than or equal to) > (greater than), >= (greater than or equal to) == (equal to), != (not equal to) 7 && (logical AND), || (logical OR) Precedence may be overridden by use of parenthesis. Note the modulus squared operator |z| is also parenthetic and always sets the imaginary component to zero. This means 'c * |z - 4|' first subtracts 4 from z, calculates the modulus squared then multiplies times 'c'. Nested modulus squared operators require overriding parenthesis: c * |z + (|pixel|)| The functions fn1(...) to fn4(...) are variable functions - when used, the user is prompted at run time (on the <Z> screen) to specify one of sin, cos, sinh, cosh, exp, log, sqr, etc. for each required variable function. The formulas are performed using either integer or floating point mathematics depending on the <F> floating point toggle. If you do not have an FPU then type MPC math is performed in lieu of traditional floating point. The 'rand' predefined variable is changed with each iteration to a new random number with the real and imaginary components containing a value between zero and 1. Use the srand() function to initialize the random numbers to a consistent random number sequence. If a formula does not contain the srand() function, then the formula compiler will use the system time to initialize the sequence. This could cause a different fractal to be generated each time the formula is used depending on how the formula is written. Remember that when using integer math there is a limited dynamic range, so what you think may be a fractal could really be just a limitation of the integer math range. God may work with integers, but His dynamic range is many orders of magnitude greater than our puny 32 bit mathematics! Always verify with the floating point <F> toggle. 2.36 Julibrots (type=julibrot) The Julibrot fractal type uses a general-purpose renderer for visualizing three dimensional solid fractals. Originally Mark Peterson developed this rendering mechanism to view a 3-D sections of a 4-D structure he called a "Julibrot". This structure, also called "layered Julia set" in the fractal literature, hinges on the relationship between the Mandelbrot and Julia sets. Each Julia set is created using a fixed value c in the iterated formula z^2 + c. The Julibrot is created by layering Julia sets in the x-y plane and continuously varying c, creating new Julia sets as z is incremented. The solid shape thus Fractint Version 18.2 Page 53 created is rendered by shading the surface using a brightness inversely proportional to the virtual viewer's eye. Starting with Fractint version 18, the Julibrot engine can be used with other Julia formulas besides the classic z^2 + c. The first field on the Julibrot parameter screen lets you select which orbit formula to use. You can also use the Julibrot renderer to visualize 3D cross sections of true four dimensional Quaternion and Hypercomplex fractals. The Julibrot Parameter Screens Orbit Algorithm - select the orbit algorithm to use. The available possibilities include 2-D Julia and both mandelbrot and Julia variants of the 4-D Quaternion and Hypercomplex fractals. Orbit parameters - the next screen lets you fill in any parameters belonging to the orbit algorithm. This list of parameters is not necessarily the same as the list normally presented for the orbit algorithm, because some of these parameters are used in the Julibrot layering process. From/To Parameters These parameters allow you to specify the "Mandelbrot" values used to generate the layered Julias. The parameter c in the Julia formulas will be incremented in steps ranging from the "from" x and y values to the "to" x and y values. If the orbit formula is one of the "true" four dimensional fractal types quat, quatj, hypercomplex, or hypercomplexj, then these numbers are used with the 3rd and 4th dimensional values. The "from/to" variables are different for the different kinds of orbit algorithm. 2D Julia sets - complex number formula z' = f(z) + c The "from/to" parameters change the values of c. 4D Julia sets - Quaternion or Hypercomplex formula z' = f(z) + c The four dimensions of c are set by the orbit parameters. The first two dimensions of z are determined by the corners values. The third and fourth dimensions of z are the "to/from" variables. 4D Mandelbrot sets - Quaternion or Hypercomplex formula z' = f(z) + c The first two dimensions of c are determined by the corners values. The third and fourth dimensions of c are the "to/from" variables. Distance between the eyes - set this to 2.5 if you want a red/blue anaglyph image, 0 for a normal greyscale image. Number of z pixels - this sets how many layers are rendered in the screen z-axis. Use a higher value with higher resolution video modes. The remainder of the parameters are needed to construct the red/blue picture so that the fractal appears with the desired depth and proper 'z' location. With the origin set to 8 inches beyond the screen plane and the depth of the fractal at 8 inches the default fractal will appear to start at 4 inches beyond the screen and extend to 12 inches if your eyeballs are 2.5 inches apart and located at a distance of 24 inches Fractint Version 18.2 Page 54 from the screen. The screen dimensions provide the reference frame. 2.37 Diffusion Limited Aggregation (type=diffusion) This type begins with a single point in the center of the screen. Subsequent points move around randomly until coming into contact with the first point, at which time their locations are fixed and they are colored randomly. This process repeats until the fractals reaches the edge of the screen. Use the show orbits function to see the points' random motion. One unfortunate problem is that on a large screen, this process will tend to take eons. To speed things up, the points are restricted to a box around the initial point. The first and only parameter to diffusion contains the size of the border between the fractal and the edge of the box. If you make this number small, the fractal will look more solid and will be generated more quickly. Diffusion was inspired by a Scientific American article a couple of years back which includes actual pictures of real physical phenomena that behave like this. Thanks to Adrian Mariano for providing the diffusion code and documentation. Juan J. Buhler added the additional options. 2.38 Magnetic Fractals (type=magnet1m/.../magnet2j) These fractals use formulae derived from the study of hierarchical lattices, in the context of magnetic renormalisation transformations. This kinda stuff is useful in an area of theoretical physics that deals with magnetic phase-transitions (predicting at which temperatures a given substance will be magnetic, or non-magnetic). In an attempt to clarify the results obtained for Real temperatures (the kind that you and I can feel), the study moved into the realm of Complex Numbers, aiming to spot Real phase-transitions by finding the intersections of lines representing Complex phase-transitions with the Real Axis. The first people to try this were two physicists called Yang and Lee, who found the situation a bit more complex than first expected, as the phase boundaries for Complex temperatures are (surprise!) fractals. And that's all the technical (?) background you're getting here! For more details (are you SERIOUS ?!) read "The Beauty of Fractals". When you understand it all, you might like to rewrite this section, before you start your new job as a professor of theoretical physics... In Fractint terms, the important bits of the above are "Fractals", "Complex Numbers", "Formulae", and "The Beauty of Fractals". Lifting the Formulae straight out of the Book and iterating them over the Complex plane (just like the Mandelbrot set) produces Fractals. Fractint Version 18.2 Page 55 The formulae are a bit more complicated than the Z^2+C used for the Mandelbrot Set, that's all. They are : [ ] 2 | Z^2 + (C-1) | MAGNET1 : | ------------- | | 2*Z + (C-2) | [ ] [ ] 2 | Z^3 + 3*(C-1)*Z + (C-1)*(C-2) | MAGNET2 : | --------------------------------------- | | 3*(Z^2) + 3*(C-2)*Z + (C-1)*(C-2) + 1 | [ ] These aren't quite as horrific as they look (oh yeah ?!) as they only involve two variables (Z and C), but cubing things, doing division, and eventually squaring the result (all in Complex Numbers) don't exactly spell S-p-e-e-d ! These are NOT the fastest fractals in Fractint ! As you might expect, for both formulae there is a single related Mandelbrot-type set (magnet1m, magnet2m) and an infinite number of related Julia-type sets (magnet1j, magnet2j), with the usual toggle between the corresponding Ms and Js via the spacebar. If you fancy delving into the Julia-types by hand, you will be prompted for the Real and Imaginary parts of the parameter denoted by C. The result is symmetrical about the Real axis (and therefore the initial image gets drawn in half the usual time) if you specify a value of Zero for the Imaginary part of C. Fractint Historical Note: Another complication (besides the formulae) in implementing these fractal types was that they all have a finite attractor (1.0 + 0.0i), as well as the usual one (Infinity). This fact spurred the development of Finite Attractor logic in Fractint. Without this code you can still generate these fractals, but you usually end up with a pretty boring image that is mostly deep blue "lake", courtesy of Fractint's standard Periodicity Logic (p. 123). See Finite Attractors (p. 139) for more information on this aspect of Fractint internals. (Thanks to Kevin Allen for Magnetic type documentation above). 2.39 L-Systems (type=lsystem) These fractals are constructed from line segments using rules specified in drawing commands. Starting with an initial string, the axiom, transformation rules are applied a specified number of times, to produce the final command string which is used to draw the image. Like the type=formula fractals, this type requires a separate data file. A sample file, FRACTINT.L, is included with this distribution. When you select type lsystem, the current lsystem file is read and you are asked for the lsystem name you wish to run. Press <F6> at this point if you Fractint Version 18.2 Page 56 wish to use a different lsystem file. After selecting an lsystem, you are asked for one parameter - the "order", or number of times to execute all the transformation rules. It is wise to start with small orders, because the size of the substituted command string grows exponentially and it is very easy to exceed your resolution. (Higher orders take longer to generate too.) The command line options "lname=" and "lfile=" can be used to over-ride the default file name and lsystem name. Each L-System entry in the file contains a specification of the angle, the axiom, and the transformation rules. Each item must appear on its own line and each line must be less than 160 characters long. The statement "angle n" sets the angle to 360/n degrees; n must be an integer greater than two and less than fifty. "Axiom string" defines the axiom. Transformation rules are specified as "a=string" and convert the single character 'a' into "string." If more than one rule is specified for a single character all of the strings will be added together. This allows specifying transformations longer than the 160 character limit. Transformation rules may operate on any characters except space, tab or '}'. Any information after a ; (semi-colon) on a line is treated as a comment. Here is a sample lsystem: Dragon { ; Name of lsystem, { indicates start Angle 8 ; Specify the angle increment to 45 degrees Axiom FX ; Starting character string F= ; First rule: Delete 'F' y=+FX--FY+ ; Change 'y' into "+fx--fy+" x=-FX++FY- ; Similar transformation on 'x' } ; final } indicates end The standard drawing commands are: F Draw forward G Move forward (without drawing) + Increase angle - Decrease angle | Try to turn 180 degrees. (If angle is odd, the turn will be the largest possible turn less than 180 degrees.) These commands increment angle by the user specified angle value. They should be used when possible because they are fast. If greater flexibility is needed, use the following commands which keep a completely separate angle pointer which is specified in degrees. D Draw forward M Move forward nn Increase angle nn degrees /nn Decrease angle nn degrees Fractint Version 18.2 Page 57 Color control: Cnn Select color nn <nn Increment color by nn >nn decrement color by nn Advanced commands: ! Reverse directions (Switch meanings of +, - and , /) @nnn Multiply line segment size by nnn nnn may be a plain number, or may be preceded by I for inverse, or Q for square root. (e.g. @IQ2 divides size by the square root of 2) [ Push. Stores current angle and position on a stack ] Pop. Return to location of last push Other characters are perfectly legal in command strings. They are ignored for drawing purposes, but can be used to achieve complex translations. 2.40 Lyapunov Fractals (type=lyapunov) The Bifurcation fractal illustrates what happens in a simple population model as the growth rate increases. The Lyapunov fractal expands that model into two dimensions by letting the growth rate vary in a periodic fashion between two values. Each pair of growth rates is run through a logistic population model and a value called the Lyapunov Exponent is calculated for each pair and is plotted. The Lyapunov Exponent is calculated by adding up log | r - 2*r*x| over many cycles of the population model and dividing by the number of cycles. Negative Lyapunov exponents indicate a stable, periodic behavior and are plotted in color. Positive Lyapunov exponents indicate chaos (or a diverging model) and are colored black. Order parameter. Each possible periodic sequence yields a two dimensional space to explore. The Order parameter selects a sequence. The default value 0 represents the sequence ab which alternates between the two values of the growth parameter. On the screen, the a values run vertically and the b values run horizontally. Here is how to calculate the space parameter for any desired sequence. Take your sequence of a's and b's and arrange it so that it starts with at least 2 a's and ends with a b. It may be necessary to rotate the sequence or swap a's and b's. Strike the first a and the last b off the list and replace each remaining a with a 1 and each remaining b with a zero. Interpret this as a binary number and convert it into decimal. An Example. I like sonnets. A sonnet is a poem with fourteen lines that has the following rhyming sequence: abba abba abab cc. Ignoring the rhyming couplet at the end, let's calculate the Order parameter for this pattern. abbaabbaabab doesn't start with at least 2 a's aabbaabababb rotate it 1001101010 drop the first and last, replace with 0's and 1's Fractint Version 18.2 Page 58 512+64+32+8+2 = 618 An Order parameter of 618 gives the Lyapunov equivalent of a sonnet. "How do I make thee? Let me count the ways..." Population Seed. When two parts of a Lyapunov overlap, which spike overlaps which is strongly dependent on the initial value of the population model. Any changes from using a different starting value between 0 and 1 may be subtle. The values 0 and 1 are interpreted in a special manner. A Seed of 1 will choose a random number between 0 and 1 at the start of each pixel. A Seed of 0 will suppress resetting the seed value between pixels unless the population model diverges in which case a random seed will be used on the next pixel. Filter Cycles. Like the Bifurcation model, the Lyapunov allow you to set the number of cycles that will be run to allow the model to approach equilibrium before the lyapunov exponent calculation is begun. The default value of 0 uses one half of the iterations before beginning the calculation of the exponent. Reference. A.K. Dewdney, Mathematical Recreations, Scientific American, Sept. 1991 2.41 fn||fn Fractals (type=lambda(fn||fn), manlam(fn||fn), julia(fn||fn), mandel(fn||fn)) Two functions=[sin|cos|sinh|cosh|exp|log|sqr|...]) are specified with these types. The two functions are alternately used in the calculation based on a comparison between the modulus of the current Z and the shift value. The first function is used if the modulus of Z is less than the shift value and the second function is used otherwise. The lambda(fn||fn) type calculates the Julia set of the formula lambda*fn(Z), for various values of the function "fn", where lambda and Z are both complex. Two values, the real and imaginary parts of lambda, should be given in the "params=" option. The third value is the shift value. The space bar will generate the corresponding "psuedo Mandelbrot" set, manlam(fn||fn). The manlam(fn||fn) type calculates the "psuedo Mandelbrot" set of the formula fn(Z)*C, for various values of the function "fn", where C and Z are both complex. Two values, the real and imaginary parts of Z(0), should be given in the "params=" option. The third value is the shift value. The space bar will generate the corresponding julia set, lamda(fn||fn). The julia(fn||fn) type calculates the Julia set of the formula fn(Z)+C, for various values of the function "fn", where C and Z are both complex. Two values, the real and imaginary parts of C, should be given in the "params=" option. The third value is the shift value. The space bar will generate the corresponding mandelbrot set, mandel(fn||fn). Fractint Version 18.2 Page 59 The mandel(fn||fn) type calculates the Mandelbrot set of the formula fn(Z)+C, for various values of the function "fn", where C and Z are both complex. Two values, the real and imaginary parts of Z(0), should be given in the "params=" option. The third value is the shift value. The space bar will generate the corresponding julia set, julia(fn||fn). 2.42 Halley (type=halley) The Halley map is an algorithm used to find the roots of polynomial equations by successive "guesses" that converge on the correct value as you feed the results of each approximation back into the formula. It works very well -- unless you are unlucky enough to pick a value that is on a line BETWEEN two actual roots. In that case, the sequence explodes into chaos, with results that diverge more and more wildly as you continue the iteration. This fractal type shows the results for the polynomial Z(Z^a - 1), which has a+1 roots in the complex plane. Use the <T>ype command and enter "halley" in response to the prompt. You will be asked for a parameter, the "order" of the equation (an integer from 2 through 10 -- 2 for Z(Z^2 - 1), 7 for Z(Z^7 - 1), etc.). A second parameter is the relaxation coefficient, and is used to control the convergence stability. A number greater than one increases the chaotic behavior and a number less than one decreases the chaotic behavior. The third parameter is the value used to determine when the formula has converged. The test for convergence is ||Z(n+1)|^2 - |Z(n)|^2| < epsilon. This convergence test produces the whisker-like projections which generally point to a root. 2.43 Dynamic System (type=dynamic, dynamic2) These fractals are based on a cyclic system of differential equations: x'(t) = -f(y(t)) y'(t) = f(x(t)) These equations are approximated by using a small time step dt, forming a time-discrete dynamic system: x(n+1) = x(n) - dt*f(y(n)) y(n+1) = y(n) + dt*f(x(n)) The initial values x(0) and y(0) are set to various points in the plane, the dynamic system is iterated, and the resulting orbit points are plotted. In fractint, the function f is restricted to: f(k) = sin(k + a*fn1(b*k)) The parameters are the spacing of the initial points, the time step dt, and the parameters (a,b,fn1) that affect the function f. Normally the orbit points are plotted individually, but for a negative spacing the points are connected. This fractal is similar to the Pickover Popcorn (p. 42). A variant is the implicit Euler approximation: y(n+1) = y(n) + dt*f(x(n)) Fractint Version 18.2 Page 60 x(n+1) = x(n) - dt*f(y(n+1)) This variant results in complex orbits. The implicit Euler approximation is selected by entering dt<0. There are two options that have unusual effects on these fractals. The Orbit Delay value controls how many initial points are computed before the orbits are displayed on the screen. This allows the orbit to settle down. The outside=summ option causes each pixel to increment color every time an orbit touches it; the resulting display is a 2-d histogram. These fractals are discussed in Chapter 14 of Pickover's "Computers, Pattern, Chaos, and Beauty". 2.44 Mandelcloud (type=mandelcloud) This fractal computes the Mandelbrot function, but displays it differently. It starts with regularly spaced initial pixels and displays the resulting orbits. This idea is somewhat similar to the Dynamic System (p. 59). There are two options that have unusual effects on this fractal. The Orbit Delay value controls how many initial points are computed before the orbits are displayed on the screen. This allows the orbit to settle down. The outside=summ option causes each pixel to increment color every time an orbit touches it; the resulting display is a 2-d histogram. This fractal was invented by Noel Giffin. 2.45 Quaternion (type=quat,quatjul) These fractals are based on quaternions. Quaternions are an extension of complex numbers, with 4 parts instead of 2. That is, a quaternion Q equals a+ib+jc+kd, where a,b,c,d are reals. Quaternions have rules for addition and multiplication. The normal Mandelbrot and Julia formulas can be generalized to use quaternions instead of complex numbers. There is one complication. Complex numbers have 2 parts, so they can be displayed on a plane. Quaternions have 4 parts, so they require 4 dimensions to view. That is, the quaternion Mandelbrot set is actually a 4-dimensional object. Each quaternion C generates a 4-dimensional Julia set. One method of displaying the 4-dimensional object is to take a 3- dimensional slice and render the resulting object in 3-dimensional perspective. Fractint isn't that sophisticated, so it merely displays a 2-dimensional slice of the resulting object. (Note: Now Fractint is that sophisticated! See the Julibrot type!) Fractint Version 18.2 Page 61 In fractint, for the Julia set, you can specify the four parameters of the quaternion constant: c=(c1,ci,cj,ck), but the 2-dimensional slice of the z-plane Julia set is fixed to (xpixel,ypixel,0,0). For the Mandelbrot set, you can specify the position of the c-plane slice: (xpixel,ypixel,cj,ck). These fractals are discussed in Chapter 10 of Pickover's "Computers, Pattern, Chaos, and Beauty". 2.46 HyperComplex (type=hypercomplex,hypercomplexj) These fractals are based on hypercomplex numbers, which like quaternions are a four dimensional generalization of complex numbers. It is not possible to fully generalize the complex numbers to four dimensions without sacrificing some of the algebraic properties shared by real and complex numbers. Quaternions violate the commutative law of multiplication, which says z1*z2 = z2*z1. Hypercomplex numbers fail the rule that says all non-zero elements have multiplicative inverses - that is, if z is not 0, there should be a number 1/z such that (1/z)*(z) = 1. This law holds most of the time but not all the time for hypercomplex numbers. However hypercomplex numbers have a wonderful property for fractal purposes. Every function defined for complex numbers has a simple generalization to hypercomplex numbers. Fractint's implementation takes advantage of this by using "fn" variables - the iteration formula is h(n+1) = fn(h(n)) + C. where "fn" is the hypercomplex generalization of sin, cos, log, sqr etc. You can see 3D versions of these fractals using fractal type Julibrot. Hypercomplex numbers were brought to our attention by Clyde Davenport, author of "A Hypercomplex Calculus with Applications to Relativity", ISBN 0-9623837-0-8. 2.47 Cellular Automata (type=cellular) These fractals are generated by 1-dimensional cellular automata. Consider a 1-dimensional line of cells, where each cell can have the value 0 or 1. In each time step, the new value of a cell is computed from the old value of the cell and the values of its neighbors. On the screen, each horizontal row shows the value of the cells at any one time. The time axis proceeds down the screen, with each row computed from the row above. Different classes of cellular automata can be described by how many different states a cell can have (k), and how many neighbors on each side are examined (r). Fractint implements the binary nearest neighbor cellular automata (k=2,r=1), the binary next-nearest neighbor cellular automata (k=2,r=2), and the ternary nearest neighbor cellular automata Fractint Version 18.2 Page 62 (k=3,r=1) and several others. The rules used here determine the next state of a given cell by using the sum of the states in the cell's neighborhood. The sum of the cells in the neighborhood are mapped by rule to the new value of the cell. For the binary nearest neighbor cellular automata, only the closest neighbor on each side is used. This results in a 4 digit rule controlling the generation of each new line: if each of the cells in the neighborhood is 1, the maximum sum is 1+1+1 = 3 and the sum can range from 0 to 3, or 4 values. This results in a 4 digit rule. For instance, in the rule 1010, starting from the right we have 0->0, 1->1, 2->0, 3->1. If the cell's neighborhood sums to 2, the new cell value would be 0. For the next-nearest cellular automata (kr = 22), each pixel is determined from the pixel value and the two neighbors on each side. This results in a 6 digit rule. For the ternary nearest neighbor cellular automata (kr = 31), each cell can have the value 0, 1, or 2. A single neighbor on each side is examined, resulting in a 7 digit rule. kr #'s in rule example rule | kr #'s in rule example rule 21 4 1010 | 42 16 2300331230331001 31 7 1211001 | 23 8 10011001 41 10 3311100320 | 33 15 021110101210010 51 13 2114220444030 | 24 10 0101001110 61 16 3452355321541340 | 25 12 110101011001 22 6 011010 | 26 14 00001100000110 32 11 21212002010 | 27 16 0010000000000110 The starting row of cells can be set to a pattern of up to 16 digits or to a random pattern. The borders are set to zeros if a pattern is entered or are set randomly if the starting row is set randomly. A zero rule will randomly generate the rule to use. Hitting the space bar toggles between continuously generating the cellular automata and stopping at the end of the current screen. Recommended reading: "Computer Software in Science and Mathematics", Stephen Wolfram, Scientific American, September, 1984. "Abstract Mathematical Art", Kenneth E. Perry, BYTE, December, 1986. "The Armchair Universe", A. K. Dewdney, W. H. Freeman and Company, 1988. "Complex Patterns Generated by Next Nearest Neighbors Cellular Automata", Wentian Li, Computers & Graphics, Volume 13, Number 4. 2.48 Phoenix (type=phoenix, mandphoenix) The phoenix type defaults to the original phoenix curve discovered by Shigehiro Ushiki, "Phoenix", IEEE Transactions on Circuits and Systems, Vol. 35, No. 7, July 1988, pp. 788-789. These images do not have the X and Y axis swapped as is normal for this type. Fractint Version 18.2 Page 63 The mandphoenix type is the corresponding Mandelbrot set image of the phoenix type. The spacebar toggles between the two as long as the mandphoenix type has an initial Z(0) of (0,0). The mandphoenix is not an effective index to the phoenix type, so explore the wild blue yonder. To reproduce the Mandelbrot set image of the phoenix type as shown in Stevens' book, "Fractal Programming in C", set initorbit=0/0 on the command line or with the <g> key. The colors need to be rotated one position because Stevens uses the values from the previous calculation instead of the current calculation to determine when to bailout. 2.49 Frothy Basins (type=frothybasin) Frothy Basins, or Riddled Basins, were discovered by James C. Alexander of the University of Maryland. The discussion below is derived from a two page article entitled "Basins of Froth" in Science News, November 14, 1992 and from correspondence with others, including Dr. Alexander. The equations that generate this fractal are not very different from those that generate many other orbit fractals. z(0) = pixel; z(n+1) = z(n)^2 - c*conj(z(n)) where c = 1 + ai, and a = 1.02871376822... One of the things that makes this fractal so interesting is the shape of the dynamical system's attractors. It is not at all uncommon for a dynamical system to have non-point attractors. Shapes such as circles are very common. Strange attractors are attractors which are themselves fractal. What is unusual about this system, however, is that the attractors intersect. This is the first case in which such a phenomenon has been observed. The three attractors for this system are made up of line segments which overlap to form an equilateral triangle. This attractor triangle can be seen by pressing the 'o' key while the fractal is being generated to turn on the "show orbits" option. An interesting variation on this fractal can be generated by applying the above mapping twice per each iteration. The result is that each of the three attractors is split into two parts, giving the system six attractors. These are also called "Riddled Basins" because each basin is riddled with holes. Which attractor a point is eventually pulled into is extremely sensitive to its initial position. A very slight change in any direction may cause it to end up on a different attractor. As a result, the basins are thoroughly intermingled. The effect appears to be a frothy mixture that has been subjected to lots of stirring and folding. Pixel color is determined by which attractor captures the orbit. The shade of color is determined by the number of iterations required to capture the orbit. In Fractint, the actual shade of color used depends on how many colors are available in the video mode being used. Fractint Version 18.2 Page 64 If 256 colors are available, the default coloring scheme is determined by the number of iterations that were required to capture the orbit. An alternative coloring scheme can be used where the shade is determined by the iterations required divided by the maximum iterations. This method is especially useful on deeply zoomed images. If only 16 colors are available, then only the alternative coloring scheme is used. If fewer than 16 colors are available, then Fractint just colors the basins without any shading. Fractint Version 18.2 Page 65 3. Doodads, Bells, and Whistles 3.1 Drawing Method The "passes option" (<X> options screen or "passes=" parameter) selects single-pass, dual-pass, or solid-guessing (default) mode. This option applies to most fractal types. Single-pass mode ("1") draws the screen pixel by pixel. Dual-pass ("2") generates a "coarse" screen first as a preview using 2x2-pixel boxes, and then generates the rest of the dots with a second pass. Solid-guessing ("g") is the default. It performs from two to four visible passes - more in higher resolution video modes. Its first visible pass is actually two passes - one pixel per 4x4, 8x8, or 16x16 pixel box is generated, and the guessing logic is applied to fill in the blocks at the next level (2x2, 4x4, or 8x8). Subsequent passes fill in the display at the next finer resolution, skipping blocks which are surrounded by the same color. Solid-guessing can guess wrong, but it sure guesses quickly! Boundary Tracing ("b"), which only works with fractal types (such as the Mandelbrot set, but not the Newton type) that do not contain "islands" of colors, finds a color "boundary", traces it around the screen, and then "blits" in the color over the enclosed area. Tesseral ("t") is a sort of "super-solid-guessing" option that successively divides the image into subsections. It's actually slower than the solid-guessing algorithm, but it looks neat, so we left it in. The "fillcolor=" option in the <X> screen or on the command line sets a fixed color to be used by the Boundary Tracing and Tesseral calculations for filling in defined regions. The effect of this is to show off the boundaries of the areas delimited by these two methods. 3.2 Palette Maps If you have a VGA, MCGA, Super-VGA, 8514/A, XGA, TARGA, or TARGA+ video adapter, you can save and restore color palettes for use with any image. To load a palette onto an existing image, use the <L> command in color- cycling or palette-editing mode. To save a palette, use the <S> command in those modes. To change the default palette for an entire run, use the command line "map=" parameter. The default filetype for color-map files is ".MAP". These color-maps are ASCII text files set up as a series of RGB triplet values (one triplet per line, encoded as the red, green, and blue [RGB] components of the color). Fractint Version 18.2 Page 66 Note that .MAP file color values are in GIF format - values go from 0 (low) to 255 (high), so for a VGA adapter they get divided by 4 before being stuffed into the VGA's Video-DAC registers (so '6' and '7' end up referring to the same color value). Fractint is distributed with some sample .MAP files: ALTERN.MAP the famous "Peterson-Vigneau Pseudo-Grey Scale" BLUES.MAP for rainy days, by Daniel Egnor CHROMA.MAP general purpose, chromatic DEFAULT.MAP the VGA start-up values FIRESTRM.MAP general purpose, muted fire colors GAMMA1.MAP and GAMMA2.MAP Lee Crocker's response to ALTERN.MAP GLASSES1.MAP used with 3d glasses modes GLASSES2.MAP used with 3d glasses modes GOODEGA.MAP for EGA users GREEN.MAP shaded green GREY.MAP another grey variant GRID.MAP for stereo surface grid images HEADACHE.MAP major stripes, by D. Egnor (try cycling and hitting <2>) LANDSCAP.MAP Guruka Singh Khalsa's favorite map for plasma "landscapes" NEON.MAP a flashy map, by Daniel Egnor PAINTJET.MAP high resolution mode PaintJet colors ROYAL.MAP the royal purple, by Daniel Egnor TOPO.MAP Monte Davis's contribution to full color terrain VOLCANO.MAP an explosion of lava, by Daniel Egnor 3.3 Autokey Mode The autokey feature allows you to set up beautiful self-running demo "loops". You can set up hypnotic sequences to attract people to a booth, to generate sequences for special effects, to teach how Fractal exploring is done, etc. A sample autokey file (DEMO.KEY) and a batch to run it (DEMO.BAT) are included with Fractint. Type "demo" at the DOS prompt to run it. Autokey record mode is enabled with the command line parameter "AUTOKEY=RECORD". Keystrokes are saved in an intelligible text format in a file called AUTO.KEY. You can change the file name with the "AUTOKEYNAME=" parameter. Playback is enabled with the parameter "AUTOKEY=PLAY". Playback can be terminated by pressing the <Esc> key. After using record mode to capture an autokey file, you'll probably want to touch it up using your editor before playing it back. Separate lines are not necessary but you'll probably find it easier to understand an autokey file if you put each command on a separate line. Autokey files can contain the following: Quoted strings. Fractint reads whatever is between the quotes just as if you had typed it. For example, "t" "ifs" issues the "t" (type) command and then enters the letters i", "f", and "s" to select the ifs type. Fractint Version 18.2 Page 67 Symbols for function keys used to select a video mode. Examples: F3 -- Function key 3 SF3 --<Shift> and <F3> together Special keys: ENTER ESC F1 PAGEUP PAGEDOWN HOME END LEFT RIGHT UP DOWN INSERT DELETE TAB WAIT <nnn.n> -- wait nnn.n seconds before continuing CALCWAIT -- pause until the current fractal calculation or file save or restore is finished. This command makes demo files more robust since calculation times depend on the speed of the machine running the demo - a "WAIT 10" command may allow enough time to complete a fractal on one machine, but not on another. The record mode does not generate this command - it should be added by hand to the autokey file whenever there is a process that should be allowed to run to completion. GOTO target -- The autokey file continues to be read from the label "target". The label can be any word that does not duplicate a key word. It must be present somewhere in the autokey file with a colon after it. Example: MESSAGE 2 This is executed once start: MESSAGE 2 This is executed repeatedly GOTO start GOTO is mainly useful for writing continuous loop demonstrations. It can also be useful when debugging an autokey file, to skip sections of it. ; -- A semi-colon indicates that the rest of the line containing it is a comment. MESSAGE nn <Your message here> -- Places a message on the top of the screen for nn seconds Making Fractint demos can be tricky. Here are some suggestions which may help: Start Fractint with "fractint autokeyname=mydemo.key autokey=record". Use a unique name each time you run so that you don't overwrite prior files. When in record mode, avoid using the cursor keys to select filenames, fractal types, formula names, etc. Instead, try to type in names. This will ensure that the exact item you want gets chosen during playback even if the list is different then. Beware of video mode assumptions. It is safest to build a separate demo for different resolution monitors. When in the record mode, try to type names quickly, then pause. If you pause partway through a name Fractint will break up the string in the .KEY file. E.g. if you paused in the middle of typing fract001, you might get: "fract" Fractint Version 18.2 Page 68 WAIT 2.2 "001" No harm done, but messy to clean up. Fractint ignores pauses less than about 1/2 second. DO pause when you want the viewer to see what is happening during playback. When done recording, clean up your mydemo.key file. Insert a CALCWAIT after each keystroke which triggers something that takes a variable amount of time (calculating a fractal, restoring a file, saving a file). Add comments with ";" to the file so you know what is going on in future. It is a good idea to use INSERT before a GOTO which restarts the demo. The <insert> key resets Fractint as if you exited the program and restarted it. Warning: an autokey file built for this version of Fractint will probably require some retouching before it works with future releases of Fractint. We have no intention of making sure that the same sequence of keystrokes will have exactly the same effect from one version of Fractint to the next. That would require pretty much freezing Fractint development, and we just love to keep enhancing it! 3.4 Distance Estimator Method This is Phil Wilson's implementation of an alternate method for the M and J sets, based on work by mathematician John Milnor and described in "The Science of Fractal Images", p. 198. While it can take full advantage of your color palette, one of the best uses is in preparing monochrome images for a printer. Using the 1600x1200x2 disk-video mode and an HP LaserJet, we have produced pictures of quality equivalent to the black and white illustrations of the M-set in "The Beauty of Fractals." The distance estimator method widens very thin "strands" which are part of the "inside" of the set. Instead of hiding invisibly between pixels, these strands are made one pixel wide. Though this option is available with any escape time fractal type, the formula used is specific to the mandel and julia types - for most other types it doesn't do a great job. To turn on the distance estimator method with any escape time fractal type, set the "Distance Estimator" value on the <Y> options screen (or use the "distest=" command line parameter). Setting the distance estimator option to a negative value -nnn enables edge-tracing mode. The edge of the set is display as color number nnn. This option works best when the "inside" and "outside" color values are also set to some other value(s). Fractint Version 18.2 Page 69 In a 2 color (monochrome) mode, setting to any positive value results in the inside of the set being expanded to include edge points, and the outside points being displayed in the other color. In color modes, setting to value 1 causes the edge points to be displayed using the inside color and the outside points to be displayed in their usual colors. Setting to a value greater than one causes the outside points to be displayed as contours, colored according to their distance from the inside of the set. Use a higher value for narrower color bands, a lower value for wider ones. 1000 is a good value to start with. The second distance estimator parameter ("width factor") sets the distance from the inside of the set which is to be considered as part of the inside. This value is expressed as a percentage of a pixel width, the default is 71. You should use 1 or 2 pass mode with the distance estimator method, to avoid missing some of the thin strands made visible by it. For the highest quality, "maxiter" should also be set to a high value, say 1000 or so. You'll probably also want "inside" set to zero, to get a black interior. Enabling the distance estimator method automatically toggles to floating point mode. When you reset distest back to zero, remember to also turn off floating point mode if you want it off. Unfortunately, images using the distance estimator method can take many hours to calculate even on a fast machine with a coprocessor, especially if a high "maxiter" value is used. One way of dealing with this is to leave it turned off while you find and frame an image. Then hit to save the current image information in a parameter file (see Parameter Save/Restore Commands (p. 22)). Use an editor to change the parameter file entry, adding "distest=1", "video=something" to select a high- resolution monochrome disk-video mode, "maxiter=1000", and "inside=0". Run the parameter file entry with the <@> command when you won't be needing your machine for a while (over the weekend?) 3.5 Inversion Many years ago there was a brief craze for "anamorphic art": images painted and viewed with the use of a cylindrical mirror, so that they looked weirdly distorted on the canvas but correct in the distorted reflection. (This byway of art history may be a useful defense when your friends and family give you odd looks for staring at fractal images color-cycling on a CRT.) The Inversion option performs a related transformation on most of the fractal types. You define the center point and radius of a circle; Fractint maps each point inside the circle to a corresponding point outside, and vice-versa. This is known to mathematicians as inverting (or if you want to get precise, "everting") the plane, and is something they can contemplate without getting a headache. John Milnor (also mentioned in connection with the Distance Estimator Method (p. 68)), made his name in the 1950s with a method for everting a seven- Fractint Version 18.2 Page 70 dimensional sphere, so we have a lot of catching up to do. For example, if a point inside the circle is 1/3 of the way from the center to the radius, it is mapped to a point along the same radial line, but at a distance of (3 * radius) from the origin. An outside point at 4 times the radius is mapped inside at 1/4 the radius. The inversion parameters on the <Y> options screen allow entry of the radius and center coordinates of the inversion circle. A default choice of -1 sets the radius at 1/6 the smaller dimension of the image currently on the screen. The default values for Xcenter and Ycenter use the coordinates currently mapped to the center of the screen. Try this one out with a Newton (p. 34) plot, so its radial "spokes" will give you something to hang on to. Plot a Newton-method image, then set the inversion radius to 1, with default center coordinates. The center "explodes" to the periphery. Inverting through a circle not centered on the origin produces bizarre effects that we're not even going to try to describe. Aren't computers wonderful? 3.6 Decomposition You'll remember that most fractal types are calculated by iterating a simple function of a complex number, producing another complex number, until either the number exceeds some pre-defined "bailout" value, or the iteration limit is reached. The pixel corresponding to the starting point is then colored based on the result of that calculation. The decomposition option ("decomp=", on the <X> screen) toggles to another coloring protocol. Here the points are colored according to which quadrant of the complex plane (negative real/positive imaginary, positive real/positive imaginary, etc.) the final value is in. If you use 4 as the parameter, points ending up in each quadrant are given their own color; if 2 (binary decomposition), points in alternating quadrants are given 2 alternating colors. The result is a kind of warped checkerboard coloring, even in areas that would ordinarily be part of a single contour. Remember, for the M-set all points whose final values exceed 2 (by any amount) after, say, 80 iterations are normally the same color; under decomposition, Fractint runs [bailout-value] iterations and then colors according to where the actual final value falls on the complex plane. When using decomposition, a higher bailout value will give a more accurate plot, at some expense in speed. You might want to set the bailout value (in the parameters prompt following selection of a new fractal type; present for most but not all types) to a higher value than the default. A value of about 50 is a good compromise for M/J sets. Fractint Version 18.2 Page 71 3.7 Logarithmic Palettes and Color Ranges By default, Fractint maps iterations to colors 1:1. I.e. if the calculation for a fractal "escapes" (exceeds the bailout value) after N iterations, the pixel is colored as color number N. If N is greater than the number of colors available, it wraps around. So, if you are using a 16-color video mode, and you are using the default maximum iteration count of 150, your image will run through the 16-color palette 150/16 = 9.375 times. When you use Logarithmic palettes, the entire range of iteration values is compressed to map to one span of the color range. This results in spectacularly different images if you are using a high iteration limit near the current iteration maximum of 32000 and are zooming in on an area near a "lakelet". When using a compressed palette in a 256 color mode, we suggest changing your colors from the usual defaults. The last few colors in the default IBM VGA color map are black. This results in points nearest the "lake" smearing into a single dark band, with little contrast from the blue (by default) lake. Fractint has a number of types of compressed palette, selected by the "Log Palette" line on the <X> screen, or by the "logmap=" command line parameter: logmap=1: for standard logarithmic palette. logmap=-1: "old" logarithmic palette. This variant was the only one used before Fractint 14.0. It differs from logmap=1 in that some colors are not used - logmap=1 "spreads" low color numbers which are unused by logmap=-1's pure logarithmic mapping so that all colors are assigned. logmap=N (>1): Same as logmap=1, but starting from iteration count N. Pixels with iteration counts less than N are mapped to color 1. This is useful when zooming in an area near the lake where no points in the image have low iteration counts - it makes use of the low colors which would otherwise be unused. logmap=-N (<-1): Similar to logmap=N, but uses a square root distribution of the colors instead of a logarithmic one. logmap=2 or -2: Auto calculates the logmap value for maximum effect. Another way to change the 1:1 mapping of iteration counts to colors is to use the "RANGES=" parameter. It has the format: RANGES=aa/bb/cc/dd/... Iteration counts up to and including the first value are mapped to color number 0, up to and including the second value to color number 1, and so on. The values must be in ascending order. A negative value can be specified for "striping". The negative value specifies a stripe width, the value following it specifies the limit of the striped range. Two alternating colors are used within the striped Fractint Version 18.2 Page 72 range. Example: RANGES=0/10/30/-5/65/79/32000 This example maps iteration counts to colors as follows: color iterations ------------------- 0 unused (formula always iterates at least once) 1 1 to 10 2 11 to 30 3 31 to 35, 41 to 45, 51 to 55, and 61 to 65 4 36 to 40, 46 to 50, and 56 to 60 5 66 to 79 6 80 and greater Note that the maximum value in a RANGES parameter is 32767. 3.8 Biomorphs Related to Decomposition (p. 70) are the "biomorphs" invented by Clifford Pickover, and discussed by A. K. Dewdney in the July 1989 "Scientific American", page 110. These are so-named because this coloring scheme makes many fractals look like one-celled animals. The idea is simple. The escape-time algorithm terminates an iterating formula when the size of the orbit value exceeds a predetermined bailout value. Normally the pixel corresponding to that orbit is colored according to the iteration when bailout happened. To create biomorphs, this is modified so that if EITHER the real OR the imaginary component is LESS than the bailout, then the pixel is set to the "biomorph" color. The effect is a bit better with higher bailout values: the bailout is automatically set to 100 when this option is in effect. You can try other values with the "bailout=" option. The biomorph option is turned on via the "biomorph=nnn" command-line option (where "nnn" is the color to use on the affected pixels). When toggling to Julia sets, the default corners are three times bigger than normal to allow seeing the biomorph appendages. Does not work with all types - in particular it fails with any of the mandelsine family. However, if you are stuck with monochrome graphics, try it - works great in two-color modes. Try it with the marksmandel and marksjulia types. 3.9 Continuous Potential Note: This option can only be used with 256 color modes. Fractint's images are usually calculated by the "level set" method, producing bands of color corresponding to regions where the calculation gives the same value. When "3D" transformed (see "3D" Images (p. 76)), most images other than plasma clouds are like terraced landscapes: most of the surface is either horizontal or vertical. To get the best results with the "illuminated" 3D fill options 5 and 6, there is an alternative approach that yields continuous changes in colors. Fractint Version 18.2 Page 73 Continuous potential is approximated by calculating potential = log(modulus)/2^iterations where "modulus" is the orbit value (magnitude of the complex number) when the modulus bailout was exceeded, at the "iterations" iteration. Clear as mud, right? Fortunately, you don't have to understand all the details. However, there ARE a few points to understand. First, Fractint's criterion for halting a fractal calculation, the "modulus bailout value", is generally set to 4. Continuous potential is inaccurate at such a low value. The bad news is that the integer math which makes the "mandel" and "julia" types so fast imposes a hard-wired maximum value of 127. You can still make interesting images from those types, though, so don't avoid them. You will see "ridges" in the "hillsides." Some folks like the effect. The good news is that the other fractal types, particularly the (generally slower) floating point algorithms, have no such limitation. The even better news is that there is a floating-point algorithm for the "mandel" and "julia" types. To force the use of a floating-point algorithm, use Fractint with the "FLOAT=YES" command-line toggle. Only a few fractal types like plasma clouds, the Barnsley IFS type, and "test" are unaffected by this toggle. The parameters for continuous potential are: potential=maxcolor[/slope[/modulus[/16bit]]] These parameters are present on the <Y> options screen. "Maxcolor" is the color corresponding to zero potential, which plots as the TOP of the mountain. Generally this should be set to one less than the number of colors, i.e. usually 255. Remember that the last few colors of the default IBM VGA palette are BLACK, so you won't see what you are really getting unless you change to a different palette. "Slope" affects how rapidly the colors change -- the slope of the "mountains" created in 3D. If this is too low, the palette will not cover all the potential values and large areas will be black. If it is too high, the range of colors in the picture will be much less than those available. There is no easy way to predict in advance what this value should be. "Modulus" is the bailout value used to determine when an orbit has "escaped". Larger values give more accurate and smoother potential. A value of 500 gives excellent results. As noted, this value must be <128 for the integer fractal types (if you select a higher number, they will use 127). "16bit": If you transform a continuous potential image to 3D, the illumination modes 5 and 6 will work fine, but the colors will look a bit granular. This is because even with 256 colors, the continuous potential is being truncated to integers. The "16bit" option can be used to add an extra 8 bits of goodness to each stored pixel, for a much smoother result when transforming to 3D. Fractint Version 18.2 Page 74 Fractint's visible behavior is unchanged when 16bit is enabled, except that solid guessing and boundary tracing are not used. But when you save an image generated with 16bit continuous potential: o The saved file is a fair bit larger. o Fractint names the file with a .POT extension instead of .GIF, if you didn't specify an extension in "savename". o The image can be used as input to a subsequent <3> command to get the promised smoother effect. o If you happen to view the saved image with a GIF viewer other than Fractint, you'll find that it is twice as wide as it is supposed to be. (Guess where the extra goodness was stored!) Though these files are structurally legal GIF files the double-width business made us think they should perhaps not be called GIF - hence the .POT filename extension. A 16bit (.POT) file can be converted to an ordinary 8 bit GIF by <R>estoring it, changing "16bit" to "no" on the <Y> options screen, and <S>aving. You might find with 16bit continuous potential that there's a long delay at the start of an image, and disk activity during calculation. Fractint uses its disk-video cache area to store the extra 8 bits per pixel - if there isn't sufficient memory available, the cache will page to disk. The following commands can be used to recreate the image that Mark Peterson first prototyped for us, and named "MtMand": TYPE=mandel CORNERS=-0.19920/-0.11/1.0/1.06707 INSIDE=255 MAXITER=255 POTENTIAL=255/2000/1000/16bit PASSES=1 FLOAT=yes Note that prior to version 15.0, Fractint: o Produced "16 bit TGA potfiles" This format is no longer generated, but you can still (for a release or two) use <R> and <3> with those files. o Assumed "inside=maxit" for continuous potential. It now uses the current "inside=" value - to recreate prior results you must be explicit about this parameter. 3.10 Starfields Once you have generated your favorite fractal image, you can convert it into a fractal starfield with the 'a' transformation (for 'astronomy'? - once again, all of the good letters were gone already). Stars are generated on a pixel-by-pixel basis - the odds that a particular pixel will coalesce into a star are based (partially) on the color index of that pixel. (The following was supplied by Mark Peterson, the starfield author). Fractint Version 18.2 Page 75 If the screen were entirely black and the 'Star Density per Pixel' were set to 30 then a starfield transformation would create an evenly distributed starfield with an average of one star for every 30 pixels. If you're on a 320x200 screen then you have 64000 pixels and would end up with about 2100 stars. By introducing the variable of 'Clumpiness' we can create more stars in areas that have higher color values. At 100% Clumpiness a color value of 255 will change the average of finding a star at that location to 50:50. A lower clumpiness values will lower the amount of probability weighting. To create a spiral galaxy draw your favorite spiral fractal (IFS, Julia, or Mandelbrot) and perform a starfield transformation. For general starfields I'd recommend transforming a plasma fractal. Real starfields have many more dim stars than bright ones because very few stars are close enough to appear bright. To achieve this effect the program will create a bell curve based on the value of ratio of Dim stars to bright stars. After calculating the bell curve the curve is folded in half and the peak used to represent the number of dim stars. Starfields can only be shown in 256 colors. Fractint will automatically try to load ALTERN.MAP and abort if the map file cannot be found. Fractint Version 18.2 Page 76 4. "3D" Images Fractint can restore images in "3D". Important: we use quotation marks because it does not CREATE images of 3D fractal objects (there are such, but we're not there yet.) Instead, it restores .GIF images as a 3D PROJECTION or STEREO IMAGE PAIR. The iteration values you've come to know and love, the ones that determine pixel colors, are translated into "height" so that your saved screen becomes a landscape viewed in perspective. You can even wrap the landscape onto a sphere for realistic-looking planets and moons that never existed outside your PC! We suggest starting with a saved plasma-cloud screen. Hit <3> in main command mode to begin the process. Next, select the file to be transformed, and the video mode. (Usually you want the same video mode the file was generated in; other choices may or may not work.) After hitting <3>, you'll be bombarded with a long series of options. Not to worry: all of them have defaults chosen to yield an acceptable starting image, so the first time out just pump your way through with the <Enter> key. When you enter a different value for any option, that becomes the default value the next time you hit <3>, so you can change one option at a time until you get what you want. Generally <ESC> will take you back to the previous screen. Once you're familiar with the effects of the 3D option values you have a variety of options on how to specify them. You can specify them all on the command line (there ARE a lot of them so they may not all fit within the DOS command line limits), with an SSTOOLS.INI file, or with a parameter file. Here's an example for you power FRACTINTers, the command FRACTINT MYFILE SAVENAME=MY3D 3D=YES BATCH=YES would make Fractint load MYFILE.GIF, re-plot it as a 3D landscape (taking all of the defaults), save the result as MY3D.GIF, and exit to DOS. By the time you've come back with that cup of coffee, you'll have a new world to view, if not conquer. Note that the image created by 3D transformation is treated as if it were a plasma cloud - We have NO idea how to retain the ability to zoom and pan around a 3D image that has been twisted, stretched, perspective- ized, and water-leveled. Actually, we do, but it involves the kind of hardware that Industrial Light & Magic, Pixar et al. use for feature films. So if you'd like to send us a check equivalent to George Lucas' net from the "Star Wars" series... 4.1 3D Mode Selection After hitting <3> and getting past the filename prompt and video mode selection, you're presented with a "3d Mode Selection" screen. If you wish to change the default for any of the following parameters, use the cursor keys to move through the menu. When you're satisfied press <Enter>. Fractint Version 18.2 Page 77 Preview Mode: Preview mode provides a rapid look at your transformed image using by skipping a lot of rows and filling the image in. Good for quickly discovering the best parameters. Let's face it, the Fractint authors most famous for "blazingly fast" code *DIDN'T* write the 3D routines! [Pieter: "But they *are* picking away it and making some progress in each release."] Show Box: If you have selected Preview Mode you have another option to worry about. This is the option to show the image box in scaled and rotated coordinates x, y, and z. The box only appears in rectangular transformations and shows how the final image will be oriented. If you select light source in the next screen, it will also show you the light source vector so you can tell where the light is coming from in relation to your image. Sorry no head or tail on the vector yet. Coarseness: This sets how many divisions the image will be divided into in the y direction, if you select preview mode, ray tracing output, or grid fill in the "Select Fill Type" screen. Spherical Projection: The next question asks if you want a sphere projection. This will take your image and map it onto a plane if you answer "no" or a sphere if you answer "yes" as described above. Try it and you'll see what we mean. See Spherical Projection (p. 84). Stereo: Stereo sound in Fractint? Well, not yet. Fractint now allows you to create 3D images for use with red/blue glasses like 3D comics you may have seen, or images like Captain EO. Option 0 is normal old 3D you can look at with just your eyes. Options 1 and 2 require the special red/blue-green glasses. They are meant to be viewed right on the screen or on a color print off of the screen. The image can be made to hover entirely or partially in front of the screen. Great fun! These two options give a gray scale image when viewed. Option 1 gives 64 shades of gray but with half the spatial resolution you have selected. It works by writing the red and blue images on adjacent pixels, which is why it eats half your resolution. In general, we recommend you use this only with resolutions above 640x350. Use this mode for continuous potential landscapes where you *NEED* all those shades. Option "2" gives you full spatial resolution but with only 16 shades of gray. If the red and blue images overlap, the colors are mixed. Good for wire-frame images (we call them surface grids), lorenz3d and 3D IFS. Works fine in 16 color modes. Option 3 is for creating stereo pair images for view later with more specialized equipment. It allows full color images to be presented in glorious stereo. The left image presented on the screen first. You may photograph it or save it. Then the second image is presented, you may do the same as the first image. You can then take the two images and convert them to a stereo image pair as outlined by Bruce Goren Fractint Version 18.2 Page 78 (see below). Also see Stereo 3D Viewing (p. 80). Ray Tracing Output: Fractint can create files of its 3d transformations which are compatible with many ray tracing programs. Currently four are supported directly: DKB (now obsolete), VIVID, MTV, and RAYSHADE. In addition a "RAW" output is supported which can be relatively easily transformed to be usable by many other products. One other option is supported: ACROSPIN. This is not a ray tracer, but the same Fractint options apply - see Acrospin (p. 153). Option values: 0 disables the creation of ray tracing output 1 DKB format (obsolete-see below) 2 VIVID format 3 generic format (must be massaged externally) 4 MTV format 5 RAYSHADE format 6 ACROSPIN format Users of POV-Ray can use the DKB output and convert to POV-Ray with the DKB2POV utility that comes with POV-Ray. A better (faster) approach is to create a RAW output file and convert to POV-Ray with RAW2POV. A still better approach is to use POV-Ray's height field feature to directly read the fractal .GIF or .POT file and do the 3D transformation inside POV-Ray. All ray tracing files consist of triangles which follow the surface created by Fractint during the 3d transform. Triangles which lie below the "water line" are not created in order to avoid causing unnecessary work for the poor ray tracers which are already overworked. A simple plane can be substituted by the user at the waterline if needed. The size (and therefore the number) of triangles created is determined by the "coarse" parameter setting. While generating the ray tracing file, you will view the image from above and watch it partitioned into triangles. The color of each triangle is the average of the color of its verticies in the original image, unless BRIEF is selected. If BRIEF is selected, a default color is assigned at the begining of the file and is used for all triangles. Also see Interfacing with Ray Tracing Programs (p. 87). Brief output: This is a ray tracing sub-option. When it is set to yes, Fractint creates a considerably smaller and somewhat faster file. In this mode, all triangles use the default color specified at the begining of the file. This color should be edited to supply the color of your choice. Fractint Version 18.2 Page 79 Targa Output: If you want any of the 3d transforms you select to be saved as a Targa-24 file or overlayed onto one, select yes for this option. The overlay option in the final screen determines whether you will create a new file or overlay an existing one. MAP File name: Imediately after selecting the previous options, you will be given the chance to select an alternate color MAP file. The default is to use the current MAP. If you want another MAP used, then enter your selection at this point. Output File Name: This is a ray tracing sub-option, used to specify the name of the file to be written. The default name is FRACT001.RAY. The name is incremented by one each time a file is written. If you have not set "overwrite=yes" then the file name will also be automatically incremented to avoid over-writing previous files. When you are satisfied with your selections press enter to go to the next parameter screen. 4.2 Select Fill Type Screen This option exists because in the course of the 3D projection, portions of the original image may be stretched to fit the new surface. Points of an image that formerly were right next to each other, now may have a space between them. This option generally determines what to do with the space between the mapped dots. It is not used if you have selected a value for RAY other than 0. For an illustration, pick the second option "just draw the points", which just maps points to corresponding points. Generally this will leave empty space between many of the points. Therefore you can choose various algorithms that "fill in" the space between the points in various ways. Later, try the first option "make a surface grid." This option will make a grid of the surface which is as many divisions in the original "y" direction as was set in "coarse" in the first screen. It is very fast, and can give you a good idea what the final relationship of parts of your picture will look like. Later, try the second option "connect the dots (wire frame)", then "surface fills" - "colors interpolated" and "colors not interpolated", the general favorites of the authors. Solid fill, while it reveals the pseudo-geology under your pseudo-landscape, inevitably takes longer. Later, try the light source fill types. These two algorithms allow you to position the "sun" over your "landscape." Each pixel is colored according to the angle the surface makes with an imaginary light source. You will be asked to enter the three coordinates of the vector pointing Fractint Version 18.2 Page 80 toward the light in a following parameter screen - see Light Source Parameters (p. 83). "Light source before transformation" uses the illumination direction without transforming it. The light source is fixed relative to your computer screen. If you generate a sequence of images with progressive rotation, the effect is as if you and the light source are fixed and the object is rotating. Therefore as the object rotates features of the object move in and out of the light. This fill option was incorrect prior to version 16.1, and has been changed. "Light source after transformation" applies the same transformation to both the light direction and the object. Since both the light direction and the object are transformed, if you generate a sequence of images with the rotation progressively changed, the effect is as if the image and the light source are fixed in relation to each other and you orbit around the image. The illumination of features on the object is constant, but you see the object from different angles. This fill option was correct in earlier Fractint versions and has not been changed. For ease of discussion we will refer to the following fill types by these numbers: 1 - surface grid 2 - (default) - no fill at all - just draw the dots 3 - wire frame - joins points with lines 4 - surface fill - (colors interpolated) 5 - surface fill - (interpolation turned off) 6 - solid fill - draws lines from the "ground" up to the point 7 - surface fill with light model - calculated before 3D transforms 8 - surface fill with light model - calculated after 3D transforms Types 4, 7, and 8 interpolate colors when filling, making a very smooth fill if the palette is continuous. This may not be desirable if the palette is not continuous. Type 5 is the same as type 4 with interpolation turned off. You might want to use fill type 5, for example, to project a .GIF photograph onto a sphere. With type 4, you might see the filled-in points, since chances are the palette is not continuous; type 5 fills those same points in with the colors of adjacent pixels. However, for most fractal images, fill type 4 works better. This screen is not available if you have selected a ray tracing option. 4.3 Stereo 3D Viewing The "Funny Glasses" (stereo 3D) parameter screen is presented only if you select a non-zero stereo option in the prior 3D parameters. (See 3D Mode Selection (p. 76).) We suggest you definitely use defaults at first on this screen. When you look at an image with both eyes, each eye sees the image in slightly different perspective because they see it from different places. Fractint Version 18.2 Page 81 The first selection you must make is ocular separation, the distance the between the viewers eyes. This is measured as a % of screen and is an important factor in setting the position of the final stereo image in front of or behind the CRT Screen. The second selection is convergence, also as a % of screen. This tends to move the image forward and back to set where it floats. More positive values move the image towards the viewer. The value of this parameter needs to be set in conjunction with the setting of ocular separation and the perspective distance. It directly adjusts the overall separation of the two stereo images. Beginning anaglyphers love to create images floating mystically in front of the screen, but grizzled old 3D veterans look upon such antics with disdain, and believe the image should be safely inside the monitor where it belongs! Left and Right Red and Blue image crop (% of screen also) help keep the visible part of the right image the same as the visible part of the left by cropping them. If there is too much in the field of either eye that the other doesn't see, the stereo effect can be ruined. Red and Blue brightness factor. The generally available red/blue-green glasses, made for viewing on ink on paper and not the light from a CRT, let in more red light in the blue-green lens than we would like. This leaves a ghost of the red image on the blue-green image (definitely not desired in stereo images). We have countered this by adjusting the intensity of the red and blue values on the CRT. In general you should not have to adjust this. The final entry is Map file name (present only if stereo=1 or stereo=2 was selected). If you have a special map file you want to use for Stereo 3D this is the place to enter its name. Generally glasses1.map is for type 1 (alternating pixels), and glasses2.map is for type 2 (superimposed pixels). Grid.map is great for wire-frame images using 16 color modes. This screen is not available if you have selected a ray tracing option. 4.4 Rectangular Coordinate Transformation The first entries are rotation values around the X, Y, and Z axes. Think of your starting image as a flat map: the X value tilts the bottom of your monitor towards you by X degrees, the Y value pulls the left side of the monitor towards you, and the Z value spins it counter-clockwise. Note that these are NOT independent rotations: the image is rotated first along the X-axis, then along the Y-axis, and finally along the Z- axis. Those are YOUR axes, not those of your (by now hopelessly skewed) monitor. All rotations actually occur through the center of the original image. Rotation parameters are not used when a ray tracing option has been selected. Then there are three scaling factors in percent. Initially, leave the X and Y axes alone and play with Z, now the vertical axis, which translates into surface "roughness." High values of Z make spiky, on- beyond-Alpine mountains and improbably deep valleys; low values make gentle, rolling terrain. Negative roughness is legal: if you're doing an Fractint Version 18.2 Page 82 M-set image and want Mandelbrot Lake to be below the ground, instead of eerily floating above, try a roughness of about -30%. Next we need a water level -- really a minimum-color value that performs the function "if (color < waterlevel) color = waterlevel". So it plots all colors "below" the one you choose at the level of that color, with the effect of filling in "valleys" and converting them to "lakes." Now we enter a perspective distance, which you can think of as the "distance" from your eye to the image. A zero value (the default) means no perspective calculations, which allows use of a faster algorithm. Perspective distance is not available if you have selected a ray tracing option. For non-zero values, picture a box with the original X-Y plane of your flat fractal on the bottom, and your 3D fractal inside. A perspective value of 100% places your eye right at the edge of the box and yields fairly severe distortion, like a close view through a wide-angle lens. 200% puts your eye as far from the front of the box as the back is behind. 300% puts your eye twice as far from the front of the box as the back is, etc. Try about 150% for reasonable results. Much larger values put you far away for even less distortion, while values smaller than 100% put you "inside" the box. Try larger values first, and work your way in. Next, you are prompted for two types of X and Y shifts (now back in the plane of your screen) that let you move the final image around if you'd like to re-center it. The first set, x and y shift with perspective, move the image and the effect changes the perspective you see. The second set, "x and y adjust without perspective", move the image but do not change perspective. They are used just for positioning the final image on the screen. Shifting of any type is not available if you have selected a ray tracing option. 4.5 3D Color Parameters You are asked for a range of "transparent" colors, if any. This option is most useful when using the 3D Overlay Mode (p. 84). Enter the color range (minimum and maximum value) for which you do not want to overwrite whatever may already be on the screen. The default is no transparency (overwrite everything). Now, for the final option. This one will smooth the transition between colors by randomizing them and reduce the banding that occurs with some maps. Select the value of randomize to between 0 (for no effect) and 7 (to randomize your colors almost beyond use). 3 is a good starting point. That's all for this screen. Press enter for these parameters and the next and final screen will appear (honestly!). Fractint Version 18.2 Page 83 4.6 Light Source Parameters This one deals with all the aspects of light source and Targa files. You must choose the direction of the light from the light source. This will be scaled in the x, y, and z directions the same as the image. For example, 1,1,3 positions the light to come from the lower right front of the screen in relation to the untransformed image. It is important to remember that these coordinates are scaled the same as your image. Thus, "1,1,1" positions the light to come from a direction of equal distances to the right, below and in front of each pixel on the original image. However, if the x,y,z scale is set to 90,90,30 the result will be from equal distances to the right and below each pixel but from only 1/3 the distance in front of the screen i.e.. it will be low in the sky, say, afternoon or morning. Then you are asked for a smoothing factor. Unless you used Continuous Potential (p. 72) when generating the starting image, the illumination when using light source fills may appear "sparkly", like a sandy beach in bright sun. A smoothing factor of 2 or 3 will allow you to see the large-scale shapes better. Smoothing is primarily useful when doing light source fill types with plasma clouds. If your fractal is not a plasma cloud and has features with sharply defined boundaries (e.g. Mandelbrot Lake), smoothing may cause the colors to run. This is a feature, not a bug. (A copyrighted response of [your favorite commercial software company here], used by permission.) The ambient option sets the minimum light value a surface has if it has no direct lighting at all. All light values are scaled from this value to white. This effectively adjusts the depth of the shadows and sets the overall contrast of the image. If you selected the full color option, you have a few more choices. The next is the haze factor. Set this to make distant objects more hazy. Close up objects will have little effect, distant objects will have most. 0 disables the function. 100 is the maximum effect, the farthest objects will be lost in the mist. Currently, this does not really use distance from the viewer, we cheat and use the y value of the original image. So the effect really only works if the y-rotation (set earlier) is between +/- 30. Next, you can chose the name under which to save your Targa file. If you have a RAM disk handy, you might want to create the file on it, for speed. So include its full path name in this option. If you have not set "overwrite=yes" then the file name will be incremented to avoid over-writing previous files. If you are going to overlay an existing Targa file, enter its name here. Next, you may select the background color for the Targa file. The default background on the Targa file is sky blue. Enter the Red, Green, and Blue component for the background color you wish. Fractint Version 18.2 Page 84 Finally, absolutely the last option (this time we mean it): you can now choose to overlay an existing Targa-24, type 2, non mapped, top-to- bottom file, such as created by Fractint or PVRay. The Targa file specified above will be overlayed with new info just as a GIF is overlayed on screen. Note: it is not necessary to use the "O" overlay command to overlay Targa files. The Targa_Overlay option must be set to yes, however. You'll probably want to adjust the final colors for monochrome fill types using light source via color cycling (p. 16). Try one of the more continuous palettes (<F8> through <F10>), or load the GRAY palette with the <A>lternate-map command. Now, lie down for a while in a quiet room with a damp washcloth on your forehead. Feeling better? Good -- because it's time to go back almost to the top of the 3D options and just say yes to: 4.7 Spherical Projection Picture a globe lying on its side, "north" pole to the right. (It's our planet, and we'll position it the way we like.) You will be mapping the X and Y axes of the starting image to latitude and longitude on the globe, so that what was a horizontal row of pixels follows a line of longitude. The defaults exactly cover the hemisphere facing you, from longitude 180 degrees (top) to 0 degrees (bottom) and latitude -90 (left) to latitude 90 (right). By changing them you can map the image to a piece of the hemisphere or wrap it clear around the globe. The next entry is for a radius factor that controls the over-all size of the globe. All the rest of the entries are the same as in the landscape projection. You may want less surface roughness for a plausible look, unless you prefer small worlds with big topography, a la "The Little Prince." WARNING: When the "construction" process begins at the edge of the globe (default) or behind it, it's plotting points that will be hidden by subsequent points as the process sweeps around the sphere toward you. Our nifty hidden-point algorithms "know" this, and the first few dozen lines may be invisible unless a high mountain happens to poke over the horizon. If you start a spherical projection and the screen stays black, wait for a while (a longer while for higher resolution or fill type 6) to see if points start to appear. Would we lie to you? If you're still waiting hours later, first check that the power's still on, then consider a faster system. 4.8 3D Overlay Mode While the <3> command (see "3D" Images (p. 76)) creates its image on a blank screen, the <#> (or <shift-3> on some keyboards) command draws a second image over an existing displayed image. This image can be any restored image from a <R> command or the result of a just executed <3> command. So you can do a landscape, then press <#> and choose spherical projection to re-plot that image or another as a moon in the sky above Fractint Version 18.2 Page 85 the landscape. <#> can be repeated as many times as you like. It's worth noting that not all that many years ago, one of us watched Benoit Mandelbrot and fractal-graphics wizard Dick Voss creating just such a moon-over-landscape image at IBM's research center in Yorktown Heights, NY. The system was a large and impressive mainframe with floating-point facilities bigger than the average minicomputer, running LBLGRAPH -- what Mandelbrot calls "an independent-minded and often very ill-mannered heap of graphics programs that originated in work by Alex Hurwitz and Jack Wright of IBM Los Angeles." We'd like to salute LBLGRAPH, its successors, and their creators, because it was their graphic output (like "Planetrise over Labelgraph Hill," plate C9 in Mandelbrot's "Fractal Geometry of Nature") that helped turn fractal geometry from a mathematical curiosity into a phenomenon. We'd also like to point out that it wasn't as fast, flexible or pretty as Fractint on a 386/16 PC with S-VGA graphics. Now, a lot of the difference has to do with the incredible progress of micro-processor power since then, so a lot of the credit should go to Intel rather than to our highly tuned code. OK, twist our arms -- it IS awfully good code. 4.9 Special Note for CGA or Hercules Users If you are one of those unfortunates with a CGA or Hercules 2-color monochrome graphics, it is now possible for you to make 3D projection images. Try the following unfortunately circuitous approach. Invoke Fractint, making sure you have set askvideo=yes. Use a disk-video mode to create a 256 color fractal. You might want to edit the fractint.cfg file to make a disk-video mode with the same pixel dimensions as your normal video. Using the "3" command, enter the file name of the saved 256 color file, then select your 2 or 4 color mode, and answer the other 3D prompts. You will then see a 3D projection of the fractal. Another example of Stone Soup responsiveness to our fan mail! 4.10 Making Terrains If you enjoy using Fractint for making landscapes, we have several new features for you to work with. When doing 3d transformations banding tends to occur because all pixels of a given height end up the same color. Now, colors can be randomized to make the transitions between different colors at different altitudes smoother. Use the new "RANDOMIZE= " variable to accomplish this. If your light source images all look like lunar landscapes since they are all monochrome and have very dark shadows, we now allow you to set the ambient light for adjusting the contrast of the final image. Use the "Ambient= " variable. In addition to being able to create scenes with light sources in monochrome, you can now do it in full color as well. Setting fullcolor=1 will generate a Targa-24 file with a full color image which will be a combination of the original colors of the source image (or map file if you select map=something) and the amount of light which reflects off a given point on the surface. Since there can be 256 different colors in the original image and 256 levels of light, you can now generate an Fractint Version 18.2 Page 86 image with *lots* of colors. To convert it to a GIF if you can't view Targa files directly, you can use PICLAB (see Other Programs (p. 153)), and the following commands: SET PALETTE 256 SET CREZ 8 TLOAD yourfile.tga MAKEPAL MAP GSAVE yourfile.gif EXIT Using the full color option allows you to also set a haze factor with the "haze= " variable to make more distant objects more hazy. As a default, full color files also have the background set to sky blue. Warning, the files which are created with the full color option are very large, 3 bytes per pixel. So be sure to use a disk with enough space. The file is created using Fractint's disk-video caching, but is always created on real disk (expanded or extended memory is not used.) Try the following settings of the new variables in sequence to get a feel for the effect of each one: ;use this with any filltype map=topo randomize=3; adjusting this smooths color transitions ;now add this using filltype 5 or 6 ambient=20; adjusting this changes the contrast filltype=6 smoothing=2; makes the light not quite as granular as the terrain ;now add the following, and this is where it gets slow fullcolor=1; use PICLAB to reduce resulting lightfile to a GIF ;and finally this haze=20; sets the amount of haze for distant objects When full color is being used, the image you see on the screen will represent the amount of light being reflected, not the colors in the final image. Don't be disturbed if the colors look weird, they are an artifact of the process being used. The image being created in the lightfile won't look like the screen. However, if you are worried, hit ESC several times and when Fractint gets to the end of the current line it will abort. Your partial image will be there as LIGHT001.TGA or with whatever file name you selected with the lightname option. Convert it as described above and adjust any parameters you are not happy with. Its a little awkward, but we haven't figured out a better way yet. Fractint Version 18.2 Page 87 4.11 Making 3D Slides Bruce Goren, CIS's resident stereoscopic maven, contributed these tips on what to do with your 3D images (Bruce inspired and prodded us so much we automated much of what follows, allowing both this and actual on screen stereo viewing, but we included it here for reference and a brief tutorial.) "I use a Targa 32 video card and TOPAS graphic software, moving the viewport or imaginary camera left and right to create two separate views of the stationary object in x,y,z, space. The distance between the two views, known as the inter-ocular distance, toe-in or convergence angle, is critical. It makes the difference between good 3-D and headache- generating bad 3-D. "For a 3D fractal landscape, I created and photographed the left and right eye views as if flying by in an imaginary airplane and mounted the film chips for stereo viewing. To make my image, first I generated a plasma cloud based on a color map I calculated to resemble a geological survey map (available on CIS as TARGA.MAP). In the 3D reconstruction, I used a perspective value of 150 and shifted the camera -15 and +15 on the X-axis for the left and right views. All other values were left to the defaults. "The images are captured on a Matrix 3000 film recorder -- basically a box with a high-resolution (1400 lines) black and white TV and a 35mm camera (Konica FS-1) looking at the TV screen through a filter wheel. The Matrix 3000 can be calibrated for 8 different film types, but so far I have only used Kodak Ektachrome 64 daylight for slides and a few print films. I glass mount the film chips myself. "Each frame is exposed three times, once through each of the red, blue, and green filters to create a color image from computer video without the scan-lines which normally result from photographing television screens. The aspect ratio of the resulting images led me to mount the chips using the 7-sprocket Busch-European Emde masks. The best source of Stereo mounting and viewing supplies I know of is an outfit called Reel 3-D Enterprises, Inc. at P.O. Box 2368, Culver City, CA 90231, tel. 213- 837-2368. "My platform is an IBM PC/AT crystal-swapped up to 9 MHz. The math co-processor runs on a separate 8-MHz accessory sub-board. The system currently has 6.5 MB of RAM." 4.12 Interfacing with Ray Tracing Programs (Also see "Ray Tracing Output", "Brief", and "Output File Name" in "3D Mode Selection" (p. 76).) Fractint allows you to save your 3d transforms in files which may be fed to a ray tracer (or to "Acrospin"). However, they are not ready to be traced by themselves. For one thing, no light source is included. They are actually meant to be included within other ray tracing files. Since the intent is to produce an object which may be included in a larger ray tracing scene, it is expected that all rotations, shifts, and final scaling will be done by the ray tracer. Thus, in creating the Fractint Version 18.2 Page 88 images, no facilities for rotations or shifting is provided. Scaling is provided to achieve the correct aspect ratio. WARNING! The files created using the RAY option can be huge. Setting COARSE to 40 will result in over 2000 triangles. Each triangle can utilize from 50 to 200 bytes each to describe, so your ray tracing files can rapidly approach or exceed 1Meg. Make sure you have enough disk space before you start. Each file starts with a comment identifying the version of Fractint by which it was created. The file ends with a comment giving the number of triangles in the file. The files consist of long strips of adjacent triangles. Triangles are clockwise or counter clockwise depending on the target ray tracer. Currently, MTV and Rayshade are the only ones which use counter clockwise triangles. The size of the triangles is set by the COARSE setting in the main 3d menu. Color information about each individual triangle is included for all files unless in the brief mode. To keep the poor ray tracer from working too hard, if WATERLINE is set to a non zero value, no triangle which lies entirely at or below the current setting of WATERLINE is written to the ray tracing file. These may be replaced by a simple plane in the syntax of the ray tracer you are using. Fractint's coordinate system has the origin of the x-y plane at the upper left hand corner of the screen, with positive x to the right and positive y down. The ray tracing files have the origin of the x-y plane moved to the center of the screen with positive x to the right and positive y up. Increasing values of the color index are out of the screen and in the +z direction. The color index 0 will be found in the xy plane at z=-1. When x- y- and zscale are set to 100, the surface created by the triangles will fall within a box of +/- 1.0 in all 3 directions. Changing scale will change the size and/or aspect ratio of the enclosed object. We will only describe the structure of the RAW format here. If you want to understand any of the ray tracing file formats besides RAW, please see your favorite ray tracer docs. The RAW format simply consists of a series of clockwise triangles. If BRIEF=yes, Each line is a vertex with coordinates x, y, and z. Each triangle is separated by a couple of CR's from the next. If BRIEF=no, the first line in each triangle description if the r,g,b value of the triangle. Setting BRIEF=yes produces shorter files with the color of each triangle removed - all triangles will be the same color. These files are otherwise identical to normal files but will run faster than the non BRIEF files. Also, with BRIEF=yes, you may be able to get files with more triangles to run than with BRIEF=no. Fractint Version 18.2 Page 89 The DKB format is now obsolete. POV-Ray users should use the RAW output and convert to POV-Ray using the POV Group's RAW2POV utility. POV-Ray users can also do all 3D transformations within POV-Ray using height fields. Fractint Version 18.2 Page 90 5. Command Line Parameters, Parameter Files, Batch Mode Fractint accepts command-line parameters that allow you to start it with a particular video mode, fractal type, starting coordinates, and just about every other parameter and option. These parameters can also be specified in a SSTOOLS.INI file, to set them every time you run Fractint. They can also be specified as named groups in a .PAR (parameter) file which you can then call up while running Fractint by using the <@> command. In all three cases (DOS command line, SSTOOLS.INI, and parameter file) the parameters use the same syntax, usually a series of keyword=value commands like SOUND=OFF. Each parameter is described in detail in subsequent sections. 5.1 Using the DOS Command Line You can specify parameters when you start Fractint from DOS by using a command like: FRACTINT SOUND=OFF FILENAME=MYIMAGE.GIF The individual parameters are separated by one or more spaces (an parameter itself may not include spaces). Upper or lower case may be used, and parameters can be in any order. Since DOS commands are limited to 128 characters, Fractint has a special command you can use when you have a lot of startup parameters (or have a set of parameters you use frequently): FRACTINT @MYFILE When @filename is specified on the command line, Fractint reads parameters from the specified file as if they were keyed on the command line. You can create the file with a text editor, putting one "keyword=value" parameter on each line. 5.2 Setting Defaults (SSTOOLS.INI File) Every time Fractint runs, it searches the current directory, and then the directories in your DOS PATH, for a file named SSTOOLS.INI. If it finds this file, it begins by reading parameters from it. This file is useful for setting parameters you always want, such as those defining your printer setup. SSTOOLS.INI is divided into sections belonging to particular programs. Each section begins with a label in brackets. Fractint looks for the label [fractint], and ignores any lines it finds in the file belonging to any other label. If an SSTOOLS.INI file looks like this: Fractint Version 18.2 Page 91 [fractint] sound=off ; (for home use only) printer=hp ; my printer is a LaserJet inside=0 ; using "traditional" black [startrek] warp=9.5 ; Captain, I dinna think the engines can take it! Fractint will use only the second, third, and fourth lines of the file. (Why use a convention like that when Fractint is the only program you know of that uses an SSTOOLS.INI file? Because there are other programs (such as Lee Crocker's PICLAB) that now use the same file, and there may one day be other, sister programs to Fractint using that file.) 5.3 Parameter Files and the <@> Command You can change parameters on-the-fly while running Fractint by using the <@> command and a parameter file. Parameter files contain named groups of parameters, looking something like this: quickdraw { ; a set of parameters named quickdraw maxiter=150 float=no } slowdraw { ; another set of parameters maxiter=2000 float=yes } If you use the <@> command and select a parameter file containing the above example, Fractint will show two choices: quickdraw and slowdraw. You move the cursor to highlight one of the choices and press <Enter> to set the parameters specified in the file by that choice. The default parameter file name is FRACTINT.PAR. A different file can be selected with the "parmfile=" option, or by using <@> and then hitting <F6>. You can create parameter files with a text editor, or for some uses, by using the command. Parameter files can be used in a number of ways, some examples: o To save the parameters for a favorite image. Fractint can do this for you with the command. o To save favorite sets of 3D transformation parameters. Fractint can do this for you with the command. o To set up different sets of parameters you use occasionally. For instance, if you have two printers, you might want to set up a group of parameters describing each. o To save image parameters for later use in batch mode - see Batch Mode (p. 110). Fractint Version 18.2 Page 92 See "Parameter Save/Restore Commands" (p. 22) for details about the <@> and commands. 5.4 General Parameter Syntax Parameters must be separated by one or more spaces. Upper and lower case can be used in keywords and values. Anything on a line following a ; (semi-colon) is ignored, i.e. is a comment. In parameter files and SSTOOLS.INI: o Individual parameters can be entered on separate lines. o Long values can be split onto multiple lines by ending a line with a \ (backslash) - leading spaces on the following line are ignored, the information on the next line from the first non-blank character onward is appended to the prior line. Some terminology: KEYWORD=nnn enter a number in place of "nnn" KEYWORD=[filename] you supply filename KEYWORD=yes|no|whatever choose one of "yes", "no", or "whatever" KEYWORD=1st[/2nd[/3rd]] the slash-separated parameters "2nd" and "3rd" are optional 5.5 Startup Parameters @FILENAME Causes Fractint to read "filename" for parameters. When it finishes, it resumes reading its own command line -- i.e., "FRACTINT MAXITER=250 @MYFILE PASSES=1" is legal. This option is only valid on the DOS command line, as Fractint is not clever enough to deal with multiple indirection. @FILENAME/GROUPNAME Like @FILENAME, but reads a named group of parameters from a parameter file. See "Parameter Files and the <@> Command" (p. 91). FILENAME=[name] Causes Fractint to read the named file, which must either have been saved from an earlier Fractint session or be a generic GIF file, and use that as the starting point, bypassing the initial information screens. The filetype is optional and defaults to .GIF. Non-Fractint GIF files are restored as fractal type "plasma". On the DOS command line you may omit FILENAME= and just give the file name. BATCH=yes See Batch Mode (p. 110). AUTOKEY=play|record Specifying "play" runs Fractint in playback mode - keystrokes are read from the autokey file (see next parameter) and interpreted as if they're Fractint Version 18.2 Page 93 being entered from the keyboard. Specifying "record" runs in recording mode - all keystrokes are recorded in the autokey file. See also Autokey Mode (p. 66). AUTOKEYNAME=[filename] Specifies the file name to be used in autokey mode. The default file name is AUTO.KEY. FPU=387|IIT|NOIIT This parameter is useful if you have an unusual coprocessor chip. If you have a 80287 replacement chip with full 80387 functionality use "FPU=387" to inform Fractint to take advantage of those extra 387 instructions. If you have the IIT fpu, but don't have IIT's 'f4x4int.com' TSR loaded, use "FPU=IIT" to force Fractint to use that chip's matrix multiplication routine automatically to speed up 3-D transformations (if you have an IIT fpu and have that TSR loaded, Fractint will auto-detect the presence of the fpu and TSR and use its extra capabilities automatically). Since all IIT chips support 80387 instructions, enabling the IIT code also enables Fractint's use of all 387 instructions. Setting "FPU=NOIIT" disables Fractint's IIT Auto- detect capability. Warning: multi-tasking operating systems such as Windows and DesQView don't automatically save the IIT chip extra registers, so running two programs at once that both use the IIT's matrix multiply feature but don't use the handshaking provided by that 'f4x4int.com' program, errors will result. MAKEDOC[=filename] Create Fractint documentation file (for printing or viewing with a text editor) and then return to DOS. Filename defaults to FRACTINT.DOC. There's also a function in Fractint's online help which can be used to produce the documentation file - use "Printing Fractint Documentation" from the main help index. 5.6 Calculation Mode Parameters PASSES=1|2|guess|btm|tesseral Selects single-pass, dual-pass, solid-Guessing mode, Boundary Tracing, or the Tesseral algorithm. See Drawing Method (p. 65). FILLCOLOR=normal|<nnn> Sets a color to be used for block fill by Boundary Tracing and Tesseral algorithms. See Drawing Method (p. 65). FLOAT=yes Most fractal types have both a fast integer math and a floating point version. The faster, but possibly less accurate, integer version is the default. If you have a new 80486 or other fast machine with a math coprocessor, or if you are using the continuous potential option (which looks best with high bailout values not possible with our integer math implementation), you may prefer to use floating point. Just add "float=yes" to the command line to do so. Also see "Limitations of Integer Math (And How We Cope)" (p. 123). Fractint Version 18.2 Page 94 SYMMETRY=xxx Forces symmetry to None, Xaxis, Yaxis, XYaxis, Origin, or Pi symmetry. Useful for debugging. 5.7 Fractal Type Parameters TYPE=[name] Selects the fractal type to calculate. The default is type "mandel". PARAMS=n/n/n/n... Set optional (required, for some fractal types) values used in the calculations. These numbers typically represent the real and imaginary portions of some startup value, and are described in detail as needed in Fractal Types (p. 29). (Example: FRACTINT TYPE=julia PARAMS=-0.48/0.626 would wait at the opening screen for you to select a video mode, but then proceed straight to the Julia set for the stated x (real) and y (imaginary) coordinates.) FUNCTION=[fn1[/fn2[/fn3[/fn4]]]] Allows setting variable functions found in some fractal type formulae. Possible values are sin, cos, tan, cotan, sinh, cosh, tanh, cotanh, exp, log, sqr, recip (i.e. 1/z), ident (i.e. identity), and cosxx (cos with a pre version 16 bug). FORMULANAME=[formulaname] Specifies the default formula name for type=formula fractals. (I.e. the name of a formula defined in the FORMULAFILE.) Required if you want to generate one of these fractal types in batch mode, as this is the only way to specify a formula name in that case. LNAME=[lsystemname] Specifies the default L-System name. (I.e. the name of an entry in the LFILE.) Required if you want to generate one of these fractal types in batch mode, as this is the only way to specify an L-System name in that case. IFS=[ifsname] Specifies the default IFS name. (I.e. the name of an entry in the IFSFILE.) Required if you want to generate one of these fractal types in batch mode, as this is the only way to specify an IFS name in that case. 5.8 Image Calculation Parameters MAXITER=nnn Reset the iteration maximum (the number of iterations at which the program gives up and says 'OK, this point seems to be part of the set in question and should be colored [insidecolor]') from the default 150. Values range from 10 to 32000 (super-high iteration limits like 30000 are useful when using logarithmic palettes). See The Mandelbrot Set (p. 29) for a description of the iteration method of calculating fractals. "maxiter=" can also be used to adjust the number of orbits plotted for 3D "attractor" fractal types such as lorenz3d and kamtorus. Fractint Version 18.2 Page 95 CORNERS=xmin/xmax/ymin/ymax[/x3rd/y3rd] Example: corners=-0.739/-0.736/0.288/0.291 Begin with these coordinates as the range of x and y coordinates, rather than the default values of (for type=mandel) -2.0/2.0/-1.5/1.5. When you specify four values (the usual case), this defines a rectangle: x- coordinates are mapped to the screen, left to right, from xmin to xmax, y-coordinates are mapped to the screen, bottom to top, from ymin to ymax. Six parameters can be used to describe any rotated or stretched parallelogram: (xmin,ymax) are the coordinates used for the top-left corner of the screen, (xmax,ymin) for the bottom-right corner, and (x3rd,y3rd) for the bottom-left. CENTER-MAG=[Xctr/Yctr/Mag] This is an alternative way to enter corners as a center point and a magnification that is popular with some fractal programs and publications. Entering just "CENTER-MAG=" tells Fractint whether to use this form rather than corners when saving parameters with the command. The <TAB> status display shows the "corners" in both forms. Note that an aspect ratio of 1.3333 is assumed; if you have altered the zoom box proportions or rotated the zoom box, this form can no longer be used. BAILOUT=nnn Over-rides the default bailout criterion for escape-time fractals. Can also be set from the parameters screen after selecting a fractal type. See description of bailout in The Mandelbrot Set (p. 29). RESET Causes Fractint to reset all calculation related parameters to their default values. Non-calculation parameters such as "printer=", "sound=", and "savename=" are not affected. RESET should be specified at the start of each parameter file entry (used with the <@> command) which defines an image, so that the entry need not describe every possible parameter - when invoked, all parameters not specifically set by the entry will have predictable values (the defaults). INITORBIT=pixel INITORBIT=nnn/nnn Allows control over the value used to begin each Mandelbrot-type orbit. "initorbit=pixel" is the default for most types; this command initializes the orbit to the complex number corresponding to the screen pixel. The command "initorbit=nnn/nnn" uses the entered value as the initializer. See the discussion of the Mandellambda Sets (p. 35) for more on this topic. ORBITDELAY=<nn> Slows up the display of orbits using the <o> command for folks with hot new computers. Units are in 1/10000 seconds per orbit point. ORBITDELAY=10 therefore allows you to see each pixel's orbit point for about one millisecond. For best display of orbits, try passes=1 and a moderate resolution such as 320x200. Note that the first time you press the 'o' key with the 'orbitdelay' function active, your computer will pause for a half-second or so to calibrate a high-resolution timer. Fractint Version 18.2 Page 96 PERIODICITY=no|show|nnn Controls periodicity checking (see Periodicity Logic (p. 123)). "no" turns it off, "show" lets you see which pixels were painted as "inside" due to being caught by periodicity. Specifying a number causes a more conservative periodicity test (each increase of 1 divides test tolerance by 2). Entering a negative number lets you turn on "show" with that number. Type lambdafn function=exp needs periodicity turned off to be accurate -- there may be other cases. RSEED=nnnn The initial random-number "seed" for plasma clouds is taken from your PC's internal clock-timer. This argument forces a value (which you can see in the <Tab> display), and allows you to reproduce plasma clouds. A detailed discussion of why a TRULY random number may be impossible to define, let alone generate, will have to wait for "FRACTINT: The 3-MB Doc File." SHOWDOT=<nn> Colors the pixel being calculated color <nn>. Useful for very slow fractals for showing you the calculation status. 5.9 Color Parameters INSIDE=nnn|bof60|bof61|zmag|attractor|epscross|startrail|period Set the color of the interior: for example, "inside=0" makes the M-set "lake" a stylish basic black. A setting of -1 makes inside=maxiter. Three more options reveal hidden structure inside the lake. Inside=bof60 and inside=bof61, are named after the figures on pages 60 and 61 of "Beauty of Fractals". See Inside=bof60|bof61|zmag|period (p. 138) for a brilliant explanation of what these do! Inside=zmag is a method of coloring based on the magnitude of Z after the maximum iterations have been reached. The affect along the edges of the Mandelbrot is like thin-metal welded sculpture. Inside=epscross colors pixels green or yellow according to whether their orbits swing close to the Y-axis or X-axis, respectively. Inside=starcross has a coloring scheme based on clusters of points in the orbits. Best with outside=<nnn>. For more information, see Inside=epscross|startrail (p. 138). Inside=period colors pixels according to the period of their eventual orbit. Note that the "Look for finite attractor" option on the <Y> options screen will override the selected inside option if an attractor is found - see Finite Attractors (p. 139). OUTSIDE=nnn|iter|real|imag|summ|mult The classic method of coloring outside the fractal is to color according to how many iterations were required before Z reached the bailout value, usually 4. This is the method used when OUTSIDE=iter. Fractint Version 18.2 Page 97 However, when Z reaches bailout the real and imaginary components can be at very diferent values. OUTSIDE=real and OUTSIDE=imag color using the iteration value plus the real or imaginary values. OUTSIDE=summ uses the sum of all these values. These options can give a startling 3d quality to otherwise flat images and can change some boring images to wonderful ones. OUTSIDE=mult colors by multiplying the iteration by real divided by imaginary. There was no mathematical reason for this, it just seemed like a good idea. Outside=nnn sets the color of the exterior to some number of your choosing: for example, "OUTSIDE=1" makes all points not INSIDE the fractal set to color 1 (blue). Note that defining an OUTSIDE color forces any image to be a two-color one: either a point is INSIDE the set, or it's OUTSIDE it. MAP=[filename] Reads in a replacement color map from [filename]. This map replaces the default color map of your video adapter. Requires a VGA or higher adapter. The difference between this argument and an alternate map read in via <L> in color-command mode is that this one applies to the entire run. See Palette Maps (p. 65). COLORS=@filename|colorspecification Sets colors for the current image, like the <L> function in color cycling and palette editing modes. Unlike the MAP= parameter, colors set with COLORS= do not replace the default - when you next select a new video mode, colors will revert to their defaults. COLORS=@filename tells Fractint to use a color map file named "filename". See Palette Maps (p. 65). COLORS=colorspecification specifies the colors directly. The value of "colorspecification" is rather long (768 characters for 256 color modes), and its syntax is not documented here. This form of the COLORS= command is not intended for manual use - it exists for use by the command when saving the description of a nice image. CYCLERANGE=nnn/nnn Sets the range of color numbers to be animated during color cycling. The default is 1/255, i.e. just color number 0 (usually black) is not cycled. CYCLELIMIT=nnn Sets the speed of color cycling. Technically, the number of DAC registers updated during a single vertical refresh cycle. Legal values are 1 - 256, default is 55. TEXTCOLORS=mono Set text screen colors to simple black and white. TEXTCOLORS=aa/bb/cc/... Set text screen colors. Omit any value to use the default (e.g. textcolors=////50 to set just the 5th value). Each value is a 2 digit hexadecimal value; 1st digit is background color (from 0 to 7), 2nd digit is foreground color (from 0 to F). Color values are: 0 black 8 gray 1 blue 9 light blue Fractint Version 18.2 Page 98 2 green A light green 3 cyan B light cyan 4 red C light red 5 magenta D light magenta 6 brown E yellow 7 white F bright white 31 colors can be specified, their meanings are as follows: heading: 1 Fractint version info 2 heading line development info (not used in released version) help: 3 sub-heading 4 main text 5 instructions at bottom of screen 6 hotlink field 7 highlighted (current) hotlink menu, selection boxes, parameter input boxes: 8 background around box and instructions at bottom 9 emphasized text outside box 10 low intensity information in box 11 medium intensity information in box 12 high intensity information in box (e.g. heading) 13 current keyin field 14 current keyin field when it is limited to one of n values 15 current choice in multiple choice list 16 speed key prompt in multiple choice list 17 speed key keyin in multiple choice list general (tab key display, IFS parameters, "thinking" display): 18 high intensity information 19 medium intensity information 20 low intensity information 21 current keyin field disk video: 22 background around box 23 high intensity information 24 low intensity information diagnostic messages: 25 error 26 information credits screen: 27 bottom lines 28 high intensity divider line 29 low intensity divider line 30 primary authors 31 contributing authors The default is textcolors=1F/1A/2E/70/28/71/31/78/70/17/ 1F/1E/2F/3F/5F/07/0D/71/70/78/0F/ 70/0E/0F/4F/20/17/20/28/0F/07 (In a real command file, all values must be on one line.) Fractint Version 18.2 Page 99 5.10 Doodad Parameters LOGMAP=yes|old|n Selects a compressed relationship between escape-time iterations and palette colors. See "Logarithmic Palettes and Color Ranges" (p. 71) for details. RANGES=nn/nn/nn/... Specifies ranges of escape-time iteration counts to be mapped to each color number. See "Logarithmic Palettes and Color Ranges" (p. 71) for details. DISTEST=nnn/nnn A nonzero value in the first parameter enables the distance estimator method. The second parameter specifies the "width factor", defaults to 71. See "Distance Estimator Method" (p. 68) for details. DECOMP=2|4|8|16|32|64|128|256 Invokes the corresponding decomposition coloring scheme. See Decomposition (p. 70) for details. BIOMORPH=nnn Turn on biomorph option; set affected pixels to color nnn. See Biomorphs (p. 72) for details. POTENTIAL=maxcolor[/slope[/modulus[/16bit]]] Enables the "continuous potential" coloring mode for all fractal types except plasma clouds, attractor types such as lorenz, and IFS. The four arguments define the maximum color value, the slope of the potential curve, the modulus "bailout" value, and whether 16 bit values are to be calculated. Example: "POTENTIAL=240/2000/40/16bit". The Mandelbrot and Julia types ignore the modulus bailout value and use their own hardwired value of 4.0 instead. See Continuous Potential (p. 72) for details. INVERT=nn/nn/nn Turns on inversion. The parameters are radius of inversion, x-coordinate of center, and y-coordinate of center. -1 as the first parameter sets the radius to 1/6 the smaller screen dimension; no x/y parameters defaults to center of screen. The values are displayed with the <Tab> command. See Inversion (p. 69) for details. FINATTRACT=no|yes Another option to show coloring inside some Julia "lakes" to show escape time to finite attractors. Works with lambda, magnet types, and possibly others. See Finite Attractors (p. 139) for more information. EXITNOASK=yes This option forces Fractint to bypass the final "are you sure?" exit screen when the ESCAPE key is pressed from the main image-generation screen. Added at the request of Ward Christensen. It's his funeral <grin>. Fractint Version 18.2 Page 100 5.11 File Parameters SAVENAME=[name] Set the filename to use when you <S>ave a screen. The default filename is FRACT001. The .GIF extension is optional (Example: SAVENAME=myfile) OVERWRITE=no|yes Sets the savename overwrite flag (default is 'no'). If 'yes', saved files will over-write existing files from previous sessions; otherwise the automatic incrementing of FRACTnnn.GIF will find the first unused filename. SAVETIME=nnn Tells Fractint to automatically do a save every nnn minutes while a calculation is in progress. This is mainly useful with long batches - see Batch Mode (p. 110). GIF87a=YES Backward-compatibility switch to force creation of GIF files in the GIF87a format. As of version 14, Fractint defaults to the new GIF89a format which permits storage of fractal information within the format. GIF87a=YES is only needed if you wish to view Fractint images with a GIF decoder that cannot accept the newer format. See GIF Save File Format (p. 150). DITHER=YES Dither a color file into two colors for display on a b/w display. This give a poor-quality display of gray levels. Note that if you have a 2- color display, you can create a 256-color gif with disk video and then read it back in dithered. PARMFILE=[parmfilename] Specifies the default parameter file to be used by the <@> and commands. If not specified, the default is FRACTINT.PAR. FORMULAFILE=[formulafilename] Specifies the formula file for type=formula fractals (default is FRACTINT.FRM). Handy if you want to generate one of these fractal types in batch mode. LFILE=[lsystemfile] Specifies the default L-System file for type=lsystem fractals (if not FRACTINT.L). IFSFILE=[ifsfilename] Specifies the default file for type=ifs fractals (default is FRACTINT.IFS). FILENAME=[.suffix] Sets the default file extension used for the <r> command. When this parameter is omitted, the default file mask shows .GIF and .POT files. You might want to specify this parameter and the SAVENAME= parameter in your SSTOOLS.INI file if you keep your fractal images separate from other .GIF files by using a different suffix for them. Fractint Version 18.2 Page 101 ORBITSAVE=yes Causes the file ORBITS.RAW to be opened and the points generated by orbit fractals or IFS fractals to be saved in a raw format. This file can be read by the Acrospin program which can rotate and scale the image rapidly in response to cursor-key commands. The filename ORBITS.RAW is fixed and will be overwritten each time a new fractal is generated with this option. (see Barnsley IFS Fractals (p. 39) Orbit Fractals (p. 46) Acrospin (p. 153)); 5.12 Video Parameters VIDEO=xxx Set the initial video mode (and bypass the informational screens). Handy for batch runs. (Example: VIDEO=F4 for IBM 16-color VGA.) You can obtain the current VIDEO= values (key assignments) from the "select video mode" screens inside Fractint. If you want to do a batch run with a video mode which isn't currently assigned to a key, you'll have to modify the key assignments - see "Video Mode Function Keys" (p. 27). ASKVIDEO=yes|no If "no," this eliminates the prompt asking you if a file to be restored is OK for your current video hardware. WARNING: every version of Fractint so far has had a bigger, better, but shuffled-around video table. Since calling for a mode your hardware doesn't support can leave your system in limbo, be careful about leaving the above two parameters in a command file to be used with future versions of Fractint, particularly for the super-VGA modes. ADAPTER=hgc|cga|ega|egamono|mcga|vga|ATI|Everex|Trident|NCR|Video7|Genoa| Paradise|Chipstech|Tseng3000|Tseng4000|AheadA|AheadB|Oaktech Bypasses Fractint's internal video autodetect logic and assumes that the specified kind of adapter is present. Use this parameter only if you encounter video problems without it. Specifying adapter=vga with an SVGA adapter will make its extended modes unusable with Fractint. All of the options after the "VGA" option specify specific SuperVGA chipsets which are capable of video resolutions higher than that of a "vanilla" VGA adapter. Note that Fractint cares about the Chipset your adapter uses internally, not the name of the company that sold it to you. VESADETECT=yes|no Specify no to bypass VESA video detection logic. Try this if you encounter video problems with a VESA compliant video adapter or driver. AFI=yes|8514|no Normally, when you attempt to use an 8514/A-specific video mode, Fractint first attempts to detect the presence of an 8514/A register- compatible adapter. If it fails to find one, it then attempts to detect the presence of an 8514/A-compatible API (IE, IBM's HDILOAD or its equivalent). Fractint then uses either its register-compatible or its API-compatible video logic based on the results of those tests. If you have an "8514/A-compatible" video adapter that passes Fractint's register-compatible detection logic but doesn't work correctly with Fractint's register-compatible video logic, setting "afi=yes" will force Fractint Version 18.2 Page 102 Fractint to bypass the register-compatible code and look only for the API interface. TEXTSAFE=yes|no|bios|save When you switch from a graphics image to text mode (e.g. when you use <F1> while a fractal is on display), Fractint remembers the graphics image, and restores it when you return from the text mode. This should be no big deal - there are a number of well-defined ways Fractint could do this which *should* work on any video adapter. They don't - every fast approach we've tried runs into a bug on one video adapter or another. So, we've implemented a fast way which works on most adapters in most modes as the default, and added this parameter for use when the default approach doesn't work. If you experience the following problems, please fool around with this parameter to try to fix the problem: o Garbled image, or lines or dashes on image, when returning to image after going to menu, <tab> display, or help. o Blank screen when starting Fractint. The problems most often occur in higher resolution modes. We have not encountered them at all in modes under 320x200x256 - for those modes Fractint always uses a fast image save/restore approach. Textsafe options: yes: This is the default. When switching to/from graphics, Fractint saves just that part of video memory which EGA/VGA adapters are supposed to modify during the mode changes. no: This forces use of monochrome 640x200x2 mode for text displays (when there is a high resolution graphics image to be saved.) This choice is fast but uses chunky and colorless characters. If it turns out to be the best choice for you, you might want to also specify "textcolors=mono" for a more consistent appearance in text screens. bios: This saves memory in the same way as textsafe=yes, but uses the adapter's BIOS routines to save/restore the graphics state. This approach is fast and ought to work on all adapters. Sadly, we've found that very few adapters implement this function perfectly. save: This is the last choice to try. It should work on all adapters in all modes but it is slow. It tells Fractint to save/restore the entire image. Expanded or extended memory is used for the save if you have enough available; otherwise a temporary disk file is used. The speed of textsafe=save will be acceptable on some machines but not others. The speed depends on: o Cpu and video adapter speed. o Whether enough expanded or extended memory is available. o Video mode of image being remembered. A few special modes are *very* slow compared to the rest. The slow ones are: 2 and 4 color modes with resolution higher than 640x480; custom modes for ATI EGA Wonder, Paradise EGA-480, STB, Compaq portable 386, AT&T 6300, and roll-your-own video modes implemented with customized "yourvid.c" code. If you want to tune Fractint to use different "textsafe" options for different video modes, see "Customized Video Modes, FRACTINT.CFG" (p. 115). (E.g. you might want to use the slower textsafe=save approach just for a few high-resolution modes which have problems with textsafe=yes.) Fractint Version 18.2 Page 103 EXITMODE=nn Sets the bios-supported videomode to use upon exit to the specified value. nn is in hexadecimal. The default is 3, which resets to 80x25 color text mode on exit. With Hercules Graphics Cards, and with monochrome EGA systems, the exit mode is always 7 and is unaffected by this parameter. TPLUS=yes|no For TARGA+ adapters. Setting this to 'no' pretends a TARGA+ is NOT installed. NONINTERLACED=yes|no For TARGA+ adapters. Setting this to 'yes' will configure the adapter to a non-interlaced mode whenever possible. It should only be used with a multisynch monitor. The default is no, i.e. interlaced. MAXCOLORRES=8|16|24 For TARGA+ adapters. This determines the number of bits to use for color resolution. 8 bit color is equivalent to VGA color resolution. The 16 and 24 bit color resolutions are true color video modes which are not yet supported by Fractint but are hopefully coming soon. PIXELZOOM=0|1|2|3 For TARGA+ adapters. Lowers the video mode resolution by powers of 2. For example, the 320x200 video resolution on the TARGA+ is actually the 640x400 video mode with a pixel zoom of 1. Using the 640x400 video mode with a zoom of 3 would lower the resolution by 8, which is 2 raised to the 3rd power, for a full screen resolution of 80x50 pixels. VIEWWINDOWS=xx[/xx[/yes|no[/nn[/nn]]]] Set the reduction factor, final media aspect ratio, crop starting coordinates (y/n), explicit x size, and explicit y size, see "View Window" (p. 26). 5.13 Sound Parameters SOUND=off|x|y|z We're all MUCH too busy to waste time with Fractint at work, and no doubt you are too, so "sound=off" is included only for use at home, to avoid waking the kids or your Significant Other, late at night. (By the way, didn't you tell yourself "just one more zoom on LambdaSine" an hour ago?) Suggestions for a "boss" hot-key will be cheerfully ignored, as this sucker is getting big enough without including a spreadsheet screen too. The "sound=x/y/x" options are for the "attractor" fractals, like the Lorenz fractals - they play with the sound on your PC speaker as they are generating an image, based on the X or Y or Z co-ordinate they are displaying at the moment. At the moment, "sound=x" (or y or z) really doesn't work very well when using an integer algorithm - try it with the floating-point toggle set, instead. The scope of the sound command has been extended. You can now hear the sound of fractal orbits--just turn on sound from the command line or the <X> menu, fire up a fractal, and try the <O>rbits command. Use the orbitdelay=<nnn> command (also on the <X> menu) to dramatically alter the effect, which ranges from an unearthly scream to a series of discrete tones. Not recommended when people you have to live with are Fractint Version 18.2 Page 104 nearby! Remember, we don't promise that it will sound beautiful! You can also "hear" any image that Fractint can decode; turn on sound before using <R> to read in a GIF file. We have no idea if this feature is useful. It was inspired by the comments of an on-line friend who is blind. We solicit feedback and suggestions from anyone who finds these sound features interesting or useful. The orbitdelay command also affects the sound of decoding images. HERTZ=nnn Adjusts the sound produced by the "sound=x/y/z" option. Legal values are 200 through 10000. 5.14 Printer Parameters PRINTER=type[/resolution[/port#]] Defines your printer setup. The SSTOOLS.INI file is a REAL handy place to put this option, so that it's available whenever you have that sudden, irresistible urge for hard copy. Printer types: IB IBM-compatible (default) EP Epson-compatible HP LaserJet CO Star Micronics Color printer, supposedly Epson-color-compatible PA Paintjet PS PostScript PSL Postscript, landscape mode PL Plotter using HP-GL Resolution: In dots per inch. Epson/IBM: 60, 120, 240 LaserJet: 75, 150, 300 PaintJet: 90, 180 PostScript: 10 through 600, or special value 0 to print full page to within about .4" of the edges (in portrait mode, width is full page and height is adjusted to 3:4 aspect ratio) Plotter: 1 to 10 for 1/Nth of page (e.g. 2 for 1/2 page) Port: 1, 2, 3 for LPT1-3 via BIOS 11, 12, 13, 14 for COM1-4 via BIOS 21, 22 for LPT1 or LPT2 using direct port access (faster when it works) 31, 32 for COM1 or COM2 using direct port access COMPORT=port/baud/options Serial printer port initialization. Port=1,2,3,etc. Baud=115,150,300,600,1200,2400,4800,9600 Options: 7,8 | 1,2 | e,n,o (any order). Example: comport=1/9600/n81 for COM1 set to 9600, no parity, 8 bits per character, 1 stop bit. LINEFEED=crlf|lf|cr Specifies the control characters to emit at end of each line: carriage return and linefeed, just linefeed, or just carriage return. The default is crlf. Fractint Version 18.2 Page 105 TITLE=yes If specified, title information is added to printouts. PRINTFILE=filename Causes output data for the printer to be written to the named file on disk instead of to a printer port. The filename is incremented by 1 each time an image is printed - e.g. if the name is FRAC01.PRN, the second print operation writes to FRAC02.PRN, etc. Existing files are not overwritten - if the file exists, the filename is incremented to a new name. 5.15 PostScript Parameters EPSF=1|2|3 Forces print-to-file and PostScript. If PRINTFILE is not specified, the default filename is FRACT001.EPS. The number determines how 'well- behaved' a .EPS file is. 1 means by-the-book. 2 allows some EPS 'no-nos' like settransfer and setscreen - BUT includes code that should make the code still work without affecting the rest of the non-EPS document. 3 is a free-for-all. COLORPS=YES|NO - Enable or disable the color extensions. RLEPS=YES|NO Enable or disable run length encoding of the PostScript file. Run length encoding will make the PostScript file much smaller, but it may take longer to print. The run length encoding code is based on pnmtops, which is copyright (C) 1989 by Jef Poskanzer, and carries the following notice: "Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. This software is provided "as is" without express or implied warranty." TRANSLATE=yes|-n|n Translate=yes prints the negative image of the fractal. Translate=n reduces the image to that many colors. A negative value causes a color reduction as well as a negative image. HALFTONE=frq/ang/sty[/f/a/s/f/a/s/f/a/s] Tells the PostScript printer how to define its halftone screen. The first value, frequency, defines the number of halftone lines per inch. The second chooses the angle (in degrees) that the screen lies at. The third option chooses the halftone 'spot' style. Good default frequencies are between 60 and 80; Good default angles are 45 and 0; the default style is 0. If the halftone= option is not specified, Fractint will print using the printer's default halftone screen, which should have been already set to do a fine job on the printer. These are the only three used when colorps=no. When color PS printing is being used, the other nine options specify the red, green, then blue screens. A negative number in any of these places will cause it to use the previous (or default) value for that parameter. NOTE: Especially when using color, the built-in screens in the printer's ROM may be the Fractint Version 18.2 Page 106 best choice for printing. The default values are as follows: halftone=45/45/1/45/75/1/45/15/1/45/0/1 and these will be used if Fractint's halftone is chosen over the printer's built-in screen. Current halftone styles: 0 Dot 1 Dot (Smoother) 2 Dot (Inverted) 3 Ring (Black) 4 Ring (White) 5 Triangle (Right) 6 Triangle (Isosceles) 7 Grid 8 Diamond 9 Line 10 Microwaves 11 Ellipse 12 Rounded Box 13 Custom 14 Star 15 Random 16 Line (slightly different) A note on device-resolution black and white printing ---------------------------------------------------- This mode of printing can now be done much more quickly, and takes a lot less file space. Just set EPSF=0 PRINTER=PSx/nnn COLORPS=NO RLEPS=YES TRANSLATE=m, where x is P or L for portrait/landscape, nnn is your printer's resolution, m is 2 or -2 for positive or negative printing respectively. This combination of parameters will print exactly one printer pixel per each image pixel and it will keep the proportions of the picture, if both your screen and printer have square pixels (or the same pixel-aspect). Choose a proper (read large) window size to fill as much of the paper as possible for the most spectacular results. 2048 by 2048 is barely enough to fill the width of a letter size page with 300 dpi printer resolution. For higher resolution printers, you will wish fractint supported larger window sizes (hint, hint...). Bug reports and/or suggestions should be forwarded to Yavuz Onder (post to sci.fractals, no e-mail yet). A word from the author (Scott Taylor) ------------------------------------- Color PostScript printing is new to me. I don't even have a color printer to test it on. (Don't want money. Want a Color PostScript printer!) The initial tests seem to have worked. I am still testing and don't know whether or not some sort of gamma correction will be needed. I'll have to wait and see about that one. Fractint Version 18.2 Page 107 5.16 PaintJet Parameters Note that the pixels printed by the PaintJet are square. Thus, a printout of an image created in a video mode with a 4:3 pixel ratio (such as 640x480 or 800x600) will come out matching the screen; other modes (such as 320x200) will come out stretched. Black and white images, or images using the 8 high resolution PaintJet colors, come out very nicely. Some images using the full spectrum of PaintJet colors are very nice, some are disappointing. When 180 dots per inch is selected (in PRINTER= command), high resolution 8 color printing is done. When 90 dpi is selected, low resolution printing using the full 330 dithered color palette is done. In both cases, Fractint starts by finding the nearest color supported by the PaintJet for each color in your image. The translation is then displayed (unless the current display mode is disk video). This display *should* be a fairly good match to what will be printed - it won't be perfect most of the time but should give some idea of how the output will look. At this point you can <Enter> to go ahead and print, <Esc> to cancel, or <k> to cancel and keep the adjusted colors. Note that you can use the color map PAINTJET.MAP to create images which use the 8 high resolution colors available on the PaintJet. Also, two high-resolution disk video modes are available for creating full page images. If you find that the preview image seems very wrong (doesn't match what actually gets printed) or think that Fractint could be doing a better job of picking PaintJet colors to match your image's colors, you can try playing with the following parameter. Fair warning: this is a very tricky business and you may find it a very frustrating business trying to get it right. HALFTONE=r/g/b (The parameter name is not appropriate - we appropriated a PostScript parameter for double duty here.) This separately sets the "gamma" adjustment for each of the red, green, and blue color components. Think of "gamma" as being like the contrast adjustment on your screen. Higher gamma values for all three components results in colors with more contrast being produced on the printer. Since each color component can have its gamma separately adjusted, you can change the resulting color mix subtly (or drastically!) Each gamma value entered has one implied decimal digit. The default is "halftone=21/19/16", for red 2.1, green 1.9, and blue 1.6. (A note from Pieter Branderhorst: I wrote this stuff to come out reasonably on my monitor/printer. I'm a bit suspicious of the guns on my monitor; if the colors seem ridiculously wrong on your system you might start by trying halftone=17/17/17.) Fractint Version 18.2 Page 108 5.17 Plotter Parameters Plotters which understand HP-GL commands are supported. To use a plotter, draw a SMALL image (32x20 or 64x40) using the <v>iew screen options. Put a red pen in the first holder in the plotter, green in the second, blue in the third. Now press to start plotting. Now get a cup of coffee... or two... or three. It'll take a while to plot. Experiment with different resolutions, plot areas, plotstyles, and even change pens to create weird-colored images. PLOTSTYLE=0|1|2 0: 3 parallel lines (red/green/blue) are drawn for each pixel, arranged like "///". Each bar is scaled according to the intensity of the corresponding color in the pixel. Using different pen colors (e.g. blue, green, violet) can come out nicely. The trick is to not tell anyone what color the bars are supposed to represent and they will accept these plotted colors because they do look nice... 1: Same as 0, but the lines are also twisted. This removes some of the 'order' of the image which is a nice effect. It also leaves more whitespace making the image much lighter, but colors such as yellow are actually visible. 2: Color lines are at the same angle and overlap each other. This type has the most whitespace. Quality improves as you increase the number of pixels squeezed into the same size on the plotter. 5.18 3D Parameters To stay out of trouble, specify all the 3D parameters, even if you want to use what you think are the default values. It takes a little practice to learn what the default values really are. The best way to create a set of parameters is to use the command on an image you like and then use an editor to modify the resulting parameter file. 3D=Yes 3D=Overlay Resets all 3d parameters to default values. If FILENAME= is given, forces a restore to be performed in 3D mode (handy when used with 'batch=yes' for batch-mode 3D images). If specified, 3D=Yes should come before any other 3d parameters on the command line or in a parameter file entry. The form 3D=Overlay is identical except that the previous graphics screen is not cleared, as with the <#> (<shift-3> on some keyboards) overlay command. Useful for building parameter files that use the 3D overlay feature. The options below override the 3D defaults: PREVIEW=yes Turns on 3D 'preview' default mode SHOWBOX=yes Turns on 3D 'showbox' default mode COARSE=nn Sets Preview 'coarseness' default value SPHERE=yes Turns on spherical projection mode STEREO=n Selects the type of stereo image creation RAY=nnn selects raytrace output file format BRIEF=yes selects brief or verbose file for DKB output Fractint Version 18.2 Page 109 INTEROCULAR=nn Sets the interocular distance for stereo CONVERGE=nn Determines the overall image separation CROP=nn/nn/nn/nn Trims the edges off stereo pairs BRIGHT=nn/nn Compensates funny glasses filter parameters LONGITUDE=nn/nn Longitude minimum and maximum LATITUDE=nn/nn Latitude minimum and maximum RADIUS=nn Radius scale factor ROTATION=nn[/nn[/nn]] Rotation about x,y, and z axes SCALEZYZ=nn/nn/nn X,y,and z scale factors ROUGHNESS=nn Same as z scale factor WATERLINE=nn Colors nn and below will be "inside" color FILLTYPE=nn 3D filltype PERSPECTIVE=nn Perspective distance XYSHIFT=nn/nn Shift image in x and y directions with perspective LIGHTSOURCE=nn/nn/nn Coordinates for light-source vector SMOOTHING=nn Smooths images in light-source fill modes TRANSPARENT=min/max Defines a range of colors to be treated as "transparent" when <#>Overlaying 3D images. XYADJUST=nn/nn This shifts the image in the x/y dir without perspective Below are new commands as of version 14 that support Marc Reinig's terrain features. RANDOMIZE=nnn (0 - 100) This feature randomly varies the color of a pixel to near by colors. Useful to minimize map banding in 3d transformations. Usable with all FILLTYPES. 0 disables, max values is 7. Try 3 - 5. AMBIENT=nnn (0 - 100) Set the depth of the shadows when using full color and light source filltypes. "0" disables the function, higher values lower the contrast. FULLCOLOR=yes Valid with any light source FILLTYPE. Allows you to create a Targa-24 file which uses the color of the image being transformed or the map you select and shades it as you would see it in real life. Well, its better than B&W. A good map file to use is topo HAZE=nnn (0 - 100) Gives more realistic terrains by setting the amount of haze for distant objects when using full color in light source FILLTYPES. Works only in the "y" direction currently, so don't use it with much y rotation. Try "rotation=85/0/0". 0 disables. LIGHTNAME=<filename> The name of the Targa-24 file to be created when using full color with light source. Default is light001.tga. If overwrite=no (the default), the file name will be incremented until an unused filename is found. Background in this file will be sky blue. Fractint Version 18.2 Page 110 5.19 Batch Mode It IS possible, believe it or not, to become so jaded with the screen drawing process, so familiar with the types and options, that you just want to hit a key and do something else until the final images are safe on disk. To do this, start Fractint with the BATCH=yes parameter. To set up a batch run with the parameters required for a particular image you might: o Find an interesting area. Note the parameters from the <Tab> display. Then use an editor to write a batch file. o Find an interesting area. Set all the options you'll want in the batch run. Use the command to store the parameters in a file. Then use an editor to add the additional required batch mode parameters (such as VIDEO=) to the generated parameter file entry. Then run the batch using "fractint @myname.par/myentry" (if you told the command to use file "myname" and to name the entry "myentry"). Another approach to batch mode calculations, using "FILENAME=" and resume, is described later. When modifying a parameter file entry generated by the command, the only parameters you must add for a batch mode run are "BATCH=yes", and "VIDEO=xxx" to select a video mode. You might want to also add "SAVENAME=[name]" to name the result as something other than the default FRACT001.GIF. Or, you might find it easier to leave the generated parameter file unchanged and add these parameters by using a command like: fractint @myname.par/myentry batch=y video=AF3 savename=mygif "BATCH=yes" tells Fractint to run in batch mode -- that is, Fractint draws the image using whatever other parameters you specified, then acts as if you had hit <S> to save the image, then exits to DOS. "FILENAME=" can be used with "BATCH=yes" to resume calculation of an incomplete image. For instance, you might interactively find an image you like; then select some slow options (a high resolution disk video mode, distance estimator method, high maxiter, or whatever); start the calculation; then interrupt immediately with a <S>ave. Rename the save file (fract001.gif if it is the first in the session and you didn't name it with the <X> options or "savename=") to xxx.gif. Later you can run Fractint in batch mode to finish the job: fractint batch=yes filename=xxx savename=xxx "SAVETIME=nnn" is useful with long batch calculations, to store a checkpoint every nnn minutes. If you start a many hour calculation with say "savetime=60", and a power failure occurs during the calculation, you'll have lost at most an hour of work on the image. You can resume calculation from the save file as above. Automatic saves triggered by SAVETIME do not increment the save file name. The same file is overwritten by each auto save until the image completes. But note that Fractint does not directly over-write save files. Instead, each save operation writes a temporary file FRACTINT.TMP, then deletes the prior save file, then renames FRACTINT.TMP to be the new save file. This protects against power failures which occur during a save operation - if such a power failure occurs, the prior save file is intact and there's a Fractint Version 18.2 Page 111 harmless incomplete FRACTINT.TMP on your disk. If you want to spread a many-hour image over multiple bits of free machine time you could use a command like: fractint batch=yes filename=xxx savename=xxx savetime=60 video=F3 While this batch is running, hit <S> (almost any key actually) to tell fractint to save what it has done so far and give your machine back. A status code of 2 is returned by fractint to the batch file. Kick off the batch again when you have another time slice for it. While running a batch file, pressing any key will cause Fractint to exit with an errorlevel = 2. Any error that interrupts an image save to disk will cause an exit with errorlevel = 2. Any error that prevents an image from being generated will cause an exit with errorlevel = 1. The SAVETIME= parameter, and batch resumes of partial calculations, only work with fractal types which can be resumed. See "Interrupting and Resuming" (p. 25) for information about non-resumable types. Fractint Version 18.2 Page 112 6. Hardware Support 6.1 Notes on Video Modes, "Standard" and Otherwise True to the spirit of public-domain programming, Fractint makes only a limited attempt to verify that your video adapter can run in the mode you specify, or even that an adapter is present, before writing to it. So if you use the "video=" command line parameter, check it before using a new version of Fractint - the old key combo may now call an ultraviolet holographic mode. EGA Fractint assumes that every EGA adapter has a full 256K of memory (and can therefore display 640 x 350 x 16 colors), but does nothing to verify that fact before slinging pixels. "TWEAKED" VGA MODES The IBM VGA adapter is a highly programmable device, and can be set up to display many video-mode combinations beyond those "officially" supported by the IBM BIOS. E.g. 320x400x256 and 360x480x256 (the latter is one of our favorites). These video modes are perfectly legal, but temporarily reprogram the adapter (IBM or fully register-compatible) in a non-standard manner that the BIOS does not recognize. Fractint also contains code that sets up the IBM (or any truly register- compatible) VGA adapter for several extended modes such as 704x528, 736x552, 768x576, and 800x600. It does this by programming the VGA controller to use the fastest dot-clock on the IBM adapter (28.322 MHz), throwing more pixels, and reducing the refresh rate to make up for it. These modes push many monitors beyond their rated specs, in terms of both resolution and refresh rate. Signs that your monitor is having problems with a particular "tweaked" mode include: o vertical or horizontal overscan (displaying dots beyond the edges of your visible CRT area) o flickering (caused by a too-slow refresh rate) o vertical roll or total garbage on the screen (your monitor simply can't keep up, or is attempting to "force" the image into a pre-set mode that doesn't fit). We have successfully tested the modes up to 768x576 on an IBM PS/2 Model 80 connected to IBM 8513, IBM 8514, NEC Multisync II, and Zenith 1490 monitors (all of which exhibit some overscan and flicker at the highest rates), and have tested 800x600 mode on the NEC Multisync II (although it took some twiddling of the vertical-size control). SUPER-EGA AND SUPER-VGA MODES Since version 12.0, we've used both John Bridges' SuperVGA Autodetecting logic *and* VESA adapter detection, so that many brand-specific SuperVGA modes have been combined into single video mode selection entries. There is now exactly one entry for SuperVGA 640x480x256 mode, for instance. Fractint Version 18.2 Page 113 If Fractint's automatic SuperVGA/VESA detection logic guesses wrong, and you know which SuperVGA chipset your video adapter uses, you can use the "adapter=" command-line option to force Fractint to assume the presence of a specific SuperVGA Chipset - see Video Parameters (p. 101) for details. 8514/A MODES The IBM 8514/A modes (640x480 and 1024x768) default to using the hardware registers. If an error occurs when trying to open the adapter, an attempt will be made to use IBM's software interface, and requires the preloading of IBM's HDILOAD TSR utility. The Adex 1280x1024 modes were written for and tested on an Adex Corporation 8514/A using a Brooktree DAC. The ATI GU 800x600x256 and 1280x1024x16 modes require a ROM bios version of 1.3 or higher for 800x600 and 1.4 or higher for 1280x1024. There are two sets of 8514/A modes: full sets (640x480, 800x600, 1024x768, 1280x1024) which cover the entire screen and do NOT have a border color (so that you cannot tell when you are "paused" in a color- cycling mode), and partial sets (632x474, 792x594, 1016x762, 1272x1018) with small border areas which do turn white when you are paused in color-cycling mode. Also, while these modes are declared to be 256- color, if you do not have your 8514/A adapter loaded with its full complement of memory you will actually be in 16-color mode. The hardware register 16-color modes have not been tested. If your 8514/A adapter is not truly register compatible and Fractint does not detect this, use of the adapter interface can be forced by using afi=y or afi=8514 in your SSTOOLS.INI file. Finally, because IBM's adapter interface does not handle drawing single pixels very well (we have to draw a 1x1 pixel "box"), generating the zoom box when using the interface is excruciatingly slow. Still, it works! XGA MODES The XGA adapter is supported using the VESA/SuperVGA Autodetect modes - the XGA looks like just another SuperVGA adapter to Fractint. The supported XGA modes are 640x480x256, 1024x768x16, 1024x768x256, 800x600x16, and 800x600x256. Note that the 1024x768x256 mode requires a full 1MB of adapter memory, the 1024x768 modes require a high-rez monitor, and the 800x600 modes require a multisynching monitor such as the NEC 2A. TARGA MODES TARGA support for Fractint is provided courtesy of Joe McLain and has been enhanced with the help of Bruce Goren and Richard Biddle. To use a TARGA board with Fractint, you must define two DOS environment variables, "TARGA" and "TARGASET". The definition of these variables is standardized by Truevision; if you have a TARGA board you probably already have added "SET" statements for these variables to your AUTOEXEC.BAT file. Be aware that there are a LOT of possible TARGA Fractint Version 18.2 Page 114 configurations, and a LOT of opportunities for a TARGA board and a VGA or EGA board to interfere with each other, and we may not have all of them smoothed away yet. Also, the TARGA boards have an entirely different color-map scheme than the VGA cards, and at the moment they cannot be run through the color-cycling menu. The "MAP=" argument (see Color Parameters (p. 96)), however, works with both TARGA and VGA boards and enables you to redefine the default color maps with either board. TARGA+ MODES To use the special modes supported for TARGA+ adapters, the TARGAP.SYS device driver has to be loaded, and the TPLUS.DAT file (included with Fractint) must be in the same directory as Fractint. The video modes with names containing "True Color Autodetect" can be used with the Targa+. You might want to use the command line parameters "tplus=", "noninterlaced=", "maxcolorres=", and "pixelzoom=" (see Video Parameters (p. 101)) in your SSTOOLS.INI file to modify Fractint's use of the adapter. 6.2 "Disk-Video" Modes These "video modes" do not involve a video adapter at all. They use (in order or preference) your expanded memory, your extended memory, or your disk drive (as file FRACTINT.$$$) to store the fractal image. These modes are useful for creating images beyond the capacity of your video adapter right up to the current internal limit of 2048 x 2048 x 256, e.g. for subsequent printing. They're also useful for background processing under multi-tasking DOS managers - create an image in a disk- video mode, save it, then restore it in a real video mode. While you are using a disk-video mode, your screen will display text information indicating whether memory or your disk drive is being used, and what portion of the "screen" is being read from or written to. A "Cache size" figure is also displayed. 64K is the maximum cache size. If you see a number less than this, it means that you don't have a lot of memory free, and that performance will be less than optimum. With a very low cache size such as 4 or 6k, performance gets considerably worse in cases using solid guessing, boundary tracing, plasma, or anything else which paints the screen non-linearly. If you have this problem, all we can suggest is having fewer TSR utilities loaded before starting Fractint, or changing in your config.sys file, such as reducing a very high BUFFERS value. The zoom box is disabled during disk-video modes (you couldn't see where it is anyway). So is the orbit display feature. Color Cycling (p. 16) can be used during disk-video modes, but only to load or save a color palette. When using real disk for your disk-video, Fractint will not generate some "attractor" types (e.g. lorenz) nor "IFS" images. These would kill your disk drive. Boundary tracing is allowed - it may give your drive a bit of a workout, but is generally tolerable. Fractint Version 18.2 Page 115 When using a real disk, and you are not directing the file to a RAM disk, and you aren't using a disk caching program on your machine, specifying BUFFERS=10 (or more) in your config.sys file is best for performance. BUFFERS=10,2 or even BUFFERS=10,4 is also good. It is also best to keep your disk relatively "compressed" (or "defragmented") if you have a utility to do this. In order to use extended memory, you must have HIMEM.SYS or an equivalent that supports the XMS 2.0 standard or higher. Also, you can't have a VDISK installed in extended memory. Himem.sys is distributed with Microsoft Windows 286/386 and 3.0. If you have problems using the extended memory, try rebooting with just himem.sys loaded and see if that clears up the problem. If you are running background disk-video fractals under Windows 3, and you don't have a lot of real memory (over 2Mb), you might find it best to force Fractint to use real disk for disk-video modes. (Force this by using a .pif file with extended memory and expanded memory set to zero.) Try this if your disk goes crazy when generating background images, which are supposedly using extended or expanded memory. This problem can occur because, to multi-task, sometimes Windows must page an application's expanded or extended memory to disk, in big chunks. Fractint's own cached disk access may be faster in such cases. 6.3 Customized Video Modes, FRACTINT.CFG If you have a favorite adapter/video mode that you would like to add to Fractint... if you want some new sizes of disk-video modes... if you want to remove table entries that do not apply to your system... if you want to specify different "textsafe=" options for different video modes... relief is here, and without even learning "C"! You can do these things by modifying the FRACTINT.CFG file with your text editor. Saving a backup copy of FRACTINT.CFG first is of course highly recommended! Fractint uses a video adapter table for most of what it needs to know about any particular adapter/mode combination. The table is loaded from FRACTINT.CFG each time Fractint is run. It can contain information for up to 300 adapter/mode combinations. The table entries, and the function keys they are tied to, are displayed in the "select video mode" screen. This table makes adding support for various third-party video cards and their modes much easier, at least for the ones that pretend to be standard with extra dots and/or colors. There is even a special "roll- your-own" video mode (mode 19) enabling those of you with "C" compilers and a copy of the Fractint source to generate video modes supporting whatever adapter you may have. The table as currently distributed begins with nine standard and several non-standard IBM video modes that have been exercised successfully with a PS/2 model 80. These entries, coupled with the descriptive comments in the table definition and the information supplied (or that should have been supplied!) with your video adapter, should be all you need to add your own entries. Fractint Version 18.2 Page 116 After the IBM and quasi-pseudo-demi-IBM modes, the table contains an ever-increasing number of entries for other adapters. Almost all of these entries have been added because someone like you sent us spec sheets, or modified Fractint to support them and then informed us about it. Lines in FRACTINT.CFG which begin with a semi-colon are treated as comments. The rest of the lines must have eleven fields separated by commas. The fields are defined as: 1. Key assignment. F2 to F10, SF1 to SF10, CF1 to CF10, or AF1 to AF10. Blank if no key is assigned to the mode. 2. The name of the adapter/video mode (25 chars max, no leading blanks). The adapter is set up for that mode via INT 10H, with: 3. AX = this, 4. BX = this, 5. CX = this, and 6. DX = this (hey, having all these registers wasn't OUR idea!) 7. An encoded value describing how to write to your video memory in that mode. Currently available codes are: 1) Use the BIOS (INT 10H, AH=12/13, AL=color) (last resort - SLOW!) 2) Pretend it's a (perhaps super-res) EGA/VGA 3) Pretend it's an MCGA 4) SuperVGA 256-Color mode using the Tseng Labs chipset 5) SuperVGA 256-Color mode using the Paradise chipset 6) SuperVGA 256-Color mode using the Video-7 chipset 7) Non-Standard IBM VGA 360 x 480 x 256-Color mode 8) SuperVGA 1024x768x16 mode for the Everex chipset 9) TARGA video modes 10) HERCULES video mode 11) Non-Video, i.e. "disk-video" 12) 8514/A video modes 13) CGA 320x200x4-color and 640x200x2-color modes 14) Reserved for Tandy 1000 video modes 15) SuperVGA 256-Color mode using the Trident chipset 16) SuperVGA 256-Color mode using the Chips & Tech chipset 17) SuperVGA 256-Color mode using the ATI VGA Wonder chipset 18) SuperVGA 256-Color mode using the EVEREX chipset 19) Roll-your-own video mode (as you've defined it in YOURVID.C) 20) SuperVGA 1024x768x16 mode for the ATI VGA Wonder chipset 21) SuperVGA 1024x768x16 mode for the Tseng Labs chipset 22) SuperVGA 1024x768x16 mode for the Trident chipset 23) SuperVGA 1024x768x16 mode for the Video 7 chipset 24) SuperVGA 1024x768x16 mode for the Paradise chipset 25) SuperVGA 1024x768x16 mode for the Chips & Tech chipset 26) SuperVGA 1024x768x16 mode for the Everex Chipset 27) SuperVGA Auto-Detect mode (we poke around looking for your adapter) 28) VESA modes 29) True Color Auto-Detect (currently only Targa+ supported) Add 100, 200, 300, or 400 to this code to specify an over-ride "textsafe=" option to be used with the mode. 100=yes, 200=no, 300=bios, 400=save. E.g. 428 for a VESA mode with textsafe=save forced. 8. The number of pixels across the screen (X - 160 to 2048) 9. The number of pixels down the screen (Y - 160 to 2048) 10. The number of available colors (2, 4, 16, or 256) 11. A comment describing the mode (25 chars max, leading blanks are OK) Fractint Version 18.2 Page 117 NOTE that the AX, BX, CX, and DX fields use hexadecimal notation (fifteen ==> 'f', sixteen ==> '10'), because that's the way most adapter documentation describes it. The other fields use standard decimal notation. If you look closely at the default entries, you will notice that the IBM VGA entries labeled "tweaked" and "non standard" have entries in the table with AX = BX = CX = 0, and DX = some other number. Those are special flags that we used to tell the program to custom-program the VGA adapter, and are NOT undocumented BIOS calls. Maybe they should be, but they aren't. If you have a fancy adapter and a new video mode that works on it, and it is not currently supported, PLEASE GET THAT INFORMATION TO US! We will add the video mode to the list on our next release, and give you credit for it. Which brings up another point: If you can confirm that a particular video adapter/mode works (or that it doesn't), and the program says it is UNTESTED, please get that information to us also. Thanks in advance! Fractint Version 18.2 Page 118 7. Common Problems Of course, Fractint would never stoop to having a "common" problem. These notes describe some, ahem, "special situations" which come up occasionally and which even we haven't the gall to label as "features". Hang during startup: There might be a problem with Fractint's video detection logic and your particular video adapter. Try running with "fractint adapter=xxx" where xxx is cga, ega, egamono, mcga, or vga. If "adapter=vga" works, and you really have a SuperVGA adapter capable of higher video modes, there are other "adapter=" options for a number of SuperVGA chipsets - please see the full selection in Video Parameters (p. 101) for details. If this solves the problem, create an SSTOOLS.INI file with the "adapter=xxx" command in it so that the fix will apply to every run. Another possible cause: If you install the latest Fractint in say directory "newfrac", then run it from another directory with the command "\newfrac\fractint", *and* you have an older version of fractint.exe somewhere in your DOS PATH, a silent hang is all you'll get. See the notes under the "Cannot find FRACTINT.EXE message" problem for the reason. Another possibility: try one of the "textsafe" parameter choices described in Video Parameters (p. 101). Scrambled image when returning from a text mode display: If an image which has been partly or completely generated gets partly destroyed when you return to it from the menu, help, or the information display, please try the various "textsafe" parameter options - see Video Parameters (p. 101) for details. If this cures the problem, create an SSTOOLS.INI file with the "textsafe=xxx" command so that the fix will apply to every run. "Holes" in an image while it is being drawn: Little squares colored in your "inside" color, in a pattern of every second square of that size, in solid guessing mode, both across and down (i.e., 1 out of 4), are a symptom of an image which should be calculated with more conservative periodicity checking than the default. See the Periodicity parameter under Image Calculation Parameters (p. 94). Black bar at top of screen during color cycling on 8086/8088 machines: (This might happen intermittently, not every run.) "fractint cyclelimit=10" might cure the problem. If so, increase the cyclelimit value (try increasing by 5 or 10 each time) until the problem reappears, then back off one step and add that cyclelimit value to your SSTOOLS.INI file. Other video problems: If you are using a VESA driver with your video adapter, the first thing to try is the "vesadetect=no" parameter. If that fixes the problem, add it to your SSTOOLS.INI file to make the fix permanent. Fractint Version 18.2 Page 119 It may help to explicitly specify your type of adapter - see the "adapter=" parameter in Video Parameters (p. 101). We've had one case where a video driver for Windows does not work properly with Fractint. If running under Windows, DesqView, or some other layered environment, try running Fractint directly from DOS to see if that avoids the problem. We've also had one case of a problem co-existing with "386 to the Max". We've had one report of an EGA adapter which got scrambled images in all modes until "textsafe=no" was used (see Video Parameters (p. 101) ). Also, see Video Adapter Notes (p. 112) for information about enhanced video modes - Fractint makes only limited attempts to verify that a video mode you request is actually supported by your adapter. Other Hangs and Strange Behavior: We've had some problems (hangs and solid beeps) on an FPU equipped machine when running under Windows 3's enhanced mode. The only ways around the problem we can find are to either run the Fractint image involved outside Windows, or to use the DOS command "SET NO87=nofpu" before running Fractint. (This SET command makes Fractint ignore your fpu, so things might be a lot slower as a result.) Insufficient memory: Fractint requires a fair bit of memory to run. Most machines with at least 640k (ok sticklers, make that "PC-compatible machines") will have no problem. Machines with 512k and machines with many TSR utilities and/or a LAN interface may have problems. Some Fractint features allocate memory when required during a run. If you get a message about insufficient memory, or suspect that some problem is due to a memory shortage, you could try commenting out some TSR utilities in your AUTOEXEC.BAT file, some non-critical drivers in your CONFIG.SYS file, or reducing the BUFFERS parameter in your CONFIG.SYS. "Cannot find FRACTINT.EXE" message: Fractint is an overlayed program - some parts of it are brought from disk into memory only when used. The overlay manager needs to know where to find the program. It must be named FRACTINT.EXE (which it is unless somebody renamed it), and you should either be in the directory containing it when you start Fractint, or that directory should be in your DOS PATH. "File FRACTINT.CFG is missing or invalid" message: You should either start Fractint while you are in the directory containing it, or should have that directory in your DOS PATH variable. If that isn't the problem, maybe you have a FRACTINT.CFG file from an older release of Fractint lying around? If so, best rename or delete it. If that isn't the problem either, then the FRACTINT.CFG included in the FRAINT.EXE release file has probably been changed or deleted. Best reinstall Fractint to get a fresh copy. Fractint Version 18.2 Page 120 Some other program doesn't like GIF files created by Fractint: Fractint generates nice clean GIF89A spec files, honest! But telling this to the other program isn't likely to change its mind. Instead, try an option which might get around the problem: run Fractint with the command line option "gif87a=yes" and then save an image. Fractint will store the image in the older GIF87A format, without any fractal parameters in it (so you won't be able to load the image back into Fractint and zoom into it - the fractal type, coordinates, etc. are not stored in this older format), and without an "aspect ratio" in the GIF header (we've seen one utility which doesn't like that field.) Disk video mode performance: This won't be blindingly fast at the best of times, but there are things which can slow it down and can be tuned. See "Disk-Video" Modes (p. 114) for details. Fractint Version 18.2 Page 121 8. Fractals and the PC 8.1 A Little History 8.1.1 Before Mandelbrot Like new forms of life, new branches of mathematics and science don't appear from nowhere. The ideas of fractal geometry can be traced to the late nineteenth century, when mathematicians created shapes -- sets of points -- that seemed to have no counterpart in nature. By a wonderful irony, the "abstract" mathematics descended from that work has now turned out to be MORE appropriate than any other for describing many natural shapes and processes. Perhaps we shouldn't be surprised. The Greek geometers worked out the mathematics of the conic sections for its formal beauty; it was two thousand years before Copernicus and Brahe, Kepler and Newton overcame the preconception that all heavenly motions must be circular, and found the ellipse, parabola, and hyperbola in the paths of planets, comets, and projectiles. In the 17th century Newton and Leibniz created calculus, with its techniques for "differentiating" or finding the derivative of functions -- in geometric terms, finding the tangent of a curve at any given point. True, some functions were discontinuous, with no tangent at a gap or an isolated point. Some had singularities: abrupt changes in direction at which the idea of a tangent becomes meaningless. But these were seen as exceptional, and attention was focused on the "well- behaved" functions that worked well in modeling nature. Beginning in the early 1870s, though, a 50-year crisis transformed mathematical thinking. Weierstrass described a function that was continuous but nondifferentiable -- no tangent could be described at any point. Cantor showed how a simple, repeated procedure could turn a line into a dust of scattered points, and Peano generated a convoluted curve that eventually touches every point on a plane. These shapes seemed to fall "between" the usual categories of one-dimensional lines, two- dimensional planes and three-dimensional volumes. Most still saw them as "pathological" cases, but here and there they began to find applications. In other areas of mathematics, too, strange shapes began to crop up. Poincare attempted to analyze the stability of the solar system in the 1880s and found that the many-body dynamical problem resisted traditional methods. Instead, he developed a qualitative approach, a "state space" in which each point represented a different planetary orbit, and studied what we would now call the topology -- the "connectedness" -- of whole families of orbits. This approach revealed that while many initial motions quickly settled into the familiar curves, there were also strange, "chaotic" orbits that never became periodic and predictable. Fractint Version 18.2 Page 122 Other investigators trying to understand fluctuating, "noisy" phenomena -- the flooding of the Nile, price series in economics, the jiggling of molecules in Brownian motion in fluids -- found that traditional models could not match the data. They had to introduce apparently arbitrary scaling features, with spikes in the data becoming rarer as they grew larger, but never disappearing entirely. For many years these developments seemed unrelated, but there were tantalizing hints of a common thread. Like the pure mathematicians' curves and the chaotic orbital motions, the graphs of irregular time series often had the property of self-similarity: a magnified small section looked very similar to a large one over a wide range of scales. 8.1.2 Who Is This Guy, Anyway? While many pure and applied mathematicians advanced these trends, it is Benoit Mandelbrot above all who saw what they had in common and pulled the threads together into the new discipline. He was born in Warsaw in 1924, and moved to France in 1935. In a time when French mathematical training was strongly analytic, he visualized problems whenever possible, so that he could attack them in geometric terms. He attended the Ecole Polytechnique, then Caltech, where he encountered the tangled motions of fluid turbulence. In 1958 he joined IBM, where he began a mathematical analysis of electronic "noise" -- and began to perceive a structure in it, a hierarchy of fluctuations of all sizes, that could not be explained by existing statistical methods. Through the years that followed, one seemingly unrelated problem after another was drawn into the growing body of ideas he would come to call fractal geometry. As computers gained more graphic capabilities, the skills of his mind's eye were reinforced by visualization on display screens and plotters. Again and again, fractal models produced results -- series of flood heights, or cotton prices -- that experts said looked like "the real thing." Visualization was extended to the physical world as well. In a provocative essay titled "How Long Is the Coast of Britain?" Mandelbrot noted that the answer depends on the scale at which one measures: it grows longer and longer as one takes into account every bay and inlet, every stone, every grain of sand. And he codified the "self-similarity" characteristic of many fractal shapes -- the reappearance of geometrically similar features at all scales. First in isolated papers and lectures, then in two editions of his seminal book, he argued that many of science's traditional mathematical models are ill-suited to natural forms and processes: in fact, that many of the "pathological" shapes mathematicians had discovered generations before are useful approximations of tree bark and lung tissue, clouds and galaxies. Fractint Version 18.2 Page 123 Mandelbrot was named an IBM Fellow in 1974, and continues to work at the IBM Watson Research Center. He has also been a visiting professor and guest lecturer at many universities. 8.2 A Little Code 8.2.1 Periodicity Logic The "Mandelbrot Lake" in the center of the M-set images is the traditional bane of plotting programs. It sucks up the most computer time because it always reaches the iteration limit -- and yet the most interesting areas are invariably right at the edge the lake. (See The Mandelbrot Set (p. 29) for a description of the iteration process.) Thanks to Mark Peterson for pointing out (well, he more like beat us over the head until we paid attention) that the iteration values in the middle of Mandelbrot Lake tend to decay to periodic loops (i.e., Z(n+m) == Z(n), a fact that is pointed out on pages 58-61 of "The Beauty of Fractals"). An intelligent program (like the one he wrote) would check for this periodicity once in a while, recognize that iterations caught in a loop are going to max out, and bail out early. For speed purposes, the current version of the program turns this checking algorithm on only if the last pixel generated was in the lake. (The checking itself takes a small amount of time, and the pixels on the very edge of the lake tend to decay to periodic loops very slowly, so this compromise turned out to be the fastest generic answer). Try a full M-set plot with a 1000-iteration maximum with any other program, and then try it on this one for a pretty dramatic proof of the value of periodicity checking. You can get a visual display of the periodicity effects if you press <O>rbits while plotting. This toggles display of the intermediate iterations during the generation process. It also gives you an idea of how much work your poor little PC is going through for you! If you use this toggle, it's best to disable solid-guessing first using <1> or <2> because in its second pass, solid-guessing bypasses many of the pixel calculations precisely where the orbits are most interesting. Mark was also responsible for pointing out that 16-bit integer math was good enough for the first few levels of M/J images, where the round-off errors stay well within the area covered by a single pixel. Fractint now uses 16-bit math where applicable, which makes a big difference on non- 32-bit PCs. 8.2.2 Limitations of Integer Math (And How We Cope) By default, Fractint uses 16-bit and/or 32-bit integer math to generate nearly all its fractal types. The advantage of integer math is speed: this is by far the fastest such plotter that we have ever seen on any PC. The disadvantage is an accuracy limit. Integer math represents numbers like 1.00 as 32-bit integers of the form [1.00 * (2^29)] Fractint Version 18.2 Page 124 (approximately a range of 500,000,000) for the Mandelbrot and Julia sets. Other integer fractal types use a bitshift of 24 rather than 29, so 1.0 is stored internally as [1.00 * (2^*24)]. This yields accuracy of better than 8 significant digits, and works fine... until the initial values of the calculations on consecutive pixels differ only in the ninth decimal place. At that point, if Fractint has a floating-point algorithm handy for that particular fractal type (and virtually all of the fractal types have one these days), it will silently switch over to the floating-point algorithm and keep right on going. Fair warning - if you don't have an FPU, the effect is that of a rocket sled hitting a wall of jello, and even if you do, the slowdown is noticeable. If it has no floating-point algorithm, Fractint does the best it can: it switches to its minimal drawing mode, with adjacent pixels having initial values differing by 1 (really 0.000000002). Attempts to zoom further may result in moving the image around a bit, but won't actually zoom. If you are stuck with an integer algorithm, you can reach minimal mode with your fifth consecutive "maximum zoom", each of which covers about 0.25% of the previous screen. By then your full-screen image is an area less than 1/(10^13)th [~0.0000000000001] the area of the initial screen. (If your image is rotated or stretched very slightly, you can run into the wall of jello as early as the fourth consecutive maximum zoom. Rotating or stretching by larger amounts has less impact on how soon you run into it.) Think of it this way: at minimal drawing mode, your VGA display would have to have a surface area of over one million square miles just to be able to display the entire M-set using the integer algorithms. Using the floating-point algorithms, your display would have to be big enough to fit the entire solar system out to the orbit of Saturn inside it. So there's a considerable saving on hardware, electricity and desk space involved here. Also, you don't have to take out asteroid insurance. 32 bit integers also limit the largest number which can be stored. This doesn't matter much since numbers outside the supported range (which is between -4 and +4) produce a boring single color. If you try to zoom-out to reduce the entire Mandelbrot set to a speck, or to squeeze it to a pancake, you'll find you can't do so in integer math mode. 8.2.3 The Fractint "Fractal Engine" Architecture Several of the authors would never ADMIT this, but Fractint has evolved a powerful and flexible architecture that makes adding new fractals very easy. (They would never admit this because they pride themselves on being the sort that mindlessly but happily hacks away at code and "sees if it works and doesn't hang the machine".) Many fractal calculations work by taking a rectangle in the complex plane, and, point by point, calculating a color corresponding to that point. Furthermore, the color calculation is often done by iterating a function over and over until some bailout condition is met. (See The Mandelbrot Set (p. 29) for a description of the iteration process.) Fractint Version 18.2 Page 125 In implementing such a scheme, there are three fractal-specific calculations that take place within a framework that is pretty much the same for them all. Rather than copy the same code over and over, we created a standard fractal engine that calls three functions that may be bolted in temporarily to the engine. The "bolting in" process uses the C language mechanism of variable function pointers. These three functions are: 1) a setup function that is run once per image, to do any required initialization of variables, 2) a once-per-pixel function that does whatever initialization has to be done to calculate a color for one pixel, and 3) a once-per-orbit-iteration function, which is the fundamental fractal algorithm that is repeatedly iterated in the fractal calculation. The common framework that calls these functions can contain all sorts of speedups, tricks, and options that the fractal implementor need not worry about. All that is necessary is to write the three functions in the correct way, and BINGO! - all options automatically apply. What makes it even easier is that usually one can re-use functions 1) and 2) written for other fractals, and therefore only need to write function 3). Then it occurred to us that there might be more than one sort of fractal engine, so we even allowed THAT to be bolted in. And we created a data structure for each fractal that includes pointers to these four functions, various prompts, a default region of the complex plane, and various miscellaneous bits of information that allow toggling between Julia and Mandelbrot or toggling between the various kinds of math used in implementation. That sounds pretty flexible, but there is one drawback - you have to be a C programmer and have a C compiler to make use of it! So we took it a step further, and designed a built-in high level compiler, so that you can enter the formulas for the various functions in a formula file in a straightforward algebra-like language, and Fractint will compile them and bolt them in for you! There is a terrible down side to this flexibility. Fractint users everywhere are going berserk. Fractal-inventing creativity is running rampant. Proposals for new fractal types are clogging the mail and the telephones. All we can say is that non-productivity software has never been so potent, and we're sorry, it's our fault! Fractint was compiled using Microsoft C 6.00A and Microsoft Assembler 5.1, using the "Medium" model. Note that the assembler code uses the "C" model option added to version 5.1, and must be assembled with the /MX or /ML switch to link with the "C" code. Because it has become too large to distribute comfortably as a single compressed file, and because many downloaders have no intention of ever modifying it, Fractint is now Fractint Version 18.2 Page 126 distributed as two files: one containing FRACTINT.EXE, auxiliary files and this document, and another containing complete source code (including a .MAK file and MAKEFRAC.BAT). See Distribution of Fractint (p. 145). Fractint Version 18.2 Page 127 Appendix A Mathematics of the Fractal Types SUMMARY OF FRACTAL TYPES barnsleyj1 (p. 38) z(0) = pixel; z(n+1) = (z-1)*c if real(z) >= 0, else z(n+1) = (z+1)*c Two parameters: real and imaginary parts of c barnsleyj2 (p. 38) z(0) = pixel; if real(z(n)) * imag(c) + real(c) * imag(z((n)) >= 0 z(n+1) = (z(n)-1)*c else z(n+1) = (z(n)+1)*c Two parameters: real and imaginary parts of c barnsleyj3 (p. 38) z(0) = pixel; if real(z(n) > 0 then z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1) + i * (2*real(z((n)) * imag(z((n))) else z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1 + real(c) * real(z(n)) + i * (2*real(z((n)) * imag(z((n)) + imag(c) * real(z(n)) Two parameters: real and imaginary parts of c. barnsleym1 (p. 38) z(0) = c = pixel; if real(z) >= 0 then z(n+1) = (z-1)*c else z(n+1) = (z+1)*c. Parameters are perturbations of z(0) barnsleym2 (p. 38) z(0) = c = pixel; if real(z)*imag(c) + real(c)*imag(z) >= 0 z(n+1) = (z-1)*c else z(n+1) = (z+1)*c Parameters are perturbations of z(0) barnsleym3 (p. 38) z(0) = c = pixel; if real(z(n) > 0 then z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1) + i * (2*real(z((n)) * imag(z((n))) else z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1 + real(c) * real(z(n)) + i * (2*real(z((n)) * imag(z((n)) + imag(c) * real(z(n)) Parameters are perturbations of z(0) bifurcation (p. 44) Pictorial representation of a population growth model. Let P = new population, p = oldpopulation, r = growth rate The model is: P = p + r*fn(p)*(1-fn(p)). Three parameters: Filter Cycles, Seed Population, and Function. bif+sinpi (p. 44) Bifurcation variation: model is: P = p + r*fn(PI*p). Three parameters: Filter Cycles, Seed Population, and Function. bif=sinpi (p. 44) Bifurcation variation: model is: P = r*fn(PI*p). Three parameters: Filter Cycles, Seed Population, and Function. Fractint Version 18.2 Page 128 biflambda (p. 44) Bifurcation variation: model is: P = r*fn(p)*(1-fn(p)). Three parameters: Filter Cycles, Seed Population, and Function. bifstewart (p. 44) Bifurcation variation: model is: P = (r*fn(p)*fn(p)) - 1. Three parameters: Filter Cycles, Seed Population, and Function. bifmay (p. 44) Bifurcation variation: model is: P = r*p / ((1+p)^beta). Three parameters: Filter Cycles, Seed Population, and Beta. cellular (p. 61) One-dimensional cellular automata or line automata. The type of CA is given by kr, where k is the number of different states of the automata and r is the radius of the neighborhood. The next generation is determined by the sum of the neighborhood and the specified rule. Four parameters: Initial String, Rule, Type, and Starting Row Number. For Type = 21, 31, 41, 51, 61, 22, 32, 42, 23, 33, 24, 25, 26, 27 Rule = 4, 7, 10, 13, 16, 6, 11, 16, 8, 15, 10, 12, 14, 16 digits circle (p. 36) Circle pattern by John Connett x + iy = pixel z = a*(x^2 + y^2) c = integer part of z color = c modulo(number of colors) cmplxmarksjul (p. 42) A generalization of the marksjulia fractal. z(0) = pixel; z(n+1) = (c^exp)*z(n)^2 + c. Four parameters: real and imaginary parts of c and exp. cmplxmarksmand (p. 42) A generalization of the marksmandel fractal. z(0) = c = pixel; z(n+1) = (c^exp)*z(n)^2 + c. Four parameters: real and imaginary parts of perturbation of z(0) and exp. complexnewton, complexbasin (p. 34) Newton fractal types extended to complex degrees. Complexnewton colors pixels according to the number of iterations required to escape to a root. Complexbasin colors pixels according to which root captures the orbit. The equation is based on the newton formula for solving the equation z^p = r z(0) = pixel; z(n+1) = ((p - 1) * z(n)^p + r)/(p * z(n)^(p - 1)). Four parameters: real & imaginary parts of degree p and root r diffusion (p. 54) Diffusion Limited Aggregation. Randomly moving points accumulate. Two parameters: border width (default 10), type dynamic (p. 59) Time-discrete dynamic system. x(0) = y(0) = start position. Fractint Version 18.2 Page 129 y(n+1) = y(n) + f( x(n) ) x(n+1) = x(n) - f( y(n) ) f(k) = sin(k + a*fn1(b*k)) For implicit Euler approximation: x(n+1) = x(n) - f( y(n+1) ) Five parameters: start position step, dt, a, b, and the function fn1. fn+fn(pix) (p. 43) c = z(0) = pixel; z(n+1) = fn1(z) + p*fn2(c) Six parameters: real and imaginary parts of the perturbation of z(0) and factor p, and the functions fn1, and fn2. fn(z*z) (p. 43) z(0) = pixel; z(n+1) = fn(z(n)*z(n)) One parameter: the function fn. fn*fn (p. 43) z(0) = pixel; z(n+1) = fn1(n)*fn2(n) Two parameters: the functions fn1 and fn2. fn*z+z (p. 43) z(0) = pixel; z(n+1) = p1*fn(z(n))*z(n) + p2*z(n) Five parameters: the real and imaginary components of p1 and p2, and the function fn. fn+fn (p. 43) z(0) = pixel; z(n+1) = p1*fn1(z(n))+p2*fn2(z(n)) Six parameters: The real and imaginary components of p1 and p2, and the functions fn1 and fn2. formula (p. 51) Formula interpreter - write your own formulas as text files! frothybasin (p. 63) Pixel color is determined by which attractor captures the orbit. The shade of color is determined by the number of iterations required to capture the orbit. z(0) = pixel; z(n+1) = z(n)^2 - c*conj(z(n)) where c = 1 + ai, and a = 1.02871376822... gingerbread (p. 49) Orbit in two dimensions defined by: x(n+1) = 1 - y(n) + |x(n)| y(n+1) = x(n) Two parameters: initial values of x(0) and y(0). halley (p. 59) Halley map for the function: F = z(z^a - 1) = 0 z(0) = pixel; z(n+1) = z(n) - R * F / [F' - (F" * F / 2 * F')] bailout when: abs(mod(z(n+1)) - mod(z(n)) < epsilon Three parameters: order of z (a), relaxation coefficient (R), small number for bailout (epsilon). henon (p. 48) Fractint Version 18.2 Page 130 Orbit in two dimensions defined by: x(n+1) = 1 + y(n) - a*x(n)*x(n) y(n+1) = b*x(n) Two parameters: a and b hopalong (p. 49) Hopalong attractor by Barry Martin - orbit in two dimensions. z(0) = y(0) = 0; x(n+1) = y(n) - sign(x(n))*sqrt(abs(b*x(n)-c)) y(n+1) = a - x(n) Parameters are a, b, and c. hypercomplex (p. 61) HyperComplex Mandelbrot set. h(0) = (0,0,0,0) h(n+1) = fn(h(n)) + C. where "fn" is sin, cos, log, sqr etc. Two parameters: cj, ck C = (xpixel,ypixel,cj,ck) hypercomplexj (p. 61) HyperComplex Julia set. h(0) = (xpixel,ypixel,zj,zk) h(n+1) = fn(h(n)) + C. where "fn" is sin, cos, log, sqr etc. Six parameters: c1, ci, cj, ck C = (c1,ci,cj,ck) icon, icon3d (p. 50) Orbit in three dimensions defined by: p = lambda + alpha * magnitude + beta * (x(n)*zreal - y(n)*zimag) x(n+1) = p * x(n) + gamma * zreal - omega * y(n) y(n+1) = p * y(n) - gamma * zimag + omega * x(n) (3D version uses magnitude for z) Parameters: Lambda, Alpha, Beta, Gamma, Omega, and Degree IFS (p. 39) Barnsley IFS (Iterated Function System) fractals. Apply contractive affine mappings. julfn+exp (p. 41) A generalized Clifford Pickover fractal. z(0) = pixel; z(n+1) = fn(z(n)) + e^z(n) + c. Three parameters: real & imaginary parts of c, and fn julfn+zsqrd (p. 41) z(0) = pixel; z(n+1) = fn(z(n)) + z(n)^2 + c Three parameters: real & imaginary parts of c, and fn julia (p. 30) Classic Julia set fractal. z(0) = pixel; z(n+1) = z(n)^2 + c. Two parameters: real and imaginary parts of c. Fractint Version 18.2 Page 131 julia_inverse (p. 32) Inverse Julia function - "orbit" traces Julia set in two dimensions. z(0) = a point on the Julia Set boundary; z(n+1) = +- sqrt(z(n) - c) Parameters: Real and Imaginary parts of c Maximum Hits per Pixel (similar to max iters) Breadth First, Depth First or Random Walk Tree Traversal Left or Right First Branching (in Depth First mode only) Try each traversal method, keeping everything else the same. Notice the differences in the way the image evolves. Start with a fairly low Maximum Hit limit, then increase it. The hit limit cannot be higher than the maximum colors in your video mode. julia(fn||fn) (p. 58) z(0) = pixel; if modulus(z(n)) < shift value, then z(n+1) = fn1(z(n)) + c, else z(n+1) = fn2(z(n)) + c. Five parameters: real, imaginary portions of c, shift value, fn1 and fn2. julia4 (p. 41) Fourth-power Julia set fractals, a special case of julzpower kept for speed. z(0) = pixel; z(n+1) = z(n)^4 + c. Two parameters: real and imaginary parts of c. julibrot (p. 52) 'Julibrot' 4-dimensional fractals. julzpower (p. 41) z(0) = pixel; z(n+1) = z(n)^m + c. Three parameters: real & imaginary parts of c, exponent m julzzpwr (p. 41) z(0) = pixel; z(n+1) = z(n)^z(n) + z(n)^m + c. Three parameters: real & imaginary parts of c, exponent m kamtorus, kamtorus3d (p. 44) Series of orbits superimposed. 3d version has 'orbit' the z dimension. x(0) = y(0) = orbit/3; x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a) y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a) After each orbit, 'orbit' is incremented by a step size. Parameters: a, step size, stop value for 'orbit', and points per orbit. lambda (p. 35) Classic Lambda fractal. 'Julia' variant of Mandellambda. z(0) = pixel; z(n+1) = lambda*z(n)*(1 - z(n)). Two parameters: real and imaginary parts of lambda. Fractint Version 18.2 Page 132 lambdafn (p. 37) z(0) = pixel; z(n+1) = lambda * fn(z(n)). Three parameters: real, imag portions of lambda, and fn lambda(fn||fn) (p. 58) z(0) = pixel; if modulus(z(n)) < shift value, then z(n+1) = lambda * fn1(z(n)), else z(n+1) = lambda * fn2(z(n)). Five parameters: real, imaginary portions of lambda, shift value, fn1 and fn2. lorenz, lorenz3d (p. 47) Lorenz two lobe attractor - orbit in three dimensions. In 2d the x and y components are projected to form the image. z(0) = y(0) = z(0) = 1; x(n+1) = x(n) + (-a*x(n)*dt) + ( a*y(n)*dt) y(n+1) = y(n) + ( b*x(n)*dt) - ( y(n)*dt) - (z(n)*x(n)*dt) z(n+1) = z(n) + (-c*z(n)*dt) + (x(n)*y(n)*dt) Parameters are dt, a, b, and c. lorenz3d1 (p. 47) Lorenz one lobe attractor - orbit in three dimensions. The original formulas were developed by Rick Miranda and Emily Stone. z(0) = y(0) = z(0) = 1; norm = sqrt(x(n)^2 + y(n)^2) x(n+1) = x(n) + (-a*dt-dt)*x(n) + (a*dt-b*dt)*y(n) + (dt-a*dt)*norm + y(n)*dt*z(n) y(n+1) = y(n) + (b*dt-a*dt)*x(n) - (a*dt+dt)*y(n) + (b*dt+a*dt)*norm - x(n)*dt*z(n) - norm*z(n)*dt z(n+1) = z(n) +(y(n)*dt/2) - c*dt*z(n) Parameters are dt, a, b, and c. lorenz3d3 (p. 47) Lorenz three lobe attractor - orbit in three dimensions. The original formulas were developed by Rick Miranda and Emily Stone. z(0) = y(0) = z(0) = 1; norm = sqrt(x(n)^2 + y(n)^2) x(n+1) = x(n) +(-(a*dt+dt)*x(n) + (a*dt-b*dt+z(n)*dt)*y(n))/3 + ((dt-a*dt)*(x(n)^2-y(n)^2) + 2*(b*dt+a*dt-z(n)*dt)*x(n)*y(n))/(3*norm) y(n+1) = y(n) +((b*dt-a*dt-z(n)*dt)*x(n) - (a*dt+dt)*y(n))/3 + (2*(a*dt-dt)*x(n)*y(n) + (b*dt+a*dt-z(n)*dt)*(x(n)^2-y(n)^2))/(3*norm) z(n+1) = z(n) +(3*x(n)*dt*x(n)*y(n)-y(n)*dt*y(n)^2)/2 - c*dt*z(n) Parameters are dt, a, b, and c. lorenz3d4 (p. 47) Lorenz four lobe attractor - orbit in three dimensions. The original formulas were developed by Rick Miranda and Emily Stone. z(0) = y(0) = z(0) = 1; x(n+1) = x(n) +(-a*dt*x(n)^3 + (2*a*dt+b*dt-z(n)*dt)*x(n)^2*y(n) + (a*dt-2*dt)*x(n)*y(n)^2 + (z(n)*dt-b*dt)*y(n)^3) / (2 * (x(n)^2+y(n)^2)) y(n+1) = y(n) +((b*dt-z(n)*dt)*x(n)^3 + (a*dt-2*dt)*x(n)^2*y(n) + (-2*a*dt-b*dt+z(n)*dt)*x(n)*y(n)^2 - a*dt*y(n)^3) / (2 * (x(n)^2+y(n)^2)) Fractint Version 18.2 Page 133 z(n+1) = z(n) +(2*x(n)*dt*x(n)^2*y(n) - 2*x(n)*dt*y(n)^3 - c*dt*z(n)) Parameters are dt, a, b, and c. lsystem (p. 55) Using a turtle-graphics control language and starting with an initial axiom string, carries out string substitutions the specified number of times (the order), and plots the resulting. lyapunov (p. 57) Derived from the Bifurcation fractal, the Lyapunov plots the Lyapunov Exponent for a population model where the Growth parameter varies between two values in a periodic manner. magnet1j (p. 54) z(0) = pixel; [ z(n)^2 + (c-1) ] 2 z(n+1) = | ---------------- | [ 2*z(n) + (c-2) ] Parameters: the real and imaginary parts of c magnet1m (p. 54) z(0) = 0; c = pixel; [ z(n)^2 + (c-1) ] 2 z(n+1) = | ---------------- | [ 2*z(n) + (c-2) ] Parameters: the real & imaginary parts of perturbation of z(0) magnet2j (p. 54) z(0) = pixel; [ z(n)^3 + 3*(C-1)*z(n) + (C-1)*(C-2) ] 2 z(n+1) = | -------------------------------------------- | [ 3*(z(n)^2) + 3*(C-2)*z(n) + (C-1)*(C-2) + 1 ] Parameters: the real and imaginary parts of c magnet2m (p. 54) z(0) = 0; c = pixel; [ z(n)^3 + 3*(C-1)*z(n) + (C-1)*(C-2) ] 2 z(n+1) = | -------------------------------------------- | [ 3*(z(n)^2) + 3*(C-2)*z(n) + (C-1)*(C-2) + 1 ] Parameters: the real and imaginary parts of perturbation of z(0) mandel (p. 29) Classic Mandelbrot set fractal. z(0) = c = pixel; z(n+1) = z(n)^2 + c. Two parameters: real & imaginary perturbations of z(0) mandel(fn||fn) (p. 58) c = pixel; z(0) = p1 if modulus(z(n)) < shift value, then z(n+1) = fn1(z(n)) + c, else z(n+1) = fn2(z(n)) + c. Five parameters: real, imaginary portions of p1, shift value, fn1 and fn2. Fractint Version 18.2 Page 134 mandelcloud (p. 60) Displays orbits of Mandelbrot set: z(0) = c = pixel; z(n+1) = z(n)^2 + c. One parameter: number of intervals mandel4 (p. 41) Special case of mandelzpower kept for speed. z(0) = c = pixel; z(n+1) = z(n)^4 + c. Parameters: real & imaginary perturbations of z(0) mandelfn (p. 38) z(0) = c = pixel; z(n+1) = c*fn(z(n)). Parameters: real & imaginary perturbations of z(0), and fn manlam(fn||fn) (p. 58) c = pixel; z(0) = p1 if modulus(z(n)) < shift value, then z(n+1) = fn1(z(n)) * c, else z(n+1) = fn2(z(n)) * c. Five parameters: real, imaginary parts of p1, shift value, fn1, fn2. Martin (p. 49) Attractor fractal by Barry Martin - orbit in two dimensions. z(0) = y(0) = 0; x(n+1) = y(n) - sin(x(n)) y(n+1) = a - x(n) Parameter is a (try a value near pi) mandellambda (p. 35) z(0) = .5; lambda = pixel; z(n+1) = lambda*z(n)*(1 - z(n)). Parameters: real & imaginary perturbations of z(0) mandphoenix (p. 62) z(0) = p1, y(0) = 0; For degree of Z = 0: z(n+1) = z(n)^2 + pixel.x + (pixel.y)y(n), y(n+1) = z(n) For degree of Z >= 2: z(n+1) = z(n)^degree + pz(n)^(degree-1) + qy(n), y(n+1) = z(n) For degree of Z <= -3: z(n+1) = z(n)^|degree| + pz(n)^(|degree|-2) + qy(n), y(n+1) = z(n) Three parameters: real part of z(0), imaginary part of z(0), and the degree of Z. manfn+exp (p. 41) 'Mandelbrot-Equivalent' for the julfn+exp fractal. z(0) = c = pixel; z(n+1) = fn(z(n)) + e^z(n) + C. Parameters: real & imaginary perturbations of z(0), and fn Fractint Version 18.2 Page 135 manfn+zsqrd (p. 41) 'Mandelbrot-Equivalent' for the Julfn+zsqrd fractal. z(0) = c = pixel; z(n+1) = fn(z(n)) + z(n)^2 + c. Parameters: real & imaginary perturbations of z(0), and fn manowar (p. 43) c = z1(0) = z(0) = pixel; z(n+1) = z(n)^2 + z1(n) + c; z1(n+1) = z(n); Parameters: real & imaginary perturbations of z(0) manowar (p. 43) z1(0) = z(0) = pixel; z(n+1) = z(n)^2 + z1(n) + c; z1(n+1) = z(n); Parameters: real & imaginary perturbations of c manzpower (p. 41) 'Mandelbrot-Equivalent' for julzpower. z(0) = c = pixel; z(n+1) = z(n)^exp + c; try exp = e = 2.71828... Parameters: real & imaginary perturbations of z(0), real & imaginary parts of exponent exp. manzzpwr (p. 41) 'Mandelbrot-Equivalent' for the julzzpwr fractal. z(0) = c = pixel z(n+1) = z(n)^z(n) + z(n)^exp + C. Parameters: real & imaginary perturbations of z(0), and exponent marksjulia (p. 42) A variant of the julia-lambda fractal. z(0) = pixel; z(n+1) = (c^exp)*z(n)^2 + c. Parameters: real & imaginary parts of c, and exponent marksmandel (p. 42) A variant of the mandel-lambda fractal. z(0) = c = pixel; z(n+1) = (c^exp)*z(n)^2 + c. Parameters: real & imaginary perturbations of z(0), and exponent marksmandelpwr (p. 42) The marksmandelpwr formula type generalized (it previously had fn=sqr hard coded). z(0) = pixel, c = z(0) ^ (z(0) - 1): z(n+1) = c * fn(z(n)) + pixel, Parameters: real and imaginary perturbations of z(0), and fn newtbasin (p. 33) Based on the Newton formula for finding the roots of z^p - 1. Pixels are colored according to which root captures the orbit. z(0) = pixel; z(n+1) = ((p-1)*z(n)^p + 1)/(p*z(n)^(p - 1)). Two parameters: the polynomial degree p, and a flag to turn Fractint Version 18.2 Page 136 on color stripes to show alternate iterations. newton (p. 34) Based on the Newton formula for finding the roots of z^p - 1. Pixels are colored according to the iteration when the orbit is captured by a root. z(0) = pixel; z(n+1) = ((p-1)*z(n)^p + 1)/(p*z(n)^(p - 1)). One parameter: the polynomial degree p. phoenix (p. 62) z(0) = pixel, y(0) = 0; For degree of Z = 0: z(n+1) = z(n)^2 + p + qy(n), y(n+1) = z(n) For degree of Z >= 2: z(n+1) = z(n)^degree + pz(n)^(degree-1) + qy(n), y(n+1) = z(n) For degree of Z <= -3: z(n+1) = z(n)^|degree| + pz(n)^(|degree|-2) + qy(n), y(n+1) = z(n) Three parameters: real p, real q, and the degree of Z. pickover (p. 49) Orbit in three dimensions defined by: x(n+1) = sin(a*y(n)) - z(n)*cos(b*x(n)) y(n+1) = z(n)*sin(c*x(n)) - cos(d*y(n)) z(n+1) = sin(x(n)) Parameters: a, b, c, and d. plasma (p. 36) Random, cloud-like formations. Requires 4 or more colors. A recursive algorithm repeatedly subdivides the screen and colors pixels according to an average of surrounding pixels and a random color, less random as the grid size decreases. Four parameters: 'graininess' (.5 to 50, default = 2), old/new algorithm, seed value used, 16-bit out output selection. popcorn (p. 42) The orbits in two dimensions defined by: x(0) = xpixel, y(0) = ypixel; x(n+1) = x(n) - h*sin(y(n) + tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n) + tan(3*x(n)) are plotted for each screen pixel and superimposed. One parameter: step size h. popcornjul (p. 42) Conventional Julia using the popcorn formula: x(0) = xpixel, y(0) = ypixel; x(n+1) = x(n) - h*sin(y(n) + tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n) + tan(3*x(n)) One parameter: step size h. quatjul (p. 60) Quaternion Julia set. q(0) = (xpixel,ypixel,zj,zk) q(n+1) = q(n)*q(n) + c. Four parameters: c, ci, cj, ck c = (c1,ci,cj,ck) Fractint Version 18.2 Page 137 quat (p. 60) Quaternion Mandelbrot set. q(0) = (0,0,0,0) q(n+1) = q(n)*q(n) + c. Two parameters: cj,ck c = (xpixel,ypixel,cj,ck) rossler3D (p. 48) Orbit in three dimensions defined by: x(0) = y(0) = z(0) = 1; x(n+1) = x(n) - y(n)*dt - z(n)*dt y(n+1) = y(n) + x(n)*dt + a*y(n)*dt z(n+1) = z(n) + b*dt + x(n)*z(n)*dt - c*z(n)*dt Parameters are dt, a, b, and c. sierpinski (p. 40) Sierpinski gasket - Julia set producing a 'Swiss cheese triangle' z(n+1) = (2*x,2*y-1) if y > .5; else (2*x-1,2*y) if x > .5; else (2*x,2*y) No parameters. spider (p. 43) c(0) = z(0) = pixel; z(n+1) = z(n)^2 + c(n); c(n+1) = c(n)/2 + z(n+1) Parameters: real & imaginary perturbation of z(0) sqr(1/fn) (p. 43) z(0) = pixel; z(n+1) = (1/fn(z(n))^2 One parameter: the function fn. sqr(fn) (p. 43) z(0) = pixel; z(n+1) = fn(z(n))^2 One parameter: the function fn. test (p. 50) 'test' point letting us (and you!) easily add fractal types via the c module testpt.c. Default set up is a mandelbrot fractal. Four parameters: user hooks (not used by default testpt.c). tetrate (p. 43) z(0) = c = pixel; z(n+1) = c^z(n) Parameters: real & imaginary perturbation of z(0) tim's_error (p. 42) A serendipitous coding error in marksmandelpwr brings to life an ancient pterodactyl! (Try setting fn to sqr.) z(0) = pixel, c = z(0) ^ (z(0) - 1): tmp = fn(z(n)) real(tmp) = real(tmp) * real(c) - imag(tmp) * imag(c); imag(tmp) = real(tmp) * imag(c) - imag(tmp) * real(c); z(n+1) = tmp + pixel; Fractint Version 18.2 Page 138 Parameters: real & imaginary perturbations of z(0) and function fn unity (p. 43) z(0) = pixel; x = real(z(n)), y = imag(z(n)) One = x^2 + y^2; y = (2 - One) * x; x = (2 - One) * y; z(n+1) = x + i*y No parameters. INSIDE=BOF60|BOF61|ZMAG|PERIOD Here is an *ATTEMPTED* explanation of what the inside=bof60 and inside=bof61 options do. This explanation is hereby dedicated to Adrian Mariano, who badgered it out of us! For the *REAL* explanation, see "Beauty of Fractals", page 62. Let p(z) be the function that is repeatedly iterated to generate a fractal using the escape-time algorithm. For example, p(z) = z^2+c in the case of a Julia set. Then let pk(z) be the result of iterating the function p for k iterations. (The "k" should be shown as a superscript.) We could also use the notation pkc(z) when the function p has a parameter c, as it does in our example. Now hold your breath and get your thinking cap on. Define a(c) = inf{|pck(0)|:k=1,2,3,...}. In English - a(c) is the greatest lower bound of the images of zero of as many iterations as you like. Put another way, a(c) is the closest to the origin any point in the orbit starting with 0 gets. Then the index (c) is the value of k (the iteration) when that closest point was achieved. Since there may be more than one, index(c) is the least such. Got it? Good, because the "Beauty of Fractals" explanation of this, is, ahhhh, *TERSE* ! Now for the punch line. Inside=bof60 colors the lake alternating shades according to the level sets of a(c). Each band represents solid areas of the fractal where the closest value of the orbit to the origin is the same. Inside=bof61 show domains where index(c) is constant. That is, areas where the iteration when the orbit swooped closest to the origin has the same value. Well, folks, that's the best we can do! Improved explanations will be accepted for the next edition! inside=zmag is similar. This option colors inside pixels according to the magnitude of the orbit point when maxiter was reached, using the formula color = (x^2 + y^2) * maxiter/2 + 1. inside=period colors pixels according to the length of their eventual cycle. For example, points that approach a fixed point have color=1. Points that approach a 2-cycle have color=2. Points that do not approach a cycle during the iterations performed have color=maxit. This option works best with a fairly large number of iterations. INSIDE=EPSCROSS|STARTRAIL Kenneth Hooper has written a paper entitled "A Note On Some Internal Structures Of The Mandelbrot Set" published in "Computers and Graphics", Vol 15, No.2, pp. 295-297. In that article he describes Clifford Pickover's "epsilon cross" method which creates some mysterious plant- Fractint Version 18.2 Page 139 like tendrils in the Mandelbrot set. The algorithm is this. In the escape-time calculation of a fractal, if the orbit comes within .01 of the Y-axis, the orbit is terminated and the pixel is colored green. Similarly, the pixel is colored yellow if it approaches the X-axis. Strictly speaking, this is not an "inside" option because a point destined to escape could be caught by this bailout criterion. Hooper has another coloring scheme called "star trails" that involves detecting clusters of points being traversed by the orbit. A table of tangents of each orbit point is built, and the pixel colored according to how many orbit points are near the first one before the orbit flies out of the cluster. This option looks fine with maxiter=16, which greatly speeds the calculation. Both of these options should be tried with the outside color fixed (outside=<nnn>) so that the "lake" structure revealed by the algorithms can be more clearly seen. Epsilon Cross is fun to watch with boundary tracing turned on - even though the result is incorrect it is interesting! Shucks - what does "incorrect" mean in chaos theory anyway?! FINITE ATTRACTORS Many of Fractint's fractals involve the iteration of functions of complex numbers until some "bailout" value is exceeded, then coloring the associated pixel according to the number of iterations performed. This process identifies which values tend to infinity when iterated, and gives us a rough measure of how "quickly" they get there. In dynamical terms, we say that "Infinity is an Attractor", as many initial values get "attracted" to it when iterated. The set of all points that are attracted to infinity is termed The Basin of Attraction of Infinity. The coloring algorithm used divides this Basin of Attraction into many distinct sets, each a single band of one color, representing all the points that are "attracted" to Infinity at the same "rate". These sets (bands of color) are termed "Level Sets" - all points in such a set are at the same "Level" away from the attractor, in terms of numbers of iterations required to exceed the bailout value. Thus, Fractint produces colored images of the Level Sets of the Basin of Attraction of Infinity, for all fractals that iterate functions of Complex numbers, at least. Now we have a sound mathematical definition of what Fractint's "bailout" processing generates, and we have formally introduced the terms Attractor, Basin of Attraction, and Level Set, so you should have little trouble following the rest of this section! For certain Julia-type fractals, Fractint can also display the Level Sets of Basins of Attraction of Finite Attractors. This capability is a by-product of the implementation of the MAGNETic fractal types, which always have at least one Finite Attractor. This option can be invoked by setting the "Look for finite attractor" option on the <Y> options screen, or by giving the "finattract=yes" command-line option. Fractint Version 18.2 Page 140 Most Julia-types that have a "lake" (normally colored blue by default) have a Finite Attractor within this lake, and the lake turns out to be, quite appropriately, the Basin of Attraction of this Attractor. The "finattract=yes" option (command-line or <Y> options screen) instructs Fractint to seek out and identify a possible Finite Attractor and, if found, to display the Level Sets of its Basin of Attraction, in addition to those of the Basin of Attraction of Infinity. In many cases this results in a "lake" with colored "waves" in it; in other cases there may be little change in the lake's appearance. For a quick demonstration, select a fractal type of LAMBDA, with a parameter of 0.5 + 0.5i. You will obtain an image with a large blue lake. Now set "Look for finite attractor" to 1 with the "Y" menu. The image will be re-drawn with a much more colorful lake. A Finite Attractor lives in the center of one of the resulting "ripple" patterns in the lake - turn the <O>rbits display on to see where it is - the orbits of all initial points that are in the lake converge there. Fractint tests for the presence of a Finite Attractor by iterating a Critical Value of the fractal's function. If the iteration doesn't bail out before exceeding twice the iteration limit, it is almost certain that we have a Finite Attractor - we assume that we have. Next we define a small circle around it and, after each iteration, as well as testing for the usual bailout value being exceeded, we test to see if we've hit the circle. If so, we bail out and color our pixels according to the number of iterations performed. Result - a nicely colored-in lake that displays the Level Sets of the Basin of Attraction of the Finite Attractor. Sometimes ! First exception: This does not work for the lakes of Mandel-types. Every point in a Mandel-type is, in effect, a single point plucked from one of its related Julia-types. A Mandel-type's lake has an infinite number of points, and thus an infinite number of related Julia-type sets, and consequently an infinite number of finite attractors too. It *MAY* be possible to color in such a lake, by determining the attractor for EVERY pixel, but this would probably treble (at least) the number of iterations needed to draw the image. Due to this overhead, Finite Attractor logic has not been implemented for Mandel-types. Secondly, certain Julia-types with lakes may not respond to this treatment, depending on the parameter value used. E.g., the Lambda Set for 0.5 + 0.5i responds well; the Lambda Set for 0.0 + 1.0i does not - its lake stays blue. Attractors that consist of single points, or a cycle of a finite number of points are ok. Others are not. If you're into fractal technospeke, the implemented approach fails if the Julia- type is a Parabolic case, or has Siegel Disks, or has Herman Rings. However, all the difficult cases have one thing in common - they all have a parameter value that falls exactly on the edge of the related Mandel-type's lake. You can avoid them by intelligent use of the Mandel-Julia Space-Bar toggle: Pick a view of the related Mandel-type where the center of the screen is inside the lake, but not too close to its edge, then use the space-bar toggle. You should obtain a usable Julia-type with a lake, if you follow this guideline. Fractint Version 18.2 Page 141 Thirdly, the initial implementation only works for Julia-types that use the "Standard" fractal engine in Fractint. Fractals with their own special algorithms are not affected by Finite Attractor logic, as yet. Finally, the finite attractor code will not work if it fails to detect a finite attractor. If the number of iterations is set too low, the finite attractor may be missed. Despite these restrictions, the Finite Attractor logic can produce interesting results. Just bear in mind that it is principally a bonus off-shoot from the development of the MAGNETic fractal types, and is not specifically tuned for optimal performance for other Julia types. (Thanks to Kevin Allen for the above). There is a second type of finite attractor coloring, which is selected by setting "Look for Finite Attractor" to a negative value. This colors points by the phase of the convergence to the finite attractor, instead of by the speed of convergence. For example, consider the Julia set for -0.1 + 0.7i, which is the three- lobed "rabbit" set. The Finite Attractor is an orbit of length three; call these values a, b, and c. Then, the Julia set iteration can converge to one of three sequences: a,b,c,a,b,c,..., or b,c,a,b,c,..., or c,a,b,c,a,b,... The Finite Attractor phase option colors the interior of the Julia set with three colors, depending on which of the three sequences the orbit converges to. Internally, the code determines one point of the orbit, say "a", and the length of the orbit cycle, say 3. It then iterates until the sequence converges to a, and then uses the iteration number modulo 3 to determine the color. TRIG IDENTITIES The following trig identities are invaluable for coding fractals that use complex-valued transcendental functions. e^(x+iy) = (e^x)cos(y) + i(e^x)sin(y) sin(x+iy) = sin(x)cosh(y) + icos(x)sinh(y) cos(x+iy) = cos(x)cosh(y) - isin(x)sinh(y) sinh(x+iy) = sinh(x)cos(y) + icosh(x)sin(y) cosh(x+iy) = cosh(x)cos(y) + isinh(x)sin(y) cosxx(x+iy) = cos(x)cosh(y) + isin(x)sinh(y) (cosxx is present in Fractint to provide compatibility with a bug which was in its cos calculation before version 16) ln(x+iy) = (1/2)ln(x*x + y*y) + i(arctan(y/x) + 2kPi) (k = 0, +-1, +-2, +-....) sin(2x) sinh(2y) tan(x+iy) = ------------------ + i------------------ cos(2x) + cosh(2y) cos(2x) + cosh(2y) Fractint Version 18.2 Page 142 sinh(2x) sin(2y) tanh(x+iy) = ------------------ + i------------------ cosh(2x) + cos(2y) cosh(2x) + cos(2y) sin(2x) - i*sinh(2y) cotan(x+iy) = -------------------- cosh(2y) - cos(2x) sinh(2x) - i*sin(2y) cotanh(x+iy) = -------------------- cosh(2x) - cos(2y) z^z = e^(log(z)*z) log(x+iy) = 1/2(log(x*x + y*y) + i(arc_tan(y/x)) e^(x+iy) = (cosh(x) + sinh(x)) * (cos(y) + isin(y)) = e^x * (cos(y) + isin(y)) = (e^x * cos(y)) + i(e^x * sin(y)) Fractint Version 18.2 Page 143 Appendix B Stone Soup With Pixels: The Authors THE STONE SOUP STORY Once upon a time, somewhere in Eastern Europe, there was a great famine. People jealously hoarded whatever food they could find, hiding it even from their friends and neighbors. One day a peddler drove his wagon into a village, sold a few of his wares, and began asking questions as if he planned to stay for the night. [No! No! It was three Russian Soldiers! - Lee Crocker] [Wait! I heard it was a Wandering Confessor! - Doug Quinn] [Well *my* kids have a book that uses Russian Soldiers! - Bert] [Look, who's writing this documentation, anyway? - Monte] [Ah, but who gets it *last* and gets to upload it? - Bert] "There's not a bite to eat in the whole province," he was told. "Better keep moving on." "Oh, I have everything I need," he said. "In fact, I was thinking of making some stone soup to share with all of you." He pulled an iron cauldron from his wagon, filled it with water, and built a fire under it. Then, with great ceremony, he drew an ordinary-looking stone from a velvet bag and dropped it into the water. By now, hearing the rumor of food, most of the villagers had come to the square or watched from their windows. As the peddler sniffed the "broth" and licked his lips in anticipation, hunger began to overcome their skepticism. "Ahh," the peddler said to himself rather loudly, "I do like a tasty stone soup. Of course, stone soup with CABBAGE -- that's hard to beat." Soon a villager approached hesitantly, holding a cabbage he'd retrieved from its hiding place, and added it to the pot. "Capital!" cried the peddler. "You know, I once had stone soup with cabbage and a bit of salt beef as well, and it was fit for a king." The village butcher managed to find some salt beef...and so it went, through potatoes, onions, carrots, mushrooms, and so on, until there was indeed a delicious meal for all. The villagers offered the peddler a great deal of money for the magic stone, but he refused to sell and traveled on the next day. And from that time on, long after the famine had ended, they reminisced about the finest soup they'd ever had. *** That's the way Fractint has grown, with quite a bit of magic, although without the element of deception. (You don't have to deceive programmers to make them think that hours of painstaking, often frustrating work is fun... they do it to themselves.) It wouldn't have happened, of course, without Benoit Mandelbrot and the explosion of interest in fractal graphics that has grown from his work at IBM. Or without the example of other Mandelplotters for the PC. Or without those wizards who first realized you could perform Mandelbrot Fractint Version 18.2 Page 144 calculations using integer math (it wasn't us - we just recognize good algorithms when we steal--uhh--see them). Or those graphics experts who hang around the Compuserve PICS forum and keep adding video modes to the program. Or... A WORD ABOUT THE AUTHORS Fractint is the result of a synergy between the main authors, many contributors, and published sources. All four of the main authors have had a hand in many aspects of the code. However, each author has certain areas of greater contribution and creativity. Since there is not room in the credits screen for the contributions of the main authors, we list these here to facilitate those who would like to communicate with us on particular subjects. Bert Tyler is the original author. He wrote the "blindingly fast" 386- specific 32 bit integer math code and the original video mode logic. Bert made Stone Soup possible, and provides a sense of direction when we need it. His forte is writing fast 80x86 assembler, his knowledge of a variety of video hardware, and his skill at hacking up the code we send him! Bert has a BA in mathematics from Cornell University. He has been in programming since he got a job at the computer center in his sophomore year at college - in other words, he hasn't done an honest day's work in his life. He has been known to pass himself off as a PC expert, a UNIX expert, a statistician, and even a financial modeling expert. He is currently masquerading as an independent PC consultant, supporting the PC-to-Mainframe communications environment at NIH. If you sent mail from the Internet to an NIH staffer on his 3+Mail system, it was probably Bert's code that mangled it during the Internet-to-3+Mail conversion. He also claims to support the MS-Kermit environment at NIH. Fractint is Bert's first effort at building a graphics program. Tim Wegner contributed the original implementation of palette animation, and is responsible for most of the 3D mechanisms. He provided the main outlines of the "StandardFractal" engine and data structures, and is accused by his cohorts of being "obsessed with options". Tim is quite proud of having originally integrated the 256 color super VGA modes in Fractint, especially since he knows almost nothing about it! Tim has BA and MA degrees in mathematics from Carleton College and the University of California Berkeley. He worked for 7 years overseas as a volunteer, doing things like working with Egyptian villagers building water systems. Since returning to the US in 1982, he has written shuttle navigation software, a software support environment prototype, and supported strategic information planning, all at NASA's Johnson Space Center. Mark Peterson invented the periodicity detection logic, several original fractal types, transcendental function libraries, alternate math implementations, the formula compiler, and the "Julibrot" intrinsic 3D fractals - in other words, most of the truly original ideas in Fractint! Fractint Version 18.2 Page 145 Mark's knowledge of higher mathematics and programming was achieved almost entirely through self-study. Mark has written several magazine articles on computer programming and is coauthor of a book on Fractint called Fractal Creations. Mark is also a free-lance computer consultant specializing in high performance applications. Pieter Branderhorst is a late-comer to the group who likes to distract the other authors with enhancements impacting at least half of the source at once. His contributions include super solid guessing, image rotation, resume, fast disk caching, and the new user interface. More than any of the authors, he has personally touched and massaged the entire source. Pieter left high school to work with computers, back when huge machines had 64k of core. He's been happily computing since, mostly programming and designing software from comms firmware to database and o/s, and anything between, and large scale online transaction processing applications. He has worked as a free-lance computer consultant (whatever that means) since 1983. DISTRIBUTION OF FRACTINT New versions of FRACTINT are uploaded to the CompuServe network, and make their way to other systems from that point. FRACTINT is available as two self-extracting archive files - FRAINT.EXE (executable & documentation) and FRASRC.EXE (source code). The latest version can always be found on CompuServe in the "Fractal Sources" library of the GRAPHDEV forum. If you're not a Compuserve subscriber, but wish to get more information about Compuserve and its graphics forums, feel free to call their 800 number (800-848-8199) and ask for operator number 229. If you don't have access to Compuserve, many other sites tend to carry these files shortly after their initial release (although sometimes using different naming conventions). For instance... If you speak Internet and FTP, SIMTEL20 and its various mirror sites tend to carry new versions of Fractint shortly after they are released. look in the PD:<MSDOS.GRAPHICS> directory for files named FRA*.*. Then again, if you don't speak Internet and FTP... Your favorite local BBS probably carries these files as well (although perhaps not the latest versions) using naming conventions like FRA*.ZIP. One BBS that *does* carry the latest version is the "Ideal Studies BBS" (508)757-1806, 1200/2400/9600HST. Peter Longo is the SYSOP and a true fractal fanatic. There is a very short registration, and thereafter the entire board is open to callers on the first call. Then again, if you don't even have a modem... Many Shareware/Freeware library services will ship you diskettes containing the latest versions of Fractint for a nominal fee that basically covers their cost of packaging and a small profit that we don't mind them making. One in particular is the Public (Software) Library, PO Box 35705, Houston, TX 77235-5705, USA. Their phone number is 800-242-4775 (outside the US, dial 713-524-6394). Ask for item #9112 Fractint Version 18.2 Page 146 for five 5.25" disks, #9113 for three 3.5" disks. Cost is $6.99 plus $4 S&H in the U.S./Canada, $11 S&H overseas. In Europe, the latest versions are available from another Fractint enthusiast, Jon Horner - Editor of FRAC'Cetera, a disk-based fractal/chaos resource. Disk prices for UK/Europe are: 5.25" HD BP4.50/5.00 : 3.5" HD BP (British Pounds) 5.00/5.50. Prices include p&p (airmail to Europe). Contact: Jon Horner, FRAC'Cetera, Le Mont Ardaine, Rue des Ardaines, St. Peters, Guernsey GY7 9EU, CI, UK. Phone (44) 0481 63689. CIS 100112,1700 The X Windows port of Fractint maintained by Ken Shirriff is available via FTP from sprite.berkeley.edu. CONTACTING THE AUTHORS Communication between the authors for development of the next version of Fractint takes place in GRAPHDEV (Graphics Developers) Section 4 (Fractal Sources) of CompuServe (CIS). Most of the authors have never met except on Compuserve. Access to the GRAPHDEV forum is open to any and all interested in computer generated fractals. New members are always welcome! Stop on by if you have any questions or just want to take a peek at what's getting tossed into the soup. Also, you'll find many GIF image files generated by fellow Fractint fans and many fractal programs as well in the GRAPHDEV forum's data library 5. If you're not a Compuserve subscriber, but wish to get more information about Compuserve and its graphics forums, feel free to call their 800 number (800-848-8199) and ask for operator number 229. The following authors have agreed to the distribution of their addresses. Usenet/Internet/Bitnet/Whatevernet users can reach CIS users directly if they know the user ID (i.e., Bert Tyler can be reached as 73477.433@compuserve.com). Just remember that CIS charges by the minute, so it costs us a little bit to read a message -- don't kill us with kindness. And don't send all your mail to Bert -- spread it around a little! Main authors (in historical order): Bert Tyler [73477,433] on CIS Tyler Software (which is also 73477.433@compuserve.com, if 124 Wooded Lane you're on the Internet - see above) Villanova, PA 19085 (215) 525-5478 Timothy Wegner [71320,675] on CIS 4714 Rockwood twegner@mitre.org on Internet Houston, TX 77004 (713) 747-7543 Fractint Version 18.2 Page 147 Mark Peterson [70441,3353] on CIS The Yankee Programmer 405-C Queen St., Suite #181 Southington, CT 06489 (203) 276-9721 Pieter Branderhorst [72611,2257] on CIS Amthor Computer Consultants 270 Moss St. Victoria, BC Canada, V8V 4M4 (604) 381-7164 Contributing authors (in alphabetic order); Joseph A Albrecht 9250 Old Cedar Ave #215 Bloomington, Mn 55425 (612) 884-3286 Kevin C Allen kevina@microsoft.com on Internet 9 Bowen Place Seven Hills NSW 2147 Australia +61-2-870-2297 (Work) (02) 831-4821 (Home) Rob Beyer [71021,2074] on CIS 23 Briarwood Lane Laguna Hills, CA, 92656 (714) 957-0227 (7-12pm PST & weekends) John W. Bridges (Author GRASP/Pictor, Imagetools, PICEM, VGAKIT) 2810 Serang Place Costa Mesa California 92626-4827 [75300,2137] on CIS, GENIE:JBRIDGES Juan J Buhler jbuhler@usina.org.ar Santa Fe 2227 1P "E" (54-1) 84 3528 Buenos Aires Argentina Michael D. Burkey burkey@sun9.math.utk.edu on Internet 6600 Crossgate Rd. Knoxville, TN 37912 Robin Bussell 13 Bayswater Rd Horfield Bristol Avon, England (044)-0272-514451 Fractint Version 18.2 Page 148 Lee Daniel Crocker [73407,2030] on CIS 5506 Camden Ave #D3 leecr@microsoft.com San Jose, CA 95124 (408) 267-2926 Monte Davis [71450,3542] on CIS 223 Vose Avenue South Orange, NJ 07079 (201) 378-3327 David Guenther [70531,3525] on CIS 50 Rockview Drive Irvine, CA 92715 Michael L. Kaufman kaufman@eecs.nwu.edu on INTERNET 2247 Ridge Ave, #2K (also accessible via EXEC-PC bbs) Evanston, IL, 60201 (708) 864-7916 Wesley Loewer loewer@largo.star.harc.edu on INTERNET 78 S. Circlewood Glen The Woodlands, TX 77381 (713) 292-3449 Adrian Mariano adrian@u.washington.edu on INTERNET 2729 72nd AVE SE Mercer Island, WA 98040 Joe McLain [75066,1257] on CIS McLain Imaging 2417 Venier Costa Mesa, CA 92627 (714) 642-5219 Bob Montgomery [73357,3140] on CIS (Author of VPIC) 132 Parsons Road Longwood, Fl 32779 Roy Murphy [76376,721] on CIS 9050 Ewing Ave. Evanston, IL 60203 Ethan Nagel [71062,3677] on CIS 4209 San Pedro NE #308 Albuquerque, NM 87109 (505) 884-7442 Jonathan Osuch [73277,1432] on CIS 2110 Northview Drive Marion, IA 52302 Marc Reinig [72410,77] on CIS 3415 Merrill Rd. marco@sun.com!daver!cypress on Usenet Aptos, CA. 95003 (408) 475-2132 Fractint Version 18.2 Page 149 Prof. JM Richard-Collard mpi@frmop53.bitnet on BitNet mpi@cnuvx1.cnusc.fr on Internet Lee H. Skinner [75450,3631] on CIS P.O. Box 14944 Albuquerque, NM 87191 (505) 293-5723 Dean Souleles [75115,1671] on CIS 8840 Collett Ave. Sepulveda, CA 91343 (818) 893-7558 Chris J Lusby Taylor 32 Turnpike Road Newbury, England Tel 011 44 635 33270 Scott Taylor [72401,410] on CIS 2913 Somerville Drive Apt #1 scott@bohemia.metronet.org on Internet Ft. Collins, Co 80526 DGWM18A on Prodigy (303) 221-1206 Paul Varner [73237,441] on CIS PO Box 930 Shepherdstown, WV 25443 (304) 876-2011 Phil Wilson [76247,3145] on CIS 410 State St., #55 Brooklyn, NY 11217 (718) 624-5272 Fractint Version 18.2 Page 150 Appendix C GIF Save File Format Since version 5.0, Fractint has had the <S>ave-to-disk command, which stores screen images in the extremely compact, flexible .GIF (Graphics Interchange Format) widely supported on Compuserve. Version 7.0 added the <R>estore-from-disk capability. Until version 14, Fractint saved images as .FRA files, which were a non- standard extension of the then-current GIF87a specification. The reason was that GIF87a did not offer a place to store the extra information needed by Fractint to implement the <R> feature -- i.e., the parameters that let you keep zooming, etc. as if the restored file had just been created in this session. The .FRA format worked with all of the popular GIF decoders that we tested, but these were not true GIF files. For one thing, information after the GIF terminator (which is where we put the extra info) has the potential to confuse the online GIF viewers used on Compuserve. For another, it is the opinion of some GIF developers that the addition of this extra information violates the GIF87a spec. That's why we used the default filetype .FRA instead. Since version 14, Fractint has used a genuine .GIF format, using the GIF89a spec - an upwardly compatible extension of GIF87a, released by Compuserve on August 1 1990. This new spec allows the placement of application data within "extension blocks". In version 14 we changed our default savename extension from .FRA to .GIF. There is one significant advantage to the new GIF89a format compared to the old GIF87a-based .FRA format for Fractint purposes: the new .GIF files may be uploaded to the Compuserve graphics forums (such as Fractint's home forum, GRAPHDEV) with fractal information intact. Therefore anyone downloading a Fractint image from Compuserve will also be downloading all the information needed to regenerate the image. Fractint can still read .FRA files generated by earlier versions. If for some reason you wish to save files in the older GIF87a format, for example because your favorite GIF decoder has not yet been upgraded to GIF89a, use the command-line parameter "GIF87a=yes". Then any saved files will use the original GIF87a format without any application- specific information. An easy way to convert an older .FRA file into true .GIF format suitable for uploading is something like this at the DOS prompt: FRACTINT MYFILE.FRA SAVENAME=MYFILE.GIF BATCH=YES Fractint will load MYFILE.FRA, save it in true .GIF format as MYFILE.GIF, and return to DOS. GIF and "Graphics Interchange Format" are trademarks of Compuserve Incorporated, an H&R Block Company. Fractint Version 18.2 Page 151 Appendix D Other Fractal Products Fractint Version 18.2 Page 152 Appendix E Bibliography BARNSLEY, Michael: "Fractals Everywhere," Academic Press, 1988. DEWDNEY, A. K., "Computer Recreations" columns in "Scientific American" -- 8/85, 7/87, 11/87, 12/88, 7/89. FEDER, Jens: "Fractals," Plenum, 1988. Quite technical, with good coverage of applications in fluid percolation, game theory, and other areas. GLEICK, James: "Chaos: Making a New Science," Viking Press, 1987. The best non-technical account of the revolution in our understanding of dynamical systems and its connections with fractal geometry. MANDELBROT, Benoit: "The Fractal Geometry of Nature," W. H. Freeman & Co., 1982. An even more revised and expanded version of the 1977 work. A rich and sometimes confusing stew of formal and informal mathematics, the prehistory of fractal geometry, and everything else. Best taken in small doses. MANDELBROT, Benoit: "Fractals: Form, Chance, and Dimension," W. H. Freeman & Co., 1977. A much revised translation of "Les objets fractals: forme, hasard, et dimension," Flammarion, 1975. PEITGEN, Heinz-Otto & RICHTER, Peter: "The Beauty of Fractals," Springer-Verlag, 1986. THE coffee-table book of fractal images, knowledgeable on computer graphics as well as the mathematics they portray. PEITGEN, Heinz-Otto & SAUPE, Ditmar: "The Science of Fractal Images," Springer-Verlag, 1988. A fantastic work, with a few nice pictures, but mostly filled with *equations*!!! PICKOVER, Clifford: "Computers, Pattern, Chaos, and Beauty," St. Martin's Press, 1990. SCHROEDER, Manfred: "Fractals, Chaos, Power Laws," W. H. Freeman & Co., 1991. WEGNER, Timothy & PETERSON, Mark: "Fractal Creations," Waite Group Press, 1991 (second edition, by Wegner and Tyler, due in the fall of 1993). If we tell you how *wonderful* this book is you might think we were bragging, so let's just call it: THE definitive companion to Fractint! WEGNER, Timothy & PETERSON, Mark & TYLER, Bert, & Branderhorst, Pieter: "Fractals for Windows," Waite Group Press, 1992. This book is to Winfract (the Windows version of Fractint) what "Fractals for Windows" is to Fractint. Fractint Version 18.2 Page 153 Appendix F Other Programs WINFRACT. Bert Tyler has ported Fractint to run under Windows 3! The same underlying code is used, with a Windows user interface. Winfract has almost all the functionality of Fractint - the biggest difference is the absence of a zillion weird video modes. Fractint for DOS will continue to be the definitive version. Winfract is available from CompuServe in GRAPHDEV Lib 4, as WINFRA.ZIP (executable) and WINSRC.ZIP (source). PICLAB, by Lee Crocker - a freeware image manipulation utility available from Compuserve in PICS Lib 10, as PICLAB.EXE. PICLAB can do very sophisticated resizing and color manipulation of GIF and TGA files. It can be used to reduce 24 bit TGA files generated with the Fractint "lightname" option to GIF files. FDESIGN, by Doug Nelson (CIS ID 70431,3374) - a freeware IFS fractal generator available from Compuserve in GRAPHDEV Lib 4, and probably on your local BBS. This program requires a VGA adapter and a Microsoft- compatible mouse, and a floating point coprocessor is highly recommended. It generates IFS fractals in a *much* more intuitive fashion than Fractint. It can also (beginning with version 3.0) save its IFS formulas in Fractint-style .IFS files. ACROSPIN, by David Parker - An inexpensive commercial program that reads an object definition file and creates images that can be rapidly rotated in three dimensions. The Fractint "orbitsave=yes" option creates files that this program can read for orbit-type fractals and IFS fractals. Contact: David Parker 801-966-2580 P O Box 26871 800-227-6248 Salt Lake City, UT 84126-0871 Fractint Version 18.2 Page 154 Appendix G Revision History Version 17.2, 3/92 - Fixed a bug which caused Fractint to hang when a Continuous Potential Bailout value was set (using the 'Y') screen and then the 'Z' screen was activated. - fixed a bug which caused "batch=yes" runs to abort whenever any key was pressed. - bug-fixes in the Stereo3D/Targa logic from Marc Reinig. - Fractint now works correctly again on FPU-less 8088s when zoomed deeply into the Mandelbrot/Julia sets - The current image is no longer marked as "not resumable" on a Shell-To-Dos ("D") command. - fixed a bug which prevented the "help" functions from working properly during fractal-type selection for some fractal types. Version 17.1, 3/92 - fixed a bug which caused PCs with no FPU to lock up when they attempted to use some fractal types. - fixed a color-cycling bug which caused the palette to single-step when you pressed ESCAPE to exit color-cycling. - fixed the action of the '<' and '>' keys during color-cycling. Version 17.0, 2/92 - New fractal types (but of course!): Lyapunov Fractals from Roy Murphy (see Lyapunov Fractals (p. 57) for details) 'BifStewart' (Stewart Map bifurcation) fractal type and new bifurcation parameters (filter cycles, seed population) from Kevin Allen. Lorenz3d1, Lorenz3d2, and Lorenz3d3 fractal types from Scott Taylor. Note that a bug in the Lorenz3d1 fractal prevents zooming-out from working with it at the moment. Martin, Circle, and Hopalong (culled from Dewdney's Scientific American Article) Lots of new entries in fractint.par. New ".L" files (TILING.L, PENROSE.L) New 'rand()' function added to the 'type=formula' parser - New fractal generation options: New 'Tesseral' calculation algorithm (use the 'X' option list to select it) from Chris Lusby Taylor. Fractint Version 18.2 Page 155 New 'Fillcolor=' option shows off Boundary Tracing and Tesseral structure inside=epscross and inside=startrail options taken from a paper by Kenneth Hooper, with credit also to Clifford Pickover New Color Postscript Printer support from Scott Taylor. Sound= command now works with <O>rbits and <R>ead commands. New 'orbitdelay' option in X-screen and command-line interface New "showdot=nn" command-line option that displays the pixel currently being worked on using the specified color value (useful for those lloooonngg images being calculated using solid guessing - "where is it now?"). New 'exitnoask=yes' commandline/SSTOOLS.INI option to avoid the final "are you sure?" screen New plasma-cloud options. The interface at the moment (documented here and here only because it might change later) lets you: - use an alternate drawing algorithm that gives you an earlier preview of the finished image. - re-generate your favorite plasma cloud (say, at a higher resolution) by forcing a re-select of the random seed. New 'N' (negative palette) option from Scott Taylor - the documentation at this point is: Pressing 'N' while in the palette editor will invert each color. It will convert only the current color if it is in 'x' mode, a range if in 'y' mode, and every color if not in either the 'x' or 'y' mode. - Speedups: New, faster floating-point Mandelbrot/Julia set code from Wesley Loewer, Frank Fussenegger and Chris Lusby Taylor (in separate contributions). Faster non-386 integer Mandelbrot code from Chris Lusby Taylor, Mike Gelvin and Bill Townsend (in separate contributions) New integer Lsystems logic from Nicholas Wilt Finite-Attractor fixups and Lambda/mandellambda speedups from Kevin Allen. GIF Decoder speedups from Mike Gelvin - Bug-fixes and other enhancements: Fractint now works with 8088-based AMSTRAD computers. The video logic is improved so that (we think) fewer video boards will need "textsafe=save" for correct operation. Fractint Version 18.2 Page 156 Fixed a bug in the VESA interface which effectively messed up adapters with unusual VESA-style access, such as STB's S3 chipset. Fixed a color-cycling bug that would at times restore the wrong colors to your image if you exited out of color-cycling, displayed a 'help' screen, and then returned to the image. Fixed the XGA video logic so that its 256-color modes use the same default 256 colors as the VGA adapter's 320x200x256 mode. Fixed the 3D bug that caused bright spots on surfaces to show as black blotches of color 0 when using a light source. Fixed an image-generation bug that sometimes caused image regeneration to restart even if not required if the image had been zoomed in to the point that floating-point had been automatically activated. Added autodetection and 640x480x256 support for the Compaq Advanced VGA Systems board - I wonder if it works? Added VGA register-compatible 320x240x256 video mode. Fixed the "logmap=yes" option to (again) take effect for continuous potential images. This was broken in version 15.x. The colors for the floating-point algorithm of the Julia fractal now match the colors for the integer algorithm. If the GIF Encoder (the "Save" command) runs out of disk space, it now tells you about it. If you select both the boundary-tracing algorithm and either "inside=0" or "outside=0", the algorithm will now give you an error message instead of silently failing. Updated 3D logic from Marc Reinig. Minor changes to permit IFS3D fractal types to be handled properly using the "B" command. Minor changes to the "Obtaining the latest Source" section to refer to BBS access (Peter Longo's) and mailed diskettes (the Public (Software) Library). Version 16.12, 8/91 Fix to cure some video problems reported with Amstrad 8088/8086-based PCs. Version 16.11, 7/91 SuperVGA Autodetect fixed for older Tseng 3000 adapters. Fractint Version 18.2 Page 157 New "adapter=" options to force the selection of specific SuperVGA adapter types. See Video Parameters (p. 101) for details. Integer/Floating-Point math toggle is changed only temporarily if floating-point math is forced due to deep zooming. Fractint now survives being modified by McAfee's "SCAN /AV" option. Bug Fixes for Acrospin interface, 3D "Light Source Before Transformation" fill type, and GIF decoder. New options in the <Z> parameters screen allow you to directly enter image coordinates. New "inside=zmag" and "outside=real|imag|mult|summ" options. The GIF Decoder now survives reading GIF files with a local color map. Improved IIT Math Coprocessor support. New color-cycling single-step options, '<' and '>'. Version 16.0, 6/91 Integrated online help / fractint.doc system from Ethan Nagel. To create a printable fractint.doc file see Startup Parameters (p. 92) . Over 350 screens of online help! Try pressing <F1> just about anywhere! New "autokey" feature. Type "demo" to run the included demo.bat and demo.key files for a great demonstration of Fractint. See Autokey Mode (p. 66) for details. New <@> command executes a saved set of commands. The command has changed to write the current image's parameters as a named set of commands in a structured file. Saved sets of commands can subsequently be executed with the <@> command. See Parameter Save/Restore Commands (p. 22). A default "fractint.par" file is included with the release. New <z> command allows changing fractal type-specific parameters without going back through the <t> (fractal type selection) screen. Ray tracer interface from Marc Reinig, generates 3d transform output for a number of ray tracers; see "Interfacing with Ray Tracing Programs" (p. 87) Selection of video modes and structure of "fractint.cfg" have changed. If you have a customized fractint.cfg file, you'll have to rebuild it based on this release's version. You can customize the assignment of your favorite video modes to function keys; see Video Mode Function Keys (p. 27). <delete> is a new command key which goes directly to video mode selection. Fractint Version 18.2 Page 158 New "cyclerange" option (command line and <y> options screen) from Hugh Steele. Limits color cycling to a specified range of colors. Improved Distance Estimator Method (p. 68) algorithm from Phil Wilson. New "ranges=" option from Norman Hills. See Logarithmic Palettes and Color Ranges (p. 71) for details. type=formula definitions can use "variable functions" to select sin, cos, sinh, cosh, exp, log, etc at run time; new built-ins tan, tanh, cotan, cotanh, and flip are available with type=formula; see Type Formula (p. 51) New <w> command in palette editing mode to convert image to greyscale All "fn" fractal types (e.g. fn*fn) can now use new functions tan, tanh, cotan, cotanh, recip, and ident; bug in prior cos function fixed, new function cosxx (conjugate of cos) is the old erroneous cos calculation New L-Systems from Herb Savage New IFS types from Alex Matulich Many new formulas in fractint.frm, including a large group from JM Richard-Collard Generalized type manzpwr with complex exponent per Lee Skinner's request Initial orbit parameter added to Gingerbreadman fractal type New color maps (neon, royal, volcano, blues, headache) from Daniel Egnor IFS type has changed to use a single file containing named entries (instead of a separate xxx.ifs file per type); the <z> command brings up IFS editor (used to be command). See Barnsley IFS Fractals (p. 39). Much improved support for PaintJet printers; see PaintJet Parameters (p. 107) From Scott Taylor: Support for plotters using HP-GL; see Plotter Parameters (p. 108) Lots of new PostScript halftones; see PostScript Parameters (p. 105) "printer=PS[L]/0/..." for full page PostScript; see PostScript Parameters (p. 105) Option to drive printer ports directly (faster); see Printer Parameters (p. 104) Option to change printer end of line control chars; see Printer Parameters (p. 104) Support for XGA video adapter Support for Targa+ video adapter 16 color VGA mode enhancements: Now use the first 16 colors of .map files to be more predictable Palette editor now works with these modes Color cycling now works properly with these modes Targa video Fractint Version 18.2 Page 159 adapter fixes; Fractint now uses (and requires) the "targa" and "targaset" environment variables for Targa systems "vesadetect=no" parameter to bypass use of VESA video driver; try this if you encounter video problems with a VESA driver Upgraded video adapter detect and handling from John Bridges; autodetect added for NCR, Trident 8900, Tseng 4000, Genoa (this code is from a beta release of VGAKIT, we're not sure it all works yet) Zoom box is included in saved/printed images (but, is not recognized as anything special when such an image is restored) The colors numbers reserved by the palette editor are now selectable with the new <v> palette editing mode command Option to use IIT floating point chip's special matrix arithmetic for faster 3D transforms; see "fpu=" in Startup Parameters (p. 92) Disk video cache increased to 64k; disk video does less seeking when running to real disk Faster floating point code for 287 and higher fpus, for types mandel, julia, barnsleyj1/m1/j2/m2, lambda, manowar, from Chuck Ebbert "filename=.xxx" can be used to set default <r> function file mask Selection of type formula or lsys now goes directly to entry selection (file selection step is now skipped); to change to a different file, use <F6> from the entry selection screen Three new values have been added to the textcolors= parameter; if you use this parameter you should update it by inserting values for the new 6th, 7th, 9th, and 13th positions; see "textcolors=" in Color Parameters (p. 96) The formula type's imag() function has changed to return the result as a real number Fractal type-specific parameters (entered after selecting a new fractal type with <T>) now restart at their default values each time you select a new fractal type Floating point input fields can now be entered in scientific notation (e.g. 11.234e-20). Entering the letters "e" and "p" in the first column causes the numbers e=2.71828... and pi=3.14159... to be entered. New option "orbitsave=yes" to create files for Acrospin for some types (see Barnsley IFS Fractals (p. 39), Orbit Fractals (p. 46), Acrospin (p. 153)) Bug fixes: Problem with Hercules adapter auto-detection repaired. Problems with VESA video adapters repaired (we're not sure we've got them all yet...) 3D transforms fixed to work at high resolutions (> 1000 dots). 3D parameters no longer clobbered when restoring non-3D images. L-Systems fixed to not crash when order too high for available Fractint Version 18.2 Page 160 memory. PostScript EPS file fixes. Bad leftmost pixels with floating point at 2048 dot resolution fixed. 3D transforms fixed to use current <x> screen float/integer setting. Restore of images using inversion fixed. Error in "cos" function (used with "fn" type fractals) fixed; prior incorrect function still available as "cosxx" for compatibility Old 3D=nn/nn/nn/... form of 3D transform parameters no longer supported Fractint source code now Microsoft C6.00A compatible. Version 15.11, 3/91, companion to Fractal Creations, not for general release Autokey feature, IIT fpu support, and some bug fixes publicly released in version 16. Version 15 and 15.1, 12/90 New user interface! Enjoy! Some key assignments have changed and some have been removed. New palette editing from Ethan Nagel. Reduced memory requirements - Fractint now uses overlays and will run on a 512K machine. New <v>iew command: use to get small window for fast preview, or to setup an image which will eventually be rendered on hard copy with different aspect ratio L-System fractal type from Adrian Mariano Postscript printer support from Scott Taylor Better Tandy video support and faster CGA video from Joseph A Albrecht 16 bit continuous potential files have changed considerably; see the Continuous Potential section for details. Continuous potential is now resumable. Mandelbrot calculation is faster again (thanks to Mike Gelvin) - double speed in 8086 32 bit case Compressed log palette and sqrt palette from Chuck Ebbert Calculation automatically resumes whenever current image is resumable and is not paused for a visible reason. Auto increment of savename changed to be more predictable New video modes: trident 1024x768x256 mode 320x480x256 tweak mode (good for reduced 640x480 viewing) changed NEC GB-1, hopefully it works now Integer mandelbrot and julia now work with periodicitycheck Initial zoombox color auto-picked for better contrast (usually) New adapter=cga|ega|mcga|vga for systems having trouble with auto- detect New textsafe=no|yes for systems having trouble with garbled text mode <r> and <3> commands now present list of video modes to pick from; <r> can reduce a non-standard or unviewable image size. Diffusion fractal type is now resumable after interrupt/save Exitmode=n parameter, sets video mode to n when exiting from fractint Fractint Version 18.2 Page 161 When savetime is used with 1 and 2 pass and solid guessing, saves are deferred till the beginning of a new row, so that no calculation time is lost. 3d photographer's mode now allows the first image to be saved to disk textcolors=mono|12/34/56/... -- allows setting user interface colors Code (again!) compilable under TC++ (we think!) .TIW files (from v9.3) are no longer supported as input to 3D transformations bug fixes: multiple restores (msc 6.0, fixed in 14.0r) repeating 3d loads problem; slow 3d loads of images with float=yes map= is now a real substitute for default colors starfield and julibrot no longer cause permanent color map replacement starfield parameters bug fix - if you couldn't get the starfield parameters to do anything interesting before, try again with this release Newton and newtbasin orbit display fixed Version 15.1: Fixed startup and text screen problems on systems with VESA compliant video adapters. New textsafe=save|bios options. Fixes for EGA with monochrome monitor, and for Hercules Graphics Card. Both should now be auto-detected and operate correctly in text modes. Options adapter=egamono and adapter=hgc added. Fixed color L-Systems to not use color 0 (black). PostScript printing fix. Version 14, 8/90 LAST MINUTE NEWS FLASH! Compuserve announces the GIF89a on August 1, 1990, and Fractint supports it on August 2! GIF files can now contain fractal information! Fractint now saves its files in the new GIF89a format by default, and uses .GIF rather than .FRA as a default filetype. Note that Fractint still *looks* for a .FRA file on file restores if it can't find a .GIF file, and can be coerced into using the old GIF87a format with the new 'gif87a=yes' command-line option. Pieter Branderhorst mounted a major campaign to get his name in lights: Mouse interface: Diagonals, faster movement, improved feel. Mouse button assignments have changed - see the online help. Zoom box enhancements: The zoom box can be rotated, stretched, skewed, and panned partially offscreen. See "More Zoom Box Commands". FINALLY!! You asked for it and we (eventually, by talking Pieter into it [actually he grabbed it]) did it! Images can be saved before completion, for a subsequent restore and continue. See "Interrupting and Resuming" and "Batch Mode". Off-center symmetry: Fractint now takes advantage of x or y axis symmetry anywhere on the screen to reduce drawing time. Panning: If you move an image up, down, left, or right, and don't change anything else, only the new edges are calculated. Fractint Version 18.2 Page 162 Disk-video caching - it is now possible, reasonable even, to do most things with disk video, including solid guessing, 3d, and plasma. Logarithmic palette changed to use all colors. It now matches regular palette except near the "lake". "logmap=old" gets the old way. New "savetime=nnn" parameter to save checkpoints during long calculations. Calculation time is shown in <Tab> display. Kevin C Allen Finite Attractor, Bifurcation Engine, Magnetic fractals... Made Bifurcation/Verhulst into a generalized Fractal Engine (like StandardFractal, but for Bifurcation types), and implemented periodicity checking for Bifurcation types to speed them up. Added Integer version of Verhulst Bifurcation (lots faster now). Integer is the default. The Floating-Point toggle works, too. Added NEW Fractal types BIFLAMBDA, BIF+SINPI, and BIF=SINPI. These are Bifurcation types that make use of the new Engine. Floating- point/Integer toggle is available for BIFLAMBDA. The SINPI types are Floating-Point only, at this time. Corrected the generation of the MandelLambda Set. Sorry, but it's always been wrong (up to v 12, at least). Ask Mandelbrot ! Added NEW Fractal types MAGNET1M, MAGNET1J, MAGNET2M, MAGNET2J from "The Beauty of Fractals". Floating-Point only, so far, but what do you expect with THESE formulae ?! Added new symmetry types XAXIS NOIMAG and XAXIS NOREAL, required by the new MAGNETic Fractal types. Added Finite Attractor Bailout (FAB) logic to detect when iterations are approaching a known finite attractor. This is required by the new MAGNETic Fractal types. Added Finite Attractor Detection (FAD) logic which can be used by *SOME* Julia types prior to generating an image, to test for finite attractors, and find their values, for use by FAB logic. Can be used by the new MAGNETic Fractal Types, Lambda Sets, and some other Julia types too. Mike Burkey sent us new tweaked video modes: VGA - 400x600x256 376x564x256 400x564x256 ATI VGA - 832x612x256 New HP Paintjet support from Chris Martin New "FUNCTION=" command to allow substition of different transcendental functions for variables in types (allows one type with four of these variables to represent 7*7*7*7 different types! ALL KINDS of new fractal types, some using "FUNCTION=": fn(z*z), fn*fn, fn*z+z, fn+fn, sqr(1/fn), sqr(fn), spider, tetrate, and Manowar. Most of these are generalizations of formula fractal types contributed by Scott Taylor and Lee Skinner. Distance Estimator logic can now be applied to many fractal types using distest= option. The types "demm" and "demj" have been replaced by "type=mandel distest=nnn" and "type=julia distest=nnn" Added extended memory support for diskvideo thanks to Paul Varner Added support for "center and magnification" format for corners. Color 0 is no longer generated except when specifically requested with inside= or outside=. Formula name is now included in <Tab> display and in <S>aved images. Bug fixes - formula type and diskvideo, batch file outside=-1 problem. Now you can produce your favorite fractal terrains in full color instead of boring old monochrome! Use the fullcolor option in 3d! Fractint Version 18.2 Page 163 Along with a few new 3D options. New "INITORBIT=" command to allow alternate Mandelbrot set orbit initialization. Version 13.0, 5/90 F1 was made the help key. Use F1 for help Use F9 for EGA 320x200x16 video mode Use CF4 for EGA 640x200x16 mode (if anybody uses that mode) Super-Solid-guessing (three or more passes) from Pieter Branderhorst (replaces the old solid-guessing mode) Boundary Tracing option from David Guenther ("fractint passes=btm", or use the new 'x' options screen) "outside=nnn" option sets all points not "inside" the fractal to color "nnn" (and generates a two-color image). 'x' option from the main menu brings up a full-screen menu of many popular options and toggle switches "Speed Key" feature for fractal type selection (either use the cursor keys for point-and-shoot, or just start typing the name of your favorite fractal type) "Attractor" fractals (Henon, Rossler, Pickover, Gingerbread) Diffusion fractal type by Adrian Mariano "type=formula" formulas from Scott Taylor and Lee H. Skinner. "sound=" options for attractor fractals. Sound=x plays speaker tones according to the 'x' attractor value Sound=y plays speaker tones according to the 'y' attractor value. Sound=z plays speaker tones according to the 'z' attractor value (These options are best invoked with the floating-point algorithm flag set.) "hertz=" option for adjusting the "sound=x/y/z" output. Printer support for color printers (printer=color) from Kurt Sowa Trident 4000 and Oak Technologies SuperVGA support from John Bridges Improved 8514/A support (the zoom-box keeps up with the cursor keys now!) Tandy 1000 640x200x16 mode from Brian Corbino (which does not, as yet, work with the F1(help) and TAB functions) The Julibrot fractal type and the Starmap option now automatically verify that they have been selected with a 256-color palette, and search for, and use, the appropriate GLASSESn.MAP or ALTERN.MAP palette map when invoked. *You* were supposed to be doing that manually all along, but *you* probably never read the docs, huh? Bug Fixes: TAB key now works after R(estore) commands PS/2 Model 30 (MCGA) adapters should be able to select 320x200x256 mode again (we think) Everex video adapters should work with the Autodetect modes again (we think) Version 12.0, 3/90 New SuperVGA Autodetecting and VESA Video modes (you tell us the resolution you want, and we'll figure out how to do it) New Full-Screen Entry for most prompting New Fractal formula interpreter ('type=formula') - roll your own Fractint Version 18.2 Page 164 fractals without using a "C" compiler! New 'Julibrot' fractal type Added floating point option to all remaining fractal types. Real (funny glasses) 3D - Now with "real-time" lorenz3D!! Non-Destructive <TAB> - Check out what your fractal parameters are without stopping the generation of a fractal image New Cross-Hair mode for changing individual palette colors (VGA only) Zooming beyond the limits of Integer algorithms (with automatic switchover to a floating-point algorithm when you zoom in "too far") New 'inside=bof60', 'inside=bof61' ("Beauty of Fractals, Page nn") options New starmap ('a' - for astrology? astronomy?) transformation option Restrictions on the options available when using Expanded Memory "Disk/RAM" video mode have been removed And a lot of other nice little clean-up features that we've already forgotten that we've added... Added capability to create 3D projection images (just barely) for people with 2 or 4 color video boards. Version 11.0, 1/90 More fractal types mandelsinh/lambdasinh mandelcosh/lambdacosh mansinzsqrd/julsinzsqrd mansinexp/julsinexp manzzprw/julzzpwr manzpower/julzpower lorenz (from Rob Beyer) lorenz3d complexnewton complexbasin dynamic popcorn Most fractal types given an integer and a floating point algorithm. "Float=yes" option now determines whether integer or floating-point algorithms are used for most fractal types. "F" command toggles the use of floating-point algorithms, flagged in the <Tab> status display 8/16/32/../256-Way decomposition option (from Richard Finegold) "Biomorph=", "bailout=", "symmetry=" and "askvideo=" options "T(ransform)" option in the IFS editor lets you select 3D options (used with the Lorenz3D fractal type) The "T(ype)" command uses a new "Point-and-Shoot" method of selecting fractal types rather than prompting you for a type name Bug fixes to continuous-potential algorithm on integer fractals, GIF encoder, and IFS editor Version 10.0, 11/89 Barnsley IFS type (Rob Beyer) Barnsley IFS3D type MandelSine/Cos/Exp type MandelLambda/MarksLambda/Unity type BarnsleyM1/J1/M2/J2/M3/J3 type Mandel4/Julia4 type Sierpinski gasket type Demm/Demj and bifurcation types (Phil Wilson), "test" is "mandel" again nversion command for most fractal types <Q>uaternary decomposition toggle and "DECOMP=" argument Fractint Version 18.2 Page 165 <E>ditor for Barnsley IFS parameters Command-line options for 3D parameters Spherical 3D calculations 5x faster 3D now clips properly to screen edges and works at extreme perspective "RSEED=" argument for reproducible plasma clouds Faster plasma clouds (by 40% on a 386) Sensitivity to "continuous potential" algorithm for all types except plasma and IFS Palette-map <S>ave and Restore (<M>) commands <L>ogarithmic and <N>ormal palette-mapping commands and arguments Maxiter increased to 32,000 to support log palette maps .MAP and .IFS files can now reside anywhere along the DOS path Direct-video support for Hercules adapters (Dean Souleles) Tandy 1000 160x200x16 mode (Tom Price) 320x400x256 register-compatible-VGA "tweaked" mode ATI VGA Wonder 1024x768x16 direct-video mode (Mark Peterson) 1024x768x16 direct-video mode for all supported chipsets Tseng 640x400x256 mode "Roll-your-own" video mode 19 New video-table "hot-keys" eliminate need for enhanced keyboard to access later entries Version 9.3, 8/89 rint command and "PRINTER=" argument (Matt Saucier) 8514/A video modes (Kyle Powell) SSTOOLS.INI sensitivity and '@THISFILE' argument Continuous-potential algorithm for Mandelbrot/Julia sets Light source 3D option for all fractal types "Distance estimator" M/J method (Phil Wilson) implemented as "test" type LambdaCosine and LambdaExponent types Color cycling mode for 640x350x16 EGA adapters Plasma clouds for 16-color and 4-color video modes Improved TARGA support (Joe McLain) CGA modes now use direct-video read/writes Tandy 1000 320x200x16 and 640x200x4 modes (Tom Price) TRIDENT chip-set super-VGA video modes (Lew Ramsey) Direct-access video modes for TRIDENT, Chips & Technologies, and ATI VGA WONDER adapters (John Bridges). and, unlike version 9.1, they WORK in version 9.3!) "zoom-out" (<Ctrl><Enter>) command <D>os command for shelling out 2/4/16-color Disk/RAM video mode capability and 2-color video modes supporting full-page printer graphics "INSIDE=-1" option (treated dynamically as "INSIDE=maxiter") Improved <H>elp and sound routines (even a "SOUND=off" argument) Turbo-C and TASM compatibility (really! Would we lie to you?) Version 8.1, 6/89 <3>D restore-from-disk and 3D <O>verlay commands, "3D=" argument Fast Newton algorithm including inversion option (Lee Crocker) 16-bit Mandelbrot/Julia logic for 386-class speed with non-386 PCs on Fractint Version 18.2 Page 166 "large" images (Mark Peterson) Restore now loads .GIF files (as plasma clouds) TARGA video modes and color-map file options (Joe McLain) 30 new color-cycling palette options (<Shft><F1> to <Alt><F10>) "Disk-video, RAM-video, EMS-video" modes Lambda sets now use integer math (with 80386 speedups) "WARN=yes" argument to prevent over-writing old .GIF files Version 7.0, 4/89 Restore from disk (from prior save-to-disk using v. 7.0 or later) New types: Newton, Lambda, Mandelfp, Juliafp, Plasma, Lambdasine Many new color-cycling options (for VGA adapters only) New periodicity logic (Mark Peterson) Initial displays recognize (and use) symmetry Solid-guessing option (now the default) Context-sensitive <H>elp Customizable video mode configuration file (FRACTINT.CFG) "Batch mode" option Improved super-VGA support (with direct video read/writes) Non-standard 360 x 480 x 256 color mode on a STANDARD IBM VGA! Version 6.0, 2/89 32-bit integer math emulated for non-386 processors; FRACT386 renamed FRACTINT More video modes Version 5.1, 1/89 Save to disk New! Improved! (and Incompatible!) optional arguments format "Correct" initial image aspect ratio More video modes Version 4.0, 12/88 Mouse support (Mike Kaufman) Dynamic iteration limits Color cycling Dual-pass mode More video modes, including "tweaked" modes for IBM VGA and register- compatible adapters Version 3.1, 11/88 Julia sets Fractint Version 18.2 Page 167 Version 2.1, 10/23/88 (the "debut" on CIS) Video table CPU type detector Version 2.0, 10/10/88 Zoom and pan Version 1.0, 9/88 The original, blindingly fast, 386-specific 32-bit integer algorithm Fractint Version 18.2 Page 168 Appendix H Version13 to Version 14 Type Mapping A number of types in Fractint version 13 and earlier were generalized in version 14. We added a "backward compatibility" hook that (hopefully) automatically translates these to the new form when the old files are read. Files may be converted via: FRACTINT OLDFILE.FRA SAVENAME=NEWFILE.GIF BATCH=YES In a few cases the biomorph flag was incorrectly set in older files. In that case, add "biomorph=no" to the command line. This procedure can also be used to convert any *.fra file to the new GIF89a spec, which now allows storage of fractal information. TYPES CHANGED FROM VERSION 13 - V13 NAME V14 NAME + PARAMETERS -------- -------------------------------------- LOGMAP=YES LOGMAP=OLD for identical Logmap type DEMJ JULIA DISTEST=nnn DEMM MANDEL DISTEST=nnn Note: DISTEST also available on many other types MANSINEXP MANFN+EXP FUNCTION=SIN Note: New functions for this type are cos sinh cosh exp log sqr JULSINEXP JULFN+EXP FUNCTION=SIN Note: New functions for this type are cos sinh cosh exp log sqr MANSINZSQRD MANFN+ZSQRD FUNCTION=SQR/SIN Note: New functions for this type are cos sinh cosh exp log sqr JULSINZSQRD JULFN+ZSQRD FUNCTION=SQR/SIN Note: New functions for this type are cos sinh cosh exp log sqr LAMBDACOS LAMBDAFN FUNCTION=COS LAMBDACOSH LAMBDAFN FUNCTION=COSH Fractint Version 18.2 Page 169 LAMBDAEXP LAMBDAFN FUNCTION=EXP LAMBDASINE LAMBDAFN FUNCTION=SIN LAMBDASINH LAMBDAFN FUNCTION=SINH Note: New functions for this type are log sqr MANDELCOS MANDELFN FUNCTION=COS MANDELCOSH MANDELFN FUNCTION=COSH MANDELEXP MANDELFN FUNCTION=EXP MANDELSINE MANDELFN FUNCTION=SIN MANDELSINH MANDELFN FUNCTION=SINH Note: New functions for this type are log sqr MANDELLAMBDA MANDELLAMBDA INITORBIT=PIXEL POPCORN SYMMETRY=NONE POPCORNJUL ------------------------------------------------------------- Formulas from FRACTINT.FRM in version 13 MANDELGLASS MANDELLAMBDA INITORBIT=.5/0 INVMANDEL V13 divide bug may cause some image differences. NEWTON4 V13 divide bug may cause some image differences. SPIDER V13 divide bug may cause some image differences. MANDELSINE MANDELFN FUNCTION=SIN BAILOUT=50 MANDELCOSINE MANDELFN FUNCTION=COS BAILOUT=50 MANDELHYPSINE MANDELFN FUNCTION=SINH BAILOUT=50 MANDELHYPCOSINE MANDELFN FUNCTION=COSH BAILOUT=50 SCOTTSIN PARAMS=nnn FN+FN FUNCTION=SIN/SQR BAILOUT=nnn+3 SCOTTSINH PARAMS=nnn FN+FN FUNCTION=SINH/SQR BAILOUT=nnn+3 SCOTTCOS PARAMS=nnn FN+FN FUNCTION=COS/SQR BAILOUT=nnn+3 SCOTTCOSH PARAMS=nnn FN+FN FUNCTION=COSH/SQR BAILOUT=nnn+3 Fractint Version 18.2 Page 170 SCOTTLPC PARAMS=nnn FN+FN FUNCTION=LOG/COS BAILOUT=nnn+3 SCOTTLPS PARAMS=nnn FN+FN FUNCTION=LOG/SIN BAILOUT=nnn+3 Note: New functions for this type are sin/sin sin/cos sin/sinh sin/cosh sin/exp cos/cos cos/sinh cos/cosh cos/exp sinh/sinh sinh/cosh sinh/exp sinh/log cosh/cosh cosh/exp cosh/log exp/exp exp/log exp/sqr log/log log/sqr sqr/sqr SCOTTSZSA PARAMS=nnn FN(Z*Z) FUNCTION=SIN BAILOUT=nnn+3 SCOTTCZSA PARAMS=nnn FN(Z*Z) FUNCTION=COS BAILOUT=nnn+3 Note: New functions for this type are sinh cosh exp log sqr SCOTTZSZZ PARAMS=nnn FN*Z+Z FUNCTION=SIN BAILOUT=nnn+3 SCOTTZCZZ PARAMS=nnn FN*Z+Z FUNCTION=COS BAILOUT=nnn+3 Note: New functions for this type are sinh cosh exp log sqr SCOTTSZSB PARAMS=nnn FN*FN FUNCTION=SIN/SIN BAILOUT=nnn+3 SCOTTCZSB PARAMS=nnn FN*FN FUNCTION=COS/COS BAILOUT=nnn+3 SCOTTLTS PARAMS=nnn FN*FN FUNCTION=LOG/SIN BAILOUT=nnn+3 SCOTTLTC PARAMS=nnn FN*FN FUNCTION=LOG/COS BAILOUT=nnn+3 Note: New functions for this type are sin/cos sin/sinh sin/cosh sin/exp sin/sqr cos/sinh cos/cosh cos/exp cos/sqr sinh/sinh sinh/cosh sinh/exp sinh/log sinh/sqr cosh/cosh cosh/exp cosh/log cosh/sqr exp/exp exp/log exp/sqr log/log log/sqr sqr/sqr SCOTTSIC PARAMS=nnn SQR(1/FN) FUNCTION=COS BAILOUT=nnn+3 SCOTTSIS PARAMS=nnn SQR(1/FN) FUNCTION=SIN BAILOUT=nnn+3 TETRATE PARAMS=nnn TETRATE BAILOUT=nnn+3 Note: New function type sqr(1/fn) with sin cos sinh cosh exp log sqr Note: New function type sqr(fn) with sin cos sinh cosh exp log sqr