The Game Master (3rd Edition)

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Fractint Version 13.0r NOTE: version 13.0r is just version 13.0 re-compiled under Microsoft C version 5.1 (13.0 was compiled under MSC 6.0). When compiled under MSC 6.0, the 'type=formula' code is partially broken (selecting the 'tetrate' formula, for example, hangs your PC). There is no corresponding 13.0r source-code release, because the source code hasn't changed. ************************************************************************* * WARNING WARNING WARNING * * >> F1, not 'h', is now the HELP key!!! << * * For Help at any time, press the F1 key * * For EGA 320x200x16 video mode, press the F9 (not F1) key * * For EGA 640x200x16 video mode, press the CF4 (not F9) key * ************************************************************************* CONTENTS New Features in Version 13.0 Introduction 1. Quick Start: FRACTINT Commands Plotting Colors Saving, Restoring and Printing "3D" Hints 2. Fractal Types Mandelbrot Julia Newton's Formula Lambda Plasma Clouds Lambda/Sine, Cosine, Sinh, Cosh, Exponent Mandel/Sine, Cosine, Sinh, Cosh, Exponent Barnsley Mandelbrot/Julia Barnsley IFS Sierpinski Gasket Quartic Mandelbrot/Julia Distance Estimator Mandelbrot/Julia Pickover Mandelbrot/Julia Types Peterson Variations Unity Bifurcation Lorenz, Lorenz3d Rossler Henon Pickover Gingerbreadman Test Stub Formula Julibrots Diffusion Doodads, Bells and Whistles "3D" Images Inversion Decomposition Starmaps Logarithmic Palettes Pickover Biomorphs and Popcorn 3. Command-Line Arguments, Batch Mode, and SSTOOLS.INI 4. File Formats, Colors, and All That .FRA and .GIF Files Continuous Potential Palette Maps 5. Hardware Support Notes on Modes, "Standard" and Otherwise FRACTINT.CFG 6. Fractals and the PC A Little History B.M. (Before Mandelbrot) Who Is This Guy, Anyway? A Little Code Logic Speed-ups The Limitations of Integer Math The Fractint "Fractal Engine" Architecture Appendix A: Mathematics of the Fractal Types Appendix B: Stone Soup with Pixels Appendix C: Revision History Appendix D: Bibliography Appendix E: Other Recommended Fractal Programs NEW FEATURES IN VERSION 13.0 F1 is now 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) Speed!! More speed!! We've gotten faster!! Super-Solid-guessing (three or more passes) from Pieter Branderhorst (replaces the old solid-guessing mode) New Boundary Tracing option from David Guenther ("fractint passes=btm", or use the new 'x' options screen) New "outside=nnn" option sets all points not "inside" the fractal to color "nnn" (and generates a two-color image). New 'x' option from the main menu brings up a full-screen menu of many popular options and toggle switches New "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) New "Attractor" fractals (Henon, Rossler, Pickover, Gingerbread) New diffusion fractal type by Adrian Mariano New "type=formula" formulas from Scott Taylor and Lee H. Skinner. New "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.) New "hertz=" option for adjusting the "sound=x/y/z" output. Printer support for color printers (printer=color) from Kurt Sowa New video modes: 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 for Version 12.0: 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) INTRODUCTION FRACTINT plots and manipulates images of "objects" -- actually, sets of mathematical points -- that have fractal dimension. See chapter 6 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. 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 recog- nizes 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 8514/A mode. Even "larger" images, up to 2048x2048x256, can be plotted to RAM, EMS, 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. FRACTINT is an experiment in collaboration. Many volunteers have joined Bert Tyler, the program's first author, in improving successive ver- sions. Through electronic-mail messages, first on CompuServe's PICS forum and now on COMART, 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 Appendix B for information about the authors and how to contribute your own ideas and code. The executable FRACTINT.EXE is public domain, and we encourage you to share copies and upload it to bulletin-board systems. Except for those modules marked with a copyright notice (and subject to the authors' conditions in the notices), the source code is also public domain. There is no warranty of its suitability for any purpose, nor any acceptance of liability, express or implied. Contribution policy: Don't want money. Got money. Want admiration. 1. QUICK START: FRACTINT Commands To start the program, enter FRACTINT at the DOS prompt. The program displays an initial "credits" screen, and waits for you to hit a command key that will (1) start plotting in a specific video mode or (2) set one of many options. You can also (at this screen or at any time) press a HELP key (<H> or <?>) for screens describing the commands and video modes. As soon as you select a video mode, the program begins drawing an initial display -- the "full" Mandelbrot set, if you haven't selected another fractal type. From this point on, and AT ANY TIME, you can hit any of the following keys to select a function. (For the "typewriter" keys, upper and lower case are equivalent: and , <?> and </>, have the same result.) Many commands and parameters can be passed to FRACTINT.EXE as command- line arguments or read from a configuration file; see Sec. 3 for details. Plotting Commands <F1> or <?> Bring up a help screen. The Escape key brings you back. Function keys & various combinations (see help screens) Select the corresponding video mode and redraw the screen <Page Up> Display and shrink the zoom box ("zoom in"). The LEFT button of a mouse displays the box; push the mouse away from you while holding that button down to shrink the box. <Page Down> Display and expand the zoom box ("zoom out"). Pulling the mouse towards you with the left button down has the same effect. Cursor ("arrow") keys Move ("pan") the zoom box. Moving the mouse has the same effect. With an enhanced-keyboard BIOS, <Control> plus a cursor key moves the box 5 times faster. With a mouse, the RIGHT button controls fast vs. slow panning. <Enter> ("Return") or <End> Redraw the area inside the zoom box as a full-screen image. Or press BOTH mouse buttons. (If there is no zoom box, this just re-draws the current screen.) <Control><Enter> "Zoom out," so that your screen would fill the current zoom box, and redraw. <Home> Redraw the previous image. The program tracks 50 sets of coordinates. <Tab> Display the current fractal type, parameters, screen or (if displayed) zoom-box coordinates, maximum iteration count, and other information useful in keeping track of where you are. With version 12.0, 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 now also tells you if your image is still being generated or has finished - a handy feature for those overnight, 1024x768 resolution fractal images. <Spacebar> Toggle between Mandelbrot-set images and their corresponding Julia-set images. Read the notes in Sec. 2 before trying this option if you want to see anything interesting. < or > Lower or raise the maximum iteration count (the number of times the program carries out its basic calculation before assigning colors). See Sec. 2. (Note: the maximum iteration count can now also be selected explicitly using the "X" options screen.) <1>, <2>, or <G> Select single-pass, dual-pass, or solid-guessing (default) mode, then redraw the screen. Single-pass mode draws the screen pixel by pixel. Dual-pass 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 performs from two to four visible passes - more passes in higher resolution video modes. Its first visible pass is actually two passes - one pixel per 4x4, 8x8, or 16x16 pixel box (depending on number of passes) 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! Note: these options, along with the Boundary-Tracing alternative, can now also be selected using the "X" options screen. Boundary Tracing, 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. Create a batch file, FRABATCH.BAT, that will start FRACTINT with the command-line arguments needed to draw the current image. If FRABATCH.BAT already exists, a new line is appended to it. <T> Select a fractal type. Move the cursor to your choice and hit <Enter>. You will be prompted for any parameters that type may require. <X> Select any number of eXtended options. Brings up a full-screen menu of options, any number of which you can change at will. This list currently includes the F(loating point algorithm) toggle, the image generation algorithm (passes = 1, 2, solid-guessing, or boundary tracing), the "maxiter=" iteration maximum, the "inside=" and "outside=" colors, the Logarithmic Palettes option, the Biomorph and Decomposition algorithms, the "sound=" option, and the "warn=" option. <E> Call up an editor that modifies, saves and restores various parameters. Currently you can change the coefficients used in plotting the Barnsley IFS fractal types or the 3D transformations used with the "Lorenz3D" and "IFS3D" fractal types, including the selection of "funny glasses" red/blue 3D. <F> Toggles the use of floating-point algorithms. Their use is shown on the <Tab> status screen. (Note: the floating point algorithms option can now also be turned on and off using the "X" options screen.) Apply the inversion process (Sec. 3) and redraw the screen. Not appli- cable to all fractal types. <N> or <L> Select normal (the default) or logarithmic palette mapping and redraw the screen. (See Sec. 3 for details.) (Note: the Logarithmic palette option can now also be turned on and off using the "X" options screen.) <Q> Apply the quaternary or binary decomposition coloring scheme (see Sec. 3) and redraw the screen. Not applicable to all fractal types. (Note that the decomposition scheme can now select 8, 16, 32, 64, 128, and 256-way decomposition, but we're stuck with <Q> for Quaternary.) (Note: the Decomposition option can now also be turned on and off using the "X" options screen.) <O> (the letter, not the number) If pressed while an image is being generated, toggle 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 being behind the scenes. (See "A Little Code" in Sec. 6.) <Insert> Restart at the "credits" screen <D> Shell to DOS. Return to FRACTINT by entering "exit" at a DOS prompt. <Esc> or <Delete> Exit FRACTINT and return to DOS. Color Commands The color-cycling commands are available ONLY for VGA adapters and EGA adapters in 640x350x16 mode. Some keys have different functions specif- ic to this mode. Note that the palette 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. <+> or <-> Switch to color-command mode and begin cycling the palette by shifting each color to the next "contour." <C> Switch to color-command mode but do not start cycling. The normally black "overscan" border of the screen changes to white as a reminder that you are in this command mode. <H> or <?> Bring up a HELP screen with commands specific to color command mode. ESCAPE exits. <+> or cursor->, <-> or cursor<- Reverse the cycling direction "out" or "in." 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-gener- ated color bands of short (F2) to long (F10) duration. <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) -- or you can use any non- command key to exit color-command mode so you can (S)ave the image. <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 and use a two-color cyclical assignment, e.g. red->yellow->red (not for EGA). <Alt><F1>-<F10> Pause and use a 3-color cyclical assignment, e.g. green->white->blue (not for 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. <M> Pause and prompt for the filename of an external color map. Several others are supplied with the program. (The .MAP extension is assumed; see Sec. 4 for details on creating your own map files.) <S> Pause, prompt for a filename, and save the current palette to the named file (.MAP assumed). <X> Enters the "Cross-Hair" Palette manipulation sub-mode, described below, in which you can modify your image one color (palette) at a time (not for EGA). Any other key terminates the color-cycling process (you can always return to it with another <C>, <+> or <->) and returns to the main command mode. Cross-Hair Mode (a Sub-Mode of Color-Cycling) This mode is entered from the main color-cycling mode by pressing the "X" key, which brings up a "Cross-Hair" cursor with which you point to a pixel using the particular color palette you wish to modify. Unlike the main color-cycling level, where you are modifying all of the colors at once, Cross-Hair mode modifies only one palette at a time. This feature is currently available only on VGA adapters. <Cursor ("arrow") Keys> Move the cross-hair cursor around the screen. The Control-Cursor keys move the cursor around faster. A mouse can also be used to move around, in which case holding down the right button slows down cursor movement. The palette value of the pixel in the middle of your cross-hair cursor is the palette value that you will be manipulating. <R>, <G>, Increase the red, green, or blue component of your palette color by a small amount. Note the case distinction of this vs: <r>, <g>, Decrease the red, green, or blue component of your palette color by a small amount. <+>, <-> Changes the RGB components of the palette you are pointing at to those of the next higher (or lower, for the <-> key) palette value. Useful for "erasing" bands of color. <Page Up>, <Page Down> Change the palette value of the cross-hair cursor to the next higher (or lower) palette value. A handy feature when the cross-hair cursor gets "lost" in the background. Holding down the left mouse button and moving the mouse forward and backward has the same effect. <ENTER> Does nothing at all. Added only to prevent you from accidentally exiting cross-hair mode by pressing both mouse buttons. Any other key terminates Cross-Hair sub-mode and returns to the main Color-Cycling command mode. Save/Restore/Print Commands <S> Save the screen contents to disk. Progress is marked by colored lines moving down the screen's edges. The default filename is FRACT001.FRA; multiple saves in a session are automatically incremented 002, 003... If you have files left over from previous sessions and want the program to keep incrementing rather than starting at 001 again and overwriting, see Sec. 3 for the "WARN=yes" argument (or set it with the "X" option screen). <R> Restore an image previously saved with <S>, or a .GIF image (See Sec. 4). You are prompted for the filename, .FRA assumed. Print the current fractal image on your (Laserjet or Epson-compatible) printer. See Sec. 3 for how to let FRACTINT know about your printer setup. "3D" 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; see "3D Images" in Sec. 3 for details. <O> Restore in 3D and overlay the result on the current screen. (This only works when there's a completed on screen, so it can't be confused with the <O>-for-orbits toggle that ONLY operates WHILE a fractal is being generated.) 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 re-draw 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. After you get used to the first "zoom levels," you may want to start in a low-resolution mode for quick progress, and switch to a higher- resolution mode when things get interesting. 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 with FRACTINT FRACTxxx (the .FRA extension is assumed). Or you could simply hit to create a batch file with the "recipe" for a given image, and next time start with FRABATCH to re-plot it. See Sec. 4 for more on batch mode and the command-line arguments the command writes. 2. FRACTAL TYPES This section is intended only as a quick introduction to the types; more detail on the mathematics of types other than the Mandelbrot set can be found in Appendix A, and some notes on how FRACTINT calculates them in Sec. 6. FRACTINT starts by default with the Mandelbrot set. You can change that by firing it up with the command-line argument "TYPE=" followed by one of the names in parentheses below, or within the program by hitting <T> and typing in the name. 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". THE MANDELBROT SET This set is the classic: the only one implemented in many plotting programs, and the source of most of the printed fractal images pub- lished 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 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 (default setting). 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. Some features of interest: 1. Use the > key to increase the 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. JULIA SETS These sets were named for mathematician Gaston Julia, and can be generated by a simple change in the iteration process described above. 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. 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 out- side, 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. 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 correspon- dence breaks down and the spacebar toggle no longer works. 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 parame- ter, the "order" of the equation (an integer from 3 through 10 -- 3 for x^3 - 1, 7 for x^7-1, etc.). The striped 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. NEWTON (type=newton) The generating formula here is identical to that for newtbasin, 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. There is symmetry in newtbasin, too, but the current version of the software isn't smart enough to exploit it. 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. COMPLEX NEWTON (type=complexnewton and type=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" and "newtbasin" 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! 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 accord- ing to the iteration when the sum of the squares of the real and imaginary parts exceeds 4. See this space in future versions for a profound discussion of the physical significance of this formula, if we ever figure it out! 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. 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 section on 3D), and your "cloud" will be converted to a mountain! You've GOT to try palette cycling 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 a single parameter, which determines how abruptly the colors change. A value of .5 yields bland clouds, while 50 yields very grainy ones. The default value is 2. Zooming is ignored, as each plasma-cloud screen is generated randomly. 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. Saved plasma-cloud screens are EXCELLENT starting images for fractal "landscapes" created with the "3D" options. LAMBDA*SINE/COSINE/SINH/COSH/EXPONENT (type=lambdasine, lambdacos, lambdasinh, lambdacos, lambdaexp) These types calculate the Julia set of the formulas lambda*sine(Z), lambda*cosine(Z), lambda*sinh(Z), lambda*cosh(Z), or lambda*exp(Z), 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 co-processor 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 co-processor, go on a LONG vacation! MANDEL*SINE/COSINE/SINH/COSH/EXPONENT (type=mandelsine, mandelcos, mandelsinh, mandelcosh, mandelexp) These are "pseudo-Mandelbrot" mappings for the LambdaSine/Cos/Sinh/Cosh/Exp 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 MandelExp set in particular is rather boring). Generate the appropriate "Mandeltrig" set, zoom on a likely spot where the colors are changing rapidly, and hit the space-bar key to plot the Julia set for that particular point. Try "FRACTINT TYPE=MANDELCOS CORNERS=4.68/4.76/-.03/.03" 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 Mandeltrigs was released. BARNSLEY MANDELBROT/JULIA SETS (type=barnsleym1/m2/j1/j2/m3/j3) Michael Barnsley has written a fascinating college-level text, "Fractals Everywhere," on fractal geometry and its graphic applica- tions. (See the bibliography, App. D.) 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 space bar. 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 J3, they are the values of C in the generating formulas, found in App. A. Be sure to try the <O>rbit function while plotting these types. BARNSLEY IFS FRACTALS (type=ifs, ifs3d) One of the most remarkable spin-offs of fractal geometry is the ability to "encode" realistic images in very small sets of numbers -- parame- ters 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). Invoking FRACTINT with "TYPE=ifs" defaults to an IFS "fern". Other IFS images can be created via the FERN, TRIANGLE and TREE files included with FRACTINT, or from user-written files with the default extension .IFS. Use the command-line argument "IFS=filename" or the main-menu command to use or manipulate one of these files. Specify whether 2D or "3D" if you have not already chosen an IFS type. You will then be asked to pick which function to edit. Hitting <Enter> leaves values unchanged. Each line in an IFS code file contains the parameters for one of the generating functions, e.g. in FERN.IFS: 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 "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. 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 the ENTER key to get a full-screen image. In order to fully appreciate the 3D fern, since your monitor is presum- ably 2D, we have added rotation, translation, and perspective capabili- ties. These share values with the same variables used in FRACTINT's other 3D facilities; for their meaning see "3D Images", farther on in this section. 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 "e" (for edit) from main screen or entering a "t" (for "transformations") from the IFS 3D codes-editing screen, will allow you to edit these values. The defaults are the same as for regular 3D, and are not always optimum for the 3D fern. Try rotation=30/30/30. Note that applying shift when using perspective changes the picture -- your "point of view" is moved. A truly wild variation of 3D may be seen by entering "2" for the stereo mode, putting on red/blue "funny glasses", and watching the fern develop with full depth perception right there before your eyes! This feature is STILL 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! 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 "iterat- ing"?) 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. QUARTIC MANDELBROT AND 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 "out- side" 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. DISTANCE ESTIMATOR MANDELBROT AND JULIA (type=demm, demj) These are 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. They produce more evenly spaced contours; while they take full advantage of your color palette, one of their 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 illus- trations of the M-set in "The Beauty of Fractals." Unfortunately, these images can take many hours to calculate even on a fast machine with a coprocessor. One way of dealing with this is to use the fast type=mandel and type=julia modes to find interesting areas. Frame these nicely, then hit B to save the coordinates to FRABATCH.BAT. Edit the batch file, changing TYPE to "demm" or "demj", MODE to a high- resolution monochrome disk-video mode, and MAXITER to 1000 or so. Run the batch file when you won't be needing your machine (over the week- end?) and print the resulting .FRA file at your convenience. PICKOVER MANDELBROT/JULIA TYPES (type=mansinzsqrd/julsinzsqrd, manzpowr/julzpowr,manzzpwr/julzzpwr, mansinexp/julsinexp) 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" option Pickover used to create organic-looking 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=mansinsqrd biomorph=0 corners=-8/8/-6/6 to see a big biomorph digesting little biomorphs! PICKOVER POPCORN (type=popcorn) 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. As a bonus, for reasons known only to the programmers, give the option "symmetry=none" and you will see 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 insite as to where the popcorn comes from. You may want to use a maxiter value less than normal - Pickover recommends a value of 50. All these are generated by equations 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 params=perturbx/perturby values. Some equations have an additional parameter n, which is generally an exponent. This value is entered as the third 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. PETERSON VARIATIONS (type=marksmandel, marksjulia, mandellambda, cmplxmarksmand, cmplxmarksjul) 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^n) * 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 would use null warping values, and specify the exponent with something like "PARAMS=0/0/4", which creates an unwarped, pentagonal MarksMandel set. MandelLambda maps the lambda function in the same way that the M-set maps J sets, and is linked to the lambda function with the spacebar toggle. 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 (see below) and let us know what you come up with! BIFURCATION (type=bifurcation) 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 normal- ly 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: 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 popula- tion 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 -- hot damn! -- fractal; zoom in where changing pixel colors look suspicious, and see for yourself. The default display has been filtered by allowing the population to settle from its initial value for 5000 cycles before plotting maxiter population values. To override this filter value, specify a new (small- er) one as the first "PARAMS=" value. "PARAMS=1" produces an unfiltered map. LORENZ ATTRACTORS (type=lorenz and type=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" using the transformation option of the <E>ditor 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. 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 cofficients 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. 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 Lorenz 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 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. xnew = 1 + y - a*x*x ynew = b*x The default parameters are a=1.4 and b=.3. 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 GINGERBREADMAN FRACTAL (type=gingerbread) This simple fractal is a charming example stolen from "Science of Fractal Images", p. 149. Look closely, and you will see that the gingerbreadman contains infinitely many copies of himself at all different scales. xnew = 1 - y + |x| ynew = x There are no parameters. 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 signif- icantly 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 parameters supported, preferably formatted as you see here to simplify folding it into the documentation. FORMULA (type=formula) Fractint now has a new 'type=formula' fractal type. This is a "roll-your-own" fractal interpreter - you don't even need a compiler! To run a "type=formula" fractal, you first need to build a text file containing formulas (there's a sample file - FRACTINT.FRM - included with this distribution). When you select the "formula" fractal type, you are prompted first for the name of your file. Then, after the program locates the file and scans it for formulas, you are prompted 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. There are also two new command-line options that work with type=formula ("formulafile=" and "formulaname=") if you are using this new 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 Precedence -------------------------------------------- 1 sin(), cos(), sinh(), cosh(), sqr(), log(), exp(), abs(), conj(), real(), imag() 2 - (negation), ^ (power) 3 * (multiplication), / (division) 4 + (addition), - (subtraction) 5 = (assignment) 6 < (less than), <= (less than or equal to) Precedence may be overridden by use of parenthesis. Note the modulas 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 modulas squared then multiplies times 'c'. Nested modulas squared operators require overriding parenthesis: c * |z + (|pixel|)| The formulas are performed using either integer or floating point mathematics depending on the type of math you have setup. If you do not have an FPU then type MPC math is performed in leu of traditional floating point. 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. JULIBROTS (type=julibrot) (the following documentation is supplied by Mark Peterson, who "invented" the Julibrot algorithm) There is a very close relationship between the Mandelbrot set and Julia sets of the same equation. To draw a Julia set you take the basic equation and vary the initial value according to the two dimensions of screen leaving the constant untouched. This method diagrams two dimensions of the equation, 'x' and 'iy', which I refer to as the Julia x and y. z(0) = screen coordinate (x + iy) z(1) = (z(0) * z(0)) + c, where c = (a + ib) z(2) = (z(1) * z(0)) + c z(3) = . . . . The Mandelbrot set is a composite of all the Julia sets. If you take the center pixel of each Julia set and plot it on the screen coordinate corresponding to the value of c, a + ib, then you have the Mandelbrot set. z(0) = 0 z(1) = (z(0) * z(0)) + c, where c = screen coordinate (a + ib) z(2) = (z(1) * z(1)) + c z(3) = . . . . I refer to the 'a' and 'ib' components of 'c' as the Mandelbrot 'x' and 'y'. All the 2 dimensional Julia sets correspond to a single point on the 2 dimensional Mandelbrot set, making a total of 4 dimensions associated with our equation. Visualizing 4 dimensional objects is not as difficult as it may sound at first if you consider we live in a 4 dimensional world. The room around you is three dimensions and as you read this text you are moving through the fourth dimension of time. You and everything around your are 4 dimensional objects - which is to say 3 dimensional objects moving through time. We can think of the 4 dimensions of our equation in the same manner, this is as a 3 dimensional object evolving over time - sort of a 3 dimensional fractal movie. The fun part of it is you get to pick the dimension representing time! To construct the 4 dimensional object into something you can view on the computer screen you start with the simple 2 dimensions of the Julia set. I'll treat the two Julia dimensions as the spacial dimensions of height and width, and the Mandelbrot 'y' dimension as the third spacial dimension of depth. This leaves the Mandelbrot 'x' dimension as time. Draw the Julia set associated with the Mandelbrot coordinate (-.83, -.25), but instead of setting the color according to the iteration level it bailed out on, make it a two color drawing where the pixels are black for iteration levels less than 30, and another color for iteration levels greater than or equal to 30. Now increment the Mandelbrot 'y' coordinate by a little bit, say (-.83, - .2485), and draw another Julia set in the same manner using a different color for bailout values of 30 or greater. Continue doing this until you reach (-.83, .25). You now have a three dimensional representation of the equation at time -.83. If you make the same drawings for points in time before and after -.83 you can construct a 3 dimensional movie of the equation which essentially is a full 4 dimensional representation. In the Julibrot fractal available with this release of FRACTINT the spacial dimensions of height and width are always the Julia dimensions. The dimension of depth is determined by the Mandelbrot coordinates. The program will consider the dimension of depth as the line between the two Mandelbrot points. To draw the image in our previous example you would set the 'From Mandelbrot' to (-.83, .25) and the 'To Mandelbrot' as (-.83, -.25). If you set the number of 'z' pixels to 128 then the program will draw the 128 Julia sets found between Mandelbrot points (-.83, .25) and (-.83, -.25). To speed things up the program doesn't actually calculate ALL the coordinates of the Julia sets. It starts with the a pixel a the Julia set closest to the observer and moves into the screen until it either reaches the required bailout or the limit to the range of depth. Zooming can be done in the same manner as with other fractals. The visual effect (with other values unchanged) is similar to putting the boxed section under a pair of magnifying glasses. The variable associated with penetration level is the level of bailout there you decide to make the fractal solid. In other words all bailout levels less than the penetration level are considered to be transparent, and those equal or greater to be opaque. The farther away the apparent pixel is the dimmer the color. 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 from the screen. The screen dimensions provide the reference frame. To the human eye blue appears brighter than red. The Blue:Red ratio is used to compensate for this fact. If the image appears reddish through the glasses raise this value until the image appears to be in shades of grey. If it appears bluish lower the ratio. Julibrots can only be shown in 256 red/blue colors for viewing in either stero-graphic (red/blue funny glasses) or grey-scaled. FRACTINT automatically loads either GLASSES1.MAP or ALTERN.MAP as appropriate. 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. "3D" IMAGES FRACTINT can now 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 .FRA images (or .GIF images -- see Sec. 4) 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, enter the filename. If no extension is specified, a .FRA extension is assumed. If no .FRA is found, it will assume a .GIF extension. If no match is then found, if will attempt to match the file with no extension. Then it will prompt you for the video mode if appropriate. Now, 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. 3D MODE SELECTION After the filename prompt and video mode check, you're presented with a "3d Mode Selection" screen. Each selection will have defaults entered if you wish to change any of the defaults use the cursor keys to move through the menu. When you're satisfied press <Enter>. Here are the options and what they do: 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! 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 above, or grid fill in the "Select Fill Type" screen next. 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. 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, lorenz3d and ifs3d. 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 (see below). When you are satisfied with your selections press enter to go to the next screen which will appear below this one. SELECT FILL TYPE Screen These options exist 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. These options generally determine what to do with the space between the mapped dots. 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. Now, 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. Try the second option, "connect the dots (wire frame)", then "surface fills - (colors interpolated)" and "(colors not interpolated), the general favorite of the authors. Solid fill, while it reveals the pseudo-geology under your pseudo-landscape, inevitably takes longer. Now 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 toward the light in one of the following screens. Light source before transformation calculates the illumination before doing the coordinate transformations, and is slightly faster. If you generate a sequence of images where one rotation is 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. Light source after transformation applies the transformations first, then calculates the illumination. If you generate a sequence of images with progressive rotation as above the effect is as if you and the light source are fixed and the object is rotating. For ease of discussion we will refer to the following fill types by their numbers: -1 - surface grid 0 - (default) - no fill at all - just draw the dots. 1 - wire frame - joins points with lines 2 - surface fill - (colors interpolated) 3 - surface fill - (interpolation turned off) 4 - solid fill - draws lines from the "ground" up to the point 5 - surface fill with light model - calculated before 3D transforms 6 - surface fill with light model - calculated after 3D transforms Types 2, 5, and 6 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 3 is the same as type 2 with interpolation turned off. You might want to use fill type 3, for example, to project a .GIF photograph onto a sphere. With type 2, you might see the filled-in points, since chances are the palette is not continuous; type 3 fills those same points in with the colors of adjacent pixels. However, for most fractal images, fill type 2 works better. STEREO 3D VIEWING If you selected Stereo option 1, 2 or 3 you will be presented another screen to select from. We suggest you definitely use defaults at first on this one. Funny Glasses Parameters 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. 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. 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. 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. Then there are three scaling factors in per cent. 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 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. 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. Finally, 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 Well, OK, we lied. If you selected one of the light source fill options, there are still MORE options. You must chose 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 ie. it will be low in the sky, say, afternoon or morning. Then you are asked for a smoothing factor. Unless you used continuous potential (see Sec. 4) when generating the starting image, the illumi- nation 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. But if your fractal is not a plasma cloud and has features with sharply defined boundaries (e.g. Mandelbrot Lake), the smoothing may cause the colors to run. This is a feature, not a bug. (A copyrighted response of [your favorite commercial soft-ware company here], used by permission.) Author's note - smoothing is a historical relic dating back to *BEFORE* we added continuous potential. If anyone actually *USES* this, let us know - otherwise we'll probably take it out. You'll probably want to adjust the final colors for fill types using light source via color cycling. Try one of the more continuous palettes (<F8> through <F10>), or load the GRAY palette with the <A>lternate-map command. Now, absolutely the last option (this time we mean it): you are asked for a range of "transparent" colors, if any. This option is most useful when using the <O>verlay command below. Enter the color range (minimum and maximum value) for which you do not want to over-write whatever may already be on the screen. The default is no transparency (display everything). Note that any image loaded up in 3D 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... 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: 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 plausi-ble 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. 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 command file. Here's an example for you power FRACTINTers, the command FRACTINT MYFILE SAVENAME=MY3D 3D= BATCH=YES would make FRACTINT load MYFILE.FRA, re-plot it as a 3D landscape (taking all of the defaults), save the result as MY3D.FRA, 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. 3D OVERLAY MODE While the <3> command creates its image on a blank screen, the <O> 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 hit <O> and choose spherical projection to re-plot that image or another as a moon in the sky above the landscape. <O> 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, flexi- ble 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. 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 diskvideo mode to create a 256 color fractal. You might want to edit the fractint.cfg file to make a diskvideo mode with the same pixel dimensions as your normal video (see "batch=config" command for how to create fractint.cfg). Using the "3" command, enter the file name of the saved 256 color file, say "no" to the "Legal for this machine?" prompt, selecting instead 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! HOW TO MAKE 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." 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 nversion 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, mentioned earlier in connection with the distance-estimator method, made his name in the 1950s with a method for everting a seven-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 command prompts you for 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 current screen mode. The default values for Xcenter and Ycenter use the center of the current screen. Try this one out with a Newton plot, so its radial "spokes" will give you something to hang on to. Plot a Newton-method image, then hit and use a radius of 1, default center coordinates. The center has "exploded" 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? 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 command toggles to another coloring protocol. (It's <Q> because the first implementation was 4-way or "quaternary," and <D> was already spoken for.) 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 first parameter, points ending up in each quadrant are given their own color; if 2 (binary decomposition), points in alternat- ing 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. The second parameter is the bailout value. The smaller it is, the faster (though not much) the plot will run; the larger it is, the more accurate it will be. A value of about 50 is a good compromise for M/J sets. LOGARITHMIC PALETTES By default, Fractint uses a continuous palette - ie, 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. If you elect to use Logarithmic palettes, however, the color Fractint uses for iteration N will be determined by the algorithm "palettenumber = maxcolors * log(maxiter) / log(N). 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". BIOMORPHS Related to binary decomposition 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 bail-out 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, or any of the new "biomorph" fractals. 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 documentation is supplied by Mark Peterson, who wrote the starfield transformation): 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. 3. Command-Line Arguments, Batch Mode, and SSTOOLS.INI FRACTINT accepts command-line arguments that allow you to load it with your own choice of video mode, starting coordinates, and just about every other parameter and option. These arguments can also be included in a SSTOOLS.INI startup file, or in command files invoked on the command line using LINK-style syntax ("FRACTINT @MYFILE"). The program first looks along the DOS PATH for any file called SSTOOLS.INI and reads start-up variables from that file. Then, it looks at its own command line; arguments there will over-ride those from the .INI file. The syntax is: FRACTINT argument argument argument... where the individual arguments are separated by one or more spaces (an individual argument may NOT include spaces). Either upper or lower case may be used, and arguments can be in any order. To make your file more readable, you may replace the spaces with Returns. Some terminology: COMMAND=nnn enter a number in place of "nnn" COMMAND=[filename] you supply filename COMMAND=yes|no|whatever means choose one of "yes", "no", and "whatever" COMMAND=1st[/2nd[/3rd]] - the slash-separated parameters "2nd" and "3rd" are optional FILENAME=[name] Causes FRACTINT to read the named file, which must either have been saved from an earlier FRACTINT session (version 7.0 or later) or be a generic GIF file, and use that as its starting point, bypassing the initial information screens. The filetype is optional and defaults to .FRA. Generic GIF files are restored as fractal type "plasma". On the command line you may omit FILENAME= and just give the name; the full argument is required in SSTOOLS.INI and other command files. @FILENAME Causes FRACTINT to read "filename" for arguments. 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 command line, as FRACTINT is not clever enough to deal with multiple indirection. PASSES=1|2|guess|btm Selects single-pass, dual-pass, solid-Guessing mode, or the Boundary Tracing algorithm. Again, the first two take the same total time to finish the display, while solid-guessing is faster at the risk of errors for a few pixels. Boundary tracing is sometimes faster, sometimes slower, and doesn't work for some fractal types at all (like the Newton fractals). INSIDE=nnn|bof60|bof61 Set the color of the interior: for example, "INSIDE=0" makes the M-set a stylish basic black. A setting of -1 makes inside=maxiter. We have added two more options, inside=bof61 and inside=bof62, which reveal more of the hidden structure inside the "lake". These options are named after the figures on pages 61 and 62 of "Beauty of Fractals". See appendix for a brilliant explanation of what these do! OUTSIDE=nnn Set the color of the exterior: 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. 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). ITERINCR=nnn Set the iteration increment (the change in maxiter when you hit the '<' or '>' keys). Default is 50. 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 from the "Video Modes" help screens inside Fractint. We don't duplicate them here only because they change so darned often!) 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 these arguments in a command file to be used with future versions of FRACTINT, particularly for the super-VGA modes. SAVENAME=[name] Set the filename to use when you <S>ave a screen. The default filename is FRACT001. The .FRA extension is optional (Example: SAVENAME=myfile) WARN=yes|no Sets the savename warning flag (default is 'no'). If 'yes', saved files will NOT over-write existing files from previous sessions; instead the automatic incrementing of FRACTnnn.FRA will continue. TYPE=[name] Selects the type of fractal image to display. The default is type "mandel," the M-set. PARAMS=n/n/n/n... Set optional (required, for some fractal types) values used in plotting. These numbers typically represent the real and imaginary portions of some startup value, and are described in detail as needed in Sec. 2. (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.) CORNERS=xmin/xmax/ymin/ymax 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. (Example: corners=-0.739/-0.736/0.288/0.291). POTENTIAL=maxcolor[/slope[/modulus[/potfile]]] Establishes the "continuous potential" coloring mode for all fractal types except plasma clouds, IFS and IFS3D. The four arguments define the maximum color value, the slope of the potential curve, the modulus "bailout" value, and the savefile name. Example: "POTENTIAL= 240/2000/40/potfile". The Mandelbrot and Julia types ignore the modulus bailout value and use their own hardwired value of 4.0 instead; see Sec. 4 on the continuous-potential algorithm for details. 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." DECOMP=2|4|8|16|32|64|128|256 Invokes the corresponding decomposition coloring scheme. LOGMAP=yes Switches the default color palette-map selection from normal (continu-ous) to logarithmic. IFS=[filename] Use the IFS coded values in [filename] to generate Barnsley IFS fractal images. The default extension is .IFS. See Sec. 2 for details. IFS3D=[filename] Use the IFS coded values in [filename] to generate Barnsley IFS3D fractals. 3D=Yes Replaces the default answers to the <3> command's prompts, as described in Sec. 2. If FILENAME= is given, forces the restore to be performed in 3D mode. Very handy when used with the 'batch=yes' option for batch-mode 3D images. (Note: the form "3D/=parm/parm/parm ..." for entering 3D parameters still works but will not be supported in the future. Use the variables below. Position-sensitive parameters are nasty and evil, especially when there are dozens of them!) The options below replace selected 3D defaults: sphere=yes Turns on spherical projection mode 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 scalexyz=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 <O>verlaying 3D images. xyadjust=nn/nn This shifts the image in the x/y dir without perspective stereo=n Selects the type of stereo image creation 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 for funny glasses filter parameters To stay out of trouble, use the complete list, 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 command file is to use the Batch command on an image you like and then modify the resulting parameters. INVERT=nn/nn/nn Turns on inversion. The parameters are radius of inversion, x-coordi- nate 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. 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. PRINTER=type[/resolution[/port#]] Defines your printer setup. Currently legal printer types (only the first two characters are needed): [HP] LaserJet, [EP]son-compatible, [IB]M-compatible (which is identical to the Epson driver at the moment), and [CO]lor (equivalent to the Star Micronics printer). The default printer type is the Epson/IBM. Resolution is in DPI, and currently available values are 60, 120, and 240 for the Epson/IBM and 75, 150, and 300 for the LaserJet (the default value is the lowest resolution). Legal printer port values are 1, 2, and 3 (for LPT 1-3) and 11, 12, 13, and 14 (for COM1-4). 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. CYCLELIMIT=nnn Sets the speed of color cycling. Technically, the number of DAC regis- ters updated during a single vertical refresh cycle. Legal values are 1 - 256, default is 55. MAP=[filename] Reads in a replacement color map from [filename]. This map replaces the default video color map of your VGA or TARGA card. The file must be in the format described in Sec. 4. The difference between this argument and an alternate map read in via <M> in color-command mode is that this one is permanent. WARNING: If colors 0 and 1 decode to the same color, your <H>elp screens will use the same color for both text and back- ground, which makes them kind of difficult to read. RAMVIDEO=no If set, when you select a "Disk/RAM" video mode, the check for adequate RAM is bypassed, and Fractint heads straight for your expanded memory and/or hard disk. This is useful where you may *technically* have enough RAM for a particular video mode, but you may want to use it later for something else (like shelling to DOS). FORMULAFILE=[formulafilename] Lets you specify the default formula file for type=formula fractals (the default is FRACTINT.FRM). Handy if you want to generate one of these fractal types in batch mode. FORMULANAME=[formulaname] Lets you specify the default formula name for type=formula fractals (the default is no formula at all). 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. ; As usual, indicates the rest of the command line (or the rest of the individual line inside command files) is a comment. 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" 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. HERTZ=nnn Adjusts the sound produced by the "sound=x/y/z" option. Legal values are 200 through 10000. BATCH MODE Two other arguments have special uses. 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. It's especially possible if you use <Tab> to note parameters for interesting areas, or to store them as lines in FRABATCH.BAT. BATCH=yes Run in batch mode -- that is, FRACTINT acts as if you had hit <S> to save the image after drawing it (using defaults or whatever other parameters you use here or in a command file), followed by exit to DOS. The "VIDEO=" mode option is required. BATCH=config Starts a quick batch-mode run that creates a default FRACTINT.CFG file from the full internal video table (FRACTINT.CFG is described in Sec. 5). SSTOOLS.INI If you are familiar with Microsoft's TOOLS.INI or WINDOWS.INI configu- ration files, the SSTOOLS.INI command file is used in the same way. In particular, you designate a section of SSTOOLS.INI as belonging to a particular program by beginning the section 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] sound=off ; (for home use only) printer=hp ; my printer is a LaserJet [startrek] Aye, captain, but I dinna think the engines can take it! [fractint] inside=0 ; using "traditional" black FRACTINT will use only the second, third, and last lines. (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 people working on other, sister programs to FRACTINT are going to read that file. And just when you thought it was safe to download again..!) 4. File Formats, Color Maps and All That FRACTINT AND .GIF FILES 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. Unfortunately, this creates a problem. The current official .GIF specification does not offer a place to store the small amount of extra information that FRACTINT needs in order to implement the <R> feature -- i.e., the parameters that let you keep zooming, etc. so on as if the restored file had just been created in this session. FRACTINT gets around this restriction in a non-standard manner, saving its application-specific information AFTER the official GIF terminator character. This trick works with all of the popular GIF decoders that we have tested (although some of them require renaming the file to xxxx.GIF first). We are not, however, claiming that these files are true .GIF files. For one thing, information after the .GIF terminator has the potential to confuse the on-line .GIF viewers used on Compuserve. For another, it is the opinion of some .GIF developers that the addition of this extra information violates the GIF spec. That's why we use the default filetype .FRA instead. If you wish to save FRACTINT images as true .GIF files, simply specify .GIF as the extension in the SAVENAME argument ("SAVENAME= somename.gif"). FRACTINT will detect the extension and drop the extra data. You will NOT be able to fully restore such files with <R>, because the information FRACTINT needs just isn't there. When the <R> command encounters true .GIF files (whether FRACTINT created them or not), it does the best it can and restores them as if they were un- zoomable plasma clouds. An easy way to convert an existing .FRA file into true .GIF format suitable for uploading is something like this at the DOS prompt: FRACTINT MYFILE SAVENAME=MYFILE.GIF BATCH=YES FRACTINT will load MYFILE.FRA, save it in true .GIF format as MYFILE.GIF, and return to DOS. Note that the .GIF spec is constantly being reviewed, and it's entirely possible that future enhancements will provide a standard way to store application-specific information. We intend to use any such extensions in an upward-compatible manner if and when they become available. GIF and "Graphics Interchange Format" are trademarks of Compuserve Incorporated, an H&R Block Company. CONTINUOUS POTENTIAL 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 viewed in 3D, 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. A 256-color MCGA/VGA video mode is mandatory to appreciate this effect (it's hard to show continuous variation with only 4 or 16 colors!) 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 general- ly 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/IFS3d types, and "test" are unaffected by this toggle. The parameters for continuous potential are: potential=maxcolor[/slope[/modulus[/potfile]]] "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, e.g. 255 for VGA. 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 these values 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). "Potfile" is a filename to store the potential image in compressed TARGA format, with 16 bits per pixel. If you save the image in the normal way and then view it in 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. If the "potfile" parameter is given, much smoother pictures can be obtained. But beware: even these compressed files still use up to 2 bytes per pixel, so they can get quite large. 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/potfile.tga PASSES=1 FLOAT=yes This assumes a 256-color video mode. PALETTE MAPS If you have a VGA, Super-VGA, 8514/A, or TARGA video adapter, you can save and restore your color palette using special color-map files and either the command-line "map=" option, or the "D", "A", "S", and "M" color-cycling mode commands. These color-maps are ASCII 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). Note that the color values are in TARGA/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). The default filetype for color-map files is ".MAP". The distribution file contains sample .MAP files for you to examine and modify - DEFAULT.MAP (the VGA start-up values), ALTERN.MAP (the famous "Peterson-Vigneau Pseudo-Grey Scale"), GAMMA1.MAP and GAMMA2.MAP (Lee Crocker's response to ALTERN.MAP), and LANDSCAP.MAP Guruka Singh Khalsa's favorite color-map for plasma "landscapes"). 5. Hardware Support 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. Hence the warning to check your "VIDEO=" argument before using a new version, in which the old key combo may now call an ultraviolet holo- graphic mode. FRACTINT also 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. NOTES ON MODES, "STANDARD" AND OTHERWISE FRACTINT uses a video adapter table in the "C" program for everything it needs to know about any particular adapter/mode combination. This table can contain information for up to 98 adapter/mode combinations, and is automatically tied to 84 keys (F2-F10, their Control/ Shift/Alt variants, and many Alt-x keypad combos) when the program is running. The table entries, and the function keys they are tied to, are dis- played as part of the help screens. 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 a standard adapter with more 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. You can customize the table using the external configuration file FRACTINT.CFG, described below. The table as currently distributed begins with nine standard and several non-standard IBM video modes that have been exercised success- fully 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. 320 x 400 x 256 and 360 x 480 x 256 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. 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. Because of this, the program cannot send any text to the screen while it is in one of these modes (the BIOS would garbage it). An internal flag inhibits all text output while the screen is in one of these video modes. FRACTINT's <H> and <Tab> commands still work, because they temporarily switch the screen to an alternate video mode. 8514/A MODES The IBM 8514/A modes use IBM's software interface, and require the pre- loading of IBM's HDIDLOAD TSR utility. There are two sets of 8514/A modes: full sets (640x480, 1024x768) 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, 1016x762) 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. Finally, because IBM's interface does not handle drawing single pixels very well (we have to draw a 1x1 pixel "box"), generating the zoom box is excru- ciatingly slow. Still, it works! SUPER-EGA AND SUPER-VGA MODES 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. With version 12.0, we've added both John Bridges' SuperVGA Autodetecting logic *and* VESA adapter detection, so that many of the brand-specific SuperVGA modes have been collapsed into a single function key. There is now exactly one function key for SuperVGA 640x480x256 mode, for instance. TARGA MODES TARGA support for FRACTINT is provided courtesy of Joe McLain. Be aware that there are a LOT of possible TARGA 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 (Sec. 3), however, works with both TARGA and VGA boards and enables you to redefine the default color maps with either board. "RAM-VIDEO" AND "DISK-VIDEO" MODES These "video modes" do not involve a video adapter at all. They use either your spare RAM (if you have enough), your spare expanded memory (if you have it, and have enough of it), or your disk drive (as file FRACTINT.DSK) 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, and for background processing under multi-tasking DOS managers such as DESQview. The zoom box is disabled during these modes (you couldn't see where it is anyway). To eliminate thrashing, if your "video" is really on a hard disk, so are features such as orbit display, dual-pass mode, solid-guessing, symmetry checks and (sorry) plasma clouds that require simultaneous access to several lines of the "display.". On a PC with 640K of memory running only FRACTINT under DOS, "disk- video" modes up to 640x480x256 fit into RAM, and any higher-resolution modes end up going to expanded memory or disk. While you are in these modes, your screen will display text information that lets you know whether you are using your RAM or your disk drive and what portion of the "screen" is being read from or written to. "TWEAKED" VGA MODES FRACTINT 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: * vertical or horizontal overscan (displaying dots beyond the edges of your visible CRT area) * flickering (caused by a too-slow refresh rate) * 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). FRACTINT.CFG If you have a favorite adapter/video mode that you would like to add to FRACTINT... if you want to remove table entries that do not apply to your system... if you're bothered that your favorite mode requires the <Alt><Left-Shift><Swizzlestick> key combination... or if you are using a non-enhanced keyboard so that you can't USE that combination... relief is here, and without even learning "C"! FRACTINT will create an external, editable video configuration file called FRACTINT.CFG, and from then on will use that instead of its internal table as long as it can locate it somewhere along the DOS path. Enter FRACTINT BATCH=CONFIG, and you will get a default file with an entry for every one of the internal video table entries, and you can go to work on it with a text editor. Any line in the file that begins with a tab or a space (or is empty) is treated as a comment. The rest of the lines must consist of ten fields separated by commas. The ten fields are defined as: 1. 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: 2. AX = this, 3. BX = this, 4. CX = this, and 5. DX = this (hey, having all these registers wasn't OUR idea!) 6. 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 [disk or RAM] "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 type) 28) VESA modes 7. The number of pixels across the screen (X - 160 to 2048) 8. The number of pixels down the screen (Y - 160 to 2048) 9. The number of available colors (2, 4, 16, or 256) 10. A comment describing this mode (25 chars max, leading blanks are OK) NOTE that the AX, BX, CX, and DX fields are generated (and read back) using 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! 6. Fractals and the PC A LITTLE HISTORY 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 tradi- tional 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 predict- able. 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. 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 exiting 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 provoc- ative 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 geometri- cally 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 genera- tions before are useful approximations of tree bark and lung tissue, clouds and galaxies. 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. A LITTLE CODE 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. 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. LIMITATIONS OF INTEGER MATH (and how we now get around it) 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)] (approximately 500,000,000 digits!) 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 noticable. 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), and disables zooming and panning. If you are stuch 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. 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. 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. 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 straight-forward 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 beserk. 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 5.1 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 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, it is now distrib- uted 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). APPENDIX A: MATHEMATICS OF THE FRACTAL TYPES Fractal Type(s) Formula(s) used ----------------------- --------------------------------------------- Mandel, Julia Z(n+1) = Z(n)^2 + C Newton, Newtbasin (roots of) Z^n - 1, wnere n is an integer ComplexNewton, ComplexBasin (roots of) Z^a - b, where a,b are complex plasma (see the Plasma section for details) Mandelsine, Lambdasine Z(n+1) = Lambda * sine(Z(n)) + C Mandelcos, Lambdacos Z(n+1) = Lambda * cos(Z(n)) + C Mandelexp, Lambdaexp Z(n+1) = Lambda * exp(Z(n)) + C Mandelsinh, Lambdasinh Z(n+1) = Lambda * sinh(Z(n)) + C Mandelcosh, Lambdacosh Z(n+1) = Lambda * cosh(Z(n)) + C BarnsleyM1, BarnsleyJ1 Z(n+1) = (Z(n)-1) * C if Real(z) >= 0 else (Z(n)+1) * modulus(C)/C BarnsleyM2, BarnsleyJ2 Z(n+1) = (Z(n)-1) * C if Real(Z(n))*Imag(C) +Real(C)*Imag(Z(n)) >= 0 else (Z(n)+1) * C BarnsleyM3, BarnsleyJ3 Z(n+1) = (Real(Z(n))^2 - Imag(Z(n))^2 - 1) + i * (2 * Real(Z((n)) * Imag(Z((n))) if Real(Z(n) > 0 else (Real(Z(n))^2 - Imag(Z(n))^2 - 1 + lambda * Real(Z(n)) + i * (2 * Real(Z((n)) * Imag(Z((n)) + lambda * Real(Z(n)) Sierpinski Z(n+1) = (2x, 2y - 1) if y > .5 else (2x - 1, 2y) if x > .5 else (2X, 2y) MandelLambda, Lambda Z(n+1) = (C) * (Z(n)^2) + C MarksMandel, MarksJulia Z(n+1) = (C^(Period-1)) * (Z(n)^2) + C ("Period" is a parameter) Cmplxmarksmand,Cmplxmarksjul Same as MarksMandel and MarksJulia except period parameter is complex rather than real Unity (see the Unity section for details) ifs, ifs3D (see the IFS section for details) Mandel4, Julia4 Z(n+1) = Z(n)^4 + C Test (as distributed, as simple Mandelbrot fractal) Mansinzsqrd, Julsinzsqrd Z(n+1) = Z(n)^2 + sin(Z(n)) + C Manzpower, Julzpower Z(n+1) = Z(n)^M + C (M is a parameter) Mandel4/Julia4 Special case of manzpower and julzpower (M=4) Manzzpwr, Julzzpwr Z(n+1) = Z(n)^Z(n) + Z(n)^M + C Mansinexp, Julsinexp Z(n+1) = sin(Z(n)) + e^(Z(n)) + C popcorn Z(n+1) = x(n+1) + i * y(n+1), where x(n+1) = x(n) - 0.05*sin(y(n)) + tan(3*y(n)) y(n+1) = y(n) - 0.05*sin(x(n)) + tan(3*x(n)) demm, demj (Mandelbrot, Julia fractals calculated and colored using the "Distance Estimator" method Bifurcation (see the Bifurcation section for details) Lorenz, Lorenz3d Lorenz Attractor - orbits of differential equation: dx/dt = -a*x + a*y dy/dt = b*x - y -z*x dz/dt = -c*z + x*y Solution generated by 1st order Taylor series: 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) (defaults: dt = .02, a = 5, b = 15, c = 1) (Lorenz3D is the Lorenz Attractor with the addition of 3D perspective transformations as specified by the IFS <E>ditor's transformation option) Rossler3d Orbit is solution to differential equation: dx/dt = -y -z dy/dt = x + a*y dz/dt = b + x*z - c*z Solution via first order Taylor series: xnew = x - y*dt - z*dt ynew = y + x*dt + a*y*dt znew = z + b*dt + x*z*dt - c*z*dt Default params are dt= .04, a= .2, b= .2, c= 5.7 Henon Orbit generated by: xnew = 1 + y - a*x*x ynew = b*x The default parameters are a=1.4 and b=.3. Pickover Orbit generated by: xnew = sin(a*y) - z*cos(b*x) ynew = z*sin(c*x) - cos(d*y) znew = sin(x) Default params are: a=2.24, b=.43, c=-.65, d=-2.43 Gingerbreadman Orbit generated by: xnew = 1 - y + |x| ynew = x Diffusion See diffusion type description Julibrot Same as Mandel and Julia - see type description 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! 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) 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) sinh(2x) sin(2y) tanh(x+iy) = ------------------ + i------------------ cosh(2x) + cos(2y) 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)) APPENDIX B STONE SOUP WITH PIXELS: THE AUTHORS 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 ques- tions 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'se 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 program- mers 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 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 three 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 help 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. More than any of the authors, he has personally touched and massaged the entire source. 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 sophmore 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 "Juliabrot" intrinsic 3D fractals - in other words, most of the truly original ideas in FRACTINT! Mark's knowledge of higher mathematics and programing was achieved almost entirely through self-study. Currently he works as a real-time supervisor of the electrical transmission system for Connecticut and Massachusetts. When not hard at work on programming projects he relaxes with Karate and Judo. Accessing the Authors ====================================== Communication between the authors for development of the next version of FRACTINT takes place in COMART (Computer Art) Section 15 (Fractals) of CompuServe (CIS). Access to this area is open to any and all interested in computer generated fractals. 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 that forum's data library 15. The following authors have agreed to the distribution of their address- es. 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! Bert Tyler [73477,433] on CIS Tyler Software btyler on BIX 124 Wooded Lane Villanova, PA 19085 (215) 525-6355 Timothy Wegner [71320,675] on CIS 4714 Rockwood nmittiw@mwvm.mitre.org on Internet Houston, TX 77004 twegner@mwunix.mitre.org on Internet (713) 747-7543 Mark C. Peterson [70441,3353] on CIS 253 West St., H Plantsville, CT 06479 (203) 276-9474 (voice) Rob Beyer [71021,2074] on CIS 23 Briarwood Lane Laguna Hills, CA, 92656 (714) 831-7665 (7-12pm PST & weekends) Pieter Branderhorst [72611,2257] on CIS Amthor Computer Consultants 3915 Bedford Rd. Victoria, B.C., Canada V8N 4K6 (604) 477-1197 Lee Daniel Crocker [73407,2030] on CIS 1380 Jewett Avenue lcrocker on BIX Pittsburg, CA 94565 ames!pacbell!sactoh0!siva!lee on Usenet (408) 354-8001 Monte Davis [71450,3542] on CIS 31 Washington St Brooklyn, NY 11201 (718) 625-4610 Richard Finegold [76701,153] on CIS 1414 179th Ave NE richg@blake.acs.washington.edu on InterNET Bellevue, WA 98008 uw-beaver!blake!richg on UseNET (206) 746-2694 David Guenther [70531,3525] on CIS 50 Rockview Drive Irvine, CA 92715 Michael L. Kaufman [71610,431] on CIS 2247 Ridge Ave, #2K (also accessible via EXEC-PC bbs) Evanston, IL, 60201 (708) 864-7916 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 Marc Reinig [72410,77] on CIS 3415 Merrill Rd. mqr@sun.com!daver!cypress!pld on Usenet Aptos, CA. 95003 abu on BIX (408) 475-2132 Dean Souleles [75115,1671] on CIS 8840 Collett Ave. Sepulveda, CA 91343 (818) 893-7558 Scott Taylor [72401,410] on CIS 1901 South County Road 23E Berthoud, CO 80513 (303) 651-6692 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 APPENDIX C REVISION HISTORY Features first appearing in: =========================== 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 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 popcorn Virtually every fractal type 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 <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" (<Control><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 "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 .FRA 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 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 APPENDIX D 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*!!! APPENDIX E OTHER FRACTAL PROGRAMS (This section is new, and pathetically small because we added it at the last minute. Please help us out, here...) FDESIGN, by Doug Nelson (CIS ID 70431,3374) - a freeware IFS fractal generator available from Compuserve in COMART Lib 15, and probably on your local BBS. This program requires a VGA adapter and a Microsoft- compatible mouse, but 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.