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YABMP.DOC ver. 0.97 ( for i486 or i386/387 ) Part 1. Introduction ____________ 1.0. Hardware Requirements --------------------- YABMP is a 32-bit Protected Mode program that requires an i486 CPU (or an i386 CPU with i387 FPU). It runs under DOS ( version 3.3? or greater ) with a DOS Extender, which is provided. YABMP requires a VGA or SuperVGA Color Graphics Card and a Color Monitor. Six VGA/SVGA modes are supported: 640 x 480, 800 x 600, and 1024 x 768 in both 256 and 16 colors. YABMP has been tested with ATI Wonder XL, ATI Graphics Ultra, and Paradise PVGA 1024 Video Cards, but the VGA modes should work with most cards. YABMP supports printing its figures with the Hewlett-Packard LaserJet (tm) II or III. 1.1. Why Yet Another Brooks-Matelski Process Program? ------------------------------------------------ Many programs are already available to compute and plot the MSet (the so-called "Mandelbrot Set") and its associated JSets. Their aim, however, often appears to be to produce these figures as a kind of art or entertainment and their results are often mathe- matically inaccurate and tend to obscure, rather than to reveal and to clarify, the underlying mathematics. The aim of YABMP is to produce the most mathematically accurate figures of the MSet and of its associated JSets. The visually (and mathematically!) most striking feature of the MSet is that it contains infinitely many small copies of itself, imbedded in itself near its boundary. A first requisite of a good MSet figure is that these be conspicuous. A second aspect of mathematical accuracy is the aspect-ratio (i.e., the proportions) of the figure. That is, a square in the complex plane must appear as a square in the figure. This means that we are not free to specify all four corners of a rectangular region in the plane. YABMP asks you only for the coordinates of the center of the desired figure and its height --- the appropriate width being determined so that the figure has the correct aspect-ratio. A third aspect of mathematical accuracy concerns the coloring of the figure. There are, in principle, only two kinds of points in these figures: There are points that belong to the set and there are points that belong to the complement of the set. Thus, the most mathematically accurate figures should require only two colors. In practice, however, a third color is required for pixels that correspond to points which lie very near the boundary of the set. If you have not seen them before, you will be astonished by the stark beauty of the three-color figures YABMP produces. 1.2. The YABMP Distribution --------------------- This YABMP distribution consists of the following 7 files: YABMP.EXE Requires an i486 CPU (or i386DX CPU with i387DX FPU). CPUID.ZIP Intel program to correctly determine your CPU and FPU, before attempting to run YABMP. DOS4GW.EXE Rational Systems (tm) DOS Extender. Must be in the current directory or a pathed directory. YABMP.HLP Help File containing brief instructions to be viewed from within YABMP. YABMP.DOC This Documentation File. YABMPCMD.ZIP Collection of *.CMD script files for auto- matically producing and saving the figures referred to in this documentation. YABMPPAL.ZIP Collection of *.PAL palette files. 1.3. Installation ------------ YABMP looks for all of its files (DOS4GW.EXE, BMPSCRN.CFG, YABMP.HLP, the *.BMP Figure Files it produces, the *.CMD script, and the *.PAL palette files) in the current directory. Just copy all of the distribution files to a single subdirectory on your fixed disk and run YABMP from that subdirectory. 1.4. File Compression and Printing Figures ------------------------------------- YABMP provides no data compression. Even the very best data compression and decompression routines take a significant time to process the very large figure files that YABMP can generate. Most figures are generated so quickly that it is not worthwhile to save them beyond the current session. If you should generate figures you want to keep, you can compress them with PKZIP (tm) and decompress them for viewing. YABMP is intended for use in exploring the MSet and its associated JSets and quick producing accurate detail figures for viewing. Data concerning a figure, which is displayed before a recalled figure file is displayed, is displayed in text mode and can be printed with PrintScreen. Screen figures and figures printed from the screen can both have the correct aspect ratio only if both screen and printer pixels have the same proportions and, therefore, printing the screen correctly is problematical. YABMP supports printing its figures with the Hewlett-Packard LaserJet II or III. Output may be directed to either LPT1:, to an H-P .PCL File, or to an .HP file. If you do not have a LaserJet connected to LPT1:, make an H-P .PCL file and send it to a LaserJet later with the DOS command: COPY name.PCL LPT1:/B, which produces the same result as sending output directly to LPT1:. An *.HP file is intended for inclusion in a document formatted with the TeX typesetting system, using the TeX "special" command. A conventional Graphics PrintScreen utility will print a solid black rectangle, since every pixel on the YABMP screen is colored. A PrintScreen utility, that can be gotten to print only Color 2, in the current mode, can also print YABMP figures. Part 2. Using YABMP ---------- 2.0 General. ------- When you first run YABMP, a Default Graphics Mode Selection Screen will appear. The six VGA and SuperVGA graphics modes, that are supported by YABMP, will be displayed. Those that appear to also be supported by your hardware are highlighted in red. 256 color graphics run faster than 16 color graphics and the resulting files can be compressed more. On the other the other hand, files generated in 256 color modes are twice as large as those produced in the 16 color mode of the same resolution. Thus, for exploration, when you do not intend to save the generated figures, YABMP Mode 2 ( 640 x 480 x 256 ) is the preferred mode. The higher resolution SuperVGA modes ( 800 x 600 and 1024 x 768 ) run slower and produce larger files, but generally do not produce commensurately more attractive figures on the screen. They are useful mainly for producing figures to be printed. Select a default graphics mode (Enter 1 or 2). Then save the selected mode by entering y. You can change your default Graphics Mode, either permanently or temporarily, from the Options Menu (Item 1). In its interactive mode, YABMP is operated from three menus. First, there is the Main MENU, which comes up when the program is run. Second, there is the Options Menu, which is Item (5) on the Main MENU. Finally, there is the Small Menu, which appears at the top of computed or recalled figures. You can generally escape from whatever you are doing by pressing Esc **ONCE**. This will usually return you to the previous menu or display or to the Main MENU. Pressing Esc will also interrupt computation and plotting, but this command does not take effect until YABMP finishes plotting the current row of pixels. You can also interrupt computation and plotting by pressing Enter. This allows you to save a partially completed figure, which can be completed later. 2.1 Getting Started Quickly. ----------------------- A collection of YABMP script files is provided for generating the figures referred to in this documentation. To see how a script file works, enter YABMP &JSETI.CMD. A JSet figure should be plotted and saved to your current directory as JSETI.BMP. After YABMP exits, follow the instructions in the following paragraph to view your figure. 2.2. Viewing an existing figure file. Item (2) of Main MENU. ------------------------------- A list of existing figure files, in the current directory, is displayed. Select the desired file by entering its number. Data on the selected file is displayed and this may be printed with the PrintScrn key. If the file is an only partially completed figure, you may elect to finish it. Otherwise, press any key to display the file and a Small Menu will appear. If key 'c' or 'C' is pressed at this point to continue, then a command file for making this figure, named CMD_n.CMD --- where n is a number, is written to your current directory. F2 Display just removes the Small Menu. Press any key to restore the Small Menu. F4 Zoom produces a box for selecting a subregion of the figure, to compute and plot. Move the box about with the Cursor keys and adjust the size of the box with PageUp and PageDown. Using the cursor and PageUp and Page Down keys on the numeric keypad (with NumLock set, so the keys are actually the numerals 2, 4, 6, 8, 3, and 9) adjusts the box more rapidly. Press Enter to select a region or Esc to return to the small menu. If you select a region, you will have an opportunity to edit several parameters. The sequence, from this point, is the same as if you had selected a subregion of a freshly computed figure and is described below. F5 Measure produces a graphics cursor (crosshairs) to read the true coordinates of a point in a figure. Move it about with the cursor and PageUp/PageDown keys or the keys on the numeric keypad. Return to the Small Menu with Esc. Press Enter to compute and display the period of the point, corresponding to the cursor location. If the figure is an MSet figure, Press j or J to make a JSet figure, using the cursor location as c. F6 Ray draws external rays over the figure. F7 Done returns to the list of existing figure files. The directory is not read again, at this point, so any figure files you have just made will not appear. To see these, Press Esc, to Return to the Main MENU, and then select (2) View an Existing File again. 2.3. Making a New MSet Figure. Item (3) of Main MENU. ------------------------ Begin by entering the parameters on which the computation depends. The desired region of the plane is specified by the coordinates of the center of the figure, which are ( x_center, y_center ), and the height of the figure, which is ymax - ymin. The program determines the width = xmax - xmin of the region, according to the aspect-ratio, so that the figure is always automatically proportioned correctly. See Checking the Aspect Ratio, under Options Menu below. Default values of the parameters, for the full MSet Figure, are given on the data entry screen. To accept a default value, just Press Enter. "thickness" is related to the width in pixels of the region near the boundary (plotted in yellow). If the thickness is too small, a figure that should be connected may be plotted disconnected. If the thickness is too large, "holes" in the figure, that should be plotted in blue, may be filled in as points near the boundary. Thickness = 0.25 produces a good figure of the full MSet. Smaller values, down to about thickness = 0.1, or less, are be required for some high- magnification details. Recall that the MSet is connected, so a good MSet figure must look connected. Too small a value of thickness will disconnect the figure. Too large a value of thickness will result in fat tendrils and filled-in holes. "maxiter" is the maximum number of iterations to be made. maxiter = 110 is sufficient for the full MSet figure and larger values will not make a significantly better figure. However, high-magnification details often require 10000 or more. If maxiter is set too small, many points that will eventually escape will not yet have escaped. These will be colored red, as interior points of the set, rather than yellow as they should be. If increasing maxiter does not reduce the red area further, then maxiter should be sufficiently large. "maxperiod" is the maximum period length checked for points in the interior of the MSet (i.e., red points). When a stably periodic point is detected, its distance from the boundary can be computed and the distance can be used to color an entire disk, rather than just a single pixel. maxperiod = 10 is sufficient for the full MSet. In plotting a detail, if you see many red pixels being plotted one at a time, try raising the value of maxperiod. But, setting maxperiod too high slows the program and it cannot do any good with small values of maxiter. The setting of maxperiod generally has no effect on the figure produced. The measure function (see F5 on the Small Menu, below) is provided for determining a suitable value of maxperiod. "minrecur" is the minimum radius of disks plotted, before recursion is terminated. minrecur = 1 results in the fastest plotting and generally produces a satisfactory figure. After these parameters have been entered, select the desired Complement Coloring Scheme. The main purpose of this program is to provide (1) Solid Complement and Solid Interior, and (2) Distance (disks) Complement and Solid Interior, which result in the best figures and the shortest computation time. The remaining schemes are provided mainly for comparison purposes. "Coloring Scheme". The cleanest and best-looking figures are made in the shortest time with color scheme (1) Solid Complement and Solid Interior, which colors the complement solid dark blue, or with color scheme (2) Distance (disks) Complement and Solid Interior, which colors the complement in slightly different shades of dark blue. These choices are highlighted in red and should normally be used. Several other familiar color schemes are provided. Color scheme (6) Solid Complement with Interior by Periods, produces a figure with the so-called hyperbolic components of the interior colored by period. For the full MSet, use the script MPERIODS.CMD or set maxiter = 10000 and maxperiod = 10. Finally, select the desired palette. (1) YABMP Default Palette should generally be used. A modified (2) VGA/SVGA Default Palette is provided for comparison. If a user-defined palette map has previously been loaded, using (3) Load a Palette File on the Options Menu, then (3) User-Defined Palette File is available. The computation of the figure can be interrupted. Pressing Esc terminates the computation and returns you to the Main MENU, discarding the partially completed figure. Pressing Enter terminates the computation, but allows you to save and use the partial result. Press Esc **ONLY ONCE** and wait until the current line is completed. After the figure is completed, press any key to see the Small Menu: F3 Save asks you for a filename and saves the figure as a file. If you choose an existing filename, the existing figure file is not overwritten, but the figure is saved as TMP_???. BMP. F4 Zoom produces a box for selecting a subregion of the figure, to compute and plot. Move the box about with the cursor keys and/or the cursor keys on the numeric keypad. Adjust the size of the box with PageUp and PageDown and/or with PageUp and Page Down on the numeric keypad. Press Enter to select a region or Esc to return to the Small Menu. If you select a region, you will have an opportunity to edit several parameters. You may want to decrease thickness and increase maxiter, if the selected region is small. minrecur = 1 should be retained, unless the figure is unsatisfactory. maxperiod should be increased if the region contains many interior (red) points. Use F5 Measure to determine the correct value. F5 Measure provides a crosshair to read the true coordinates of a point in a figure. Move it about with the cursor keys and/or the cursor keys on the numeric keypad. Return to the Small Menu with Esc. Press Enter to compute and display the period of a point. Suppose that you select a region that contains a small imbedded copy of the MSET and you wish to produce a blow-up in which this small copy occupies much of the screen. If you do not increase maxperiod, this small copy may be plotted a single point at a time. Determine the period of a point in the central cardioid of the small copy and use 5 times this as your new value of maxperiod. Press j or J to make a JSet figure, using the cursor location as c. F6 Ray draws external rays over the figure. F7 returns you to the Main MENU. 2.4. Making a New JSet Figure. Item (4) of Main MENU. ________________________ A JSET figure is made in essentially the same way as an MSet figure. Begin by entering the coordinates c_real and c_imag, of the point c in the complex plane that determines the figure. After a figure is computed or partially computed, you can use the Small Menu to select a subregion. The Small Menu works in the same way as described for MSet figures, above. 2.5. Options Menu. Item (5) of Main MENU. ------------ (1) Select Default Video Mode works in the same way as the initial selection, when you first run the program, as described above. You may make the selection permanent, saving it to a new BMPSCRN.CFG file, or not. You may also use this item to just view the currently selected video mode, using Esc to return to the Menu. (2) Check Aspect-Ratio should enable you to set your screen controls so that figures are proportioned correctly. You can also check the aspect-ratio by computing and printing the JSet figure for c = 0; that is, using c_real = c_imag = 0. This figure is the unit circle. By aspect-ratio, we mean the ratio of the width of an object to its height. The useful part of your screen, a single pixel, and a rectangle in the complex plane all have aspect ratios and these combine together to determine the proportions of your figures. Two displays are provided, as follows: (1) Computed Unit Circle displays a computed circle inscribed in a square. Use your screen controls to render the square accurately square (measure it!). (2) Raw pixels shows a circle in a square, which is maxypixels by maxxpixels square. For instance, in VGA 640 x 480 modes, maxypixels = 480 and maxxpixels = 640. (3) Load a Palette File allows you to select and load a *.PAL palette file from your current directory. (4) View Palette permits you to view the palette used for a given choice of color scheme and palette in the current default video mode. (5) Displays this YABMP.DOC documentation file. (6) Displays version and author information. (7) Returns to Main MENU. 2.6. Printing an existing figure file. Item (6) of Main MENU. -------------------------------- This works in the same way as Viewing an existing figure file, but you have to select the printer orientation (portrait or landscape) and the number of dots per inch. Smaller values of dpi produce larger figures. You may elect to send output to the first parallel printer port, LPT1:, to an H-P *.PCL file, or to an *.HP File. A printer output file made from File.BMP is named File.PCL or File.HP and existing duplicate printer output files are overwritten. Use COPY /B to copy a *.PCL File to a LaserJet (tm) printer connected to a printer port. *.HP files are intended for inclusion in documents formatted with the TeX typesetting system, using the TeX "special" command. In general, larger values of the parameter "thickness" should be used for printing than for viewing. 2.7. Command File Processing. ----------------------- Invoking YABMP with &Filename on the command line (Enter YABMP &Filename) causes YABMP to read the file and process the commands it contains. The format of a YABMP command file is as follows: type M /* type is M or J */ point 0.30239579 -0.02468438 /* c_real c_imag for type J */ loc -0.75 0.0 2.75 /* x_center y_center height */ thick 0.25 /* thickness */ maxi 100 /* maxiter */ maxp 10 /* maxperiod */ minr 1 /* minrecur */ color 2 /* color_scheme */ pal 1 /* palette */ mode 2 /* mode */ name mset /* filename to save, */ /* without extension */ Keywords are written in lower case, with one keyword per line. No delimiters are used. The color_scheme, palette, and video mode are specified by their selection numbers on the relevant Menus. Command files may have any name or extension. If (3) User-Defined Palette File is desired, the name of a *.PAL palette file in the current directory must be specified. The required format is: pal 3 map NEON.PAL /* palette */ Syntax checking of command files is very primitive, so care should be taken in preparing them. A collection of prepared command files, for producing the figures referred to in this documentation, is provided. 2.8. User-Defined Palette File Structure ----------------------------------- A User-Defined Palette File can be used to produce amusing effects, when Color Scheme (3) Distance (points) Complement and Solid Interior is selected in a 256 color mode. With other Color Schemes or 16 color modes, nothing very amusing results. A YABMP *.PAL Palette File is a 256 line text file. Each line consists of a triple of integer intensities: b, g, r, in the order blue, green, red, delimited by two commas. Intensities may have any value, between 0 and 255, inclusive. Comments may be included on any line, after a space after the triple, but each line must begin with the triple. See the included sample *.PAL files. Each of the 256 colors is used in a specific way, which is not user-modifiable, as follows: color( 0 ) Background color of the blank screen, when plotting begins. Its default value is 0, 0, 0 (black). color( 1 ) Complement color in Color Schemes 1 and 6. Its default value is 21, 0, 0 (dark blue). color( 2 ) Color for points very near the boundary. Its default value is 0, 155, 255 (bright yellow). color( 3 ) Interior points. Its default value is 0, 0, 255 (bright red). color( 4 ) to color( 255 ) are used in ways that depend on Color Scheme and whether the number of colors is 256 or 16. Here are the formulas used to determine the colors of points of the complement of the set: In 256 color modes: Color Scheme 1 color = 1 Color Scheme 2 color = dist mod 224 + 32 Color Scheme 3 color = dist mod 224 + 32 Color Scheme 4 color = k mod (number of colors - 3) + 3 Color Scheme 5 color = 1 or 4, accordingly as the imaginary part of the last value of z is positive or negative. Color Scheme 6 color = 1 In 16 color modes: Color Scheme 1 color = 1 Color Scheme 2 color = (dist/16) mod 12 + 4 Color Scheme 3 color = (dist/16) mod 12 + 4 Color Scheme 4 color = k mod (number of colors - 3) + 3 Color Scheme 5 color = 1 or 4, accordingly as the imaginary part of the last value of z is positive or negative. Color Scheme 6 color = 1 Here, dist is the distance in pixels from the point being computed to the boundary of the set. k is the number of iterations before escape. Part 3. A Bit of Mathematics, History, and Computer Graphics ---------------------------------------------------- To produce the most interesting mathematically correct figures, the user needs to know a bit about the underlying mathematics. 3.1 The Fatou and Julia Sets of a Complex Function ---------------------------------------------- We begin by considering a simple example. Let the function f, from the complex plane to the complex plane, be defined by f(z) = z²; that is, if z = x + i·y, then f(z) = (x² - y²) + i·(2xy). First suppose that z is inside the unit circle; i.e., x² + y² < 1. Then the iterates of f; that is, f(z), f(f(z)), f(f(f(z))), ... converge uniformly to the zero function. Next suppose that z is outside the unit circle; i.e., x² + y² > 1. Then the iterates of z converge to infinity or, to be more precise, to the function that maps each point to infinity. Each such point, either inside or outside the unit circle, has a neighborhood on which the iterates of f form a normal family and all such points comprise the Fatou Set of f. The Julia set of f is the complement of the Fatou Set which, in this case, is just the unit circle. To get YABMP to show you this Julia Set figure, first Select (4) Compute and Plot New JSet from the Main MENU. Then enter c_real = 0, c_imag = 0 (so that the function is f(z) = z² + 0), thickness = 1.5, maxiter = 50, Coloring Scheme = 1, and Palette = 1. What you will see is a yellow circle, with red interior and blue exterior. The red points converge to 0, while the blue points "converge to infinity." The Julia Set is the circle, while all the red and blue points belong to the Fatou Set. The formal definition, of the Fatou and Julia sets, is abstract and difficult: If a point z has a neighborhood U, such that the sequence of the iterates of f restricted to U form a normal family, then the point z is said to belong to the Fatou Set of f. Otherwise, if no such neighborhood exists, z is said to belong to the Julia Set, J(f), of f. In our example, the Julia Set is precisely the unit circle, but it is impossible to plot a really clean circle because the points of the circle do not exactly coincide with pixel centers. About 1906, Pierre Fatou made an astonishing discovery, concerning the rational function f(z) = z²/(z² + 1). Fatou found that almost every point z converges to zero, under the iteration of f. However, there remained an exceptional set of points that remain bounded away from zero, namely the Julia set of f. After the first world war Fatou, Gaston Julia and others took up the theoretical study of this phenomenon and obtained many interesting results. For instance, Fatou was able to prove that the Julia Set of his rational function is a Cantor Set. However, in the absence of machine computation, it was impossible to compute and plot figures. Some time before June 1978, Robert Brooks and J. Peter Matelski, then Assistant Professors of Mathematics at the State University of New York at Stony Brook, made some of the first machine computed and plotted such figures (see References [1] and [6]). They studied the function (1) f(z) = z² + c and made a number of computer plots, two of which are reproduced in [1]. Fig. 1 of [1] shows the JSet for c = 0.1 + i·0.6, which they defined as the set of points z which have a stable periodic orbit under the mapping (1). Fig. 2 of [1] is the first full MSet figure. They defined the MSet as the set of points c, which have a stable periodic orbit under the mapping (1). Brooks' and Matelski's figures were necessarily crude, since the only output device at their disposal was a line printer. Their figures consisted of asterisks printed on green striped paper. Some time later, Benoit Mandelbrot, working at IBM with much better equipment, made much higher resolution computer pictures which, however, were mathematically incorrect. Mandelbrot thought and his figures showed that his set, which he called Q, consists of many well separated components or islands. In 1982, Douady and Hubbard (see Reference [2]) defined the MSet to be the set of parameter values c, for which the map (1) has a bounded critical orbit or, equivalently, for which the map has a connected Julia Set. They proved that the MSet is connected and that the complement of the MSet is connected, thus providing a sound mathematical basis for the further study of the MSet and of the JSets of (1). 3.2 Computer Graphics ----------------- The most obvious way to attempt to compute and plot a figure of the MSet is to attempt to do it directly from its definition. To do this, we start with a point c, that corresponds to the center of a pixel on the screen. Beginning with z = 0, we just iterate z = z² + c until either (i) the modulus of z exceeds some large number M or (ii) a certain large number maxiter of iterations has been performed. In Case (i), the iterates are evidently diverging to infinity, so we can confidently color the pixel to indicate that it is an element of the complement of the MSet. In case (ii), where the modulus of the iterates has remained small after maxiter iterations have been performed, all we really know is that the iterates are not (yet) diverging to infinity. Different colors, related to the number of iterations performed before z escaped from the disk of radius M, are often used to produce a more colorful figure. In Case (ii), when maxiter iterations have been performed without z escaping from the disk of radius M, we color the pixel to indicate that the point belongs to the MSet. That is, we assume that it will never escape. Note that the point c scans the screen, while z always starts from zero. To produce a JSet figure, we select a complex number c, which remains fixed for the figure, and we speak of the JSet of the complex number c. We start with a point z, that corresponds to the center of a pixel on the screen. Then, we just iterate z = z² + c until either (i) the modulus of z exceeds some large number M or (ii) a certain large number maxiter of iterations has been performed and color the pixel accordingly. Note that here, c is fixed and z scans the screen. This naïve method suffices to produce crude figures, of the kind usually made by available programs, but it cannot produce a mathematically accurate sharp figure. There are two reasons for this. First of all, many points of JSets and many points near the boundary of the MSet take a very large number of iterations to escape. So, unless the maximum number of iterations is very large some points will be colored incorrectly. But there is a second, much more subtle reason why this method cannot produce accurate sharp figures. The boundaries of the MSet and of the JSets are extremely diaphanous. They are composed of many fine filaments, separated by mostly empty space or "holes," rather like lace. It is easily possible that, in scanning the screen, all of the points corresponding to the centers of pixels will fall in the "holes." Let us try to understand this better, with the aid of the "Spiral" figure of the file MSHVL4.BMP. In VGA 640x480 mode, this figure consists of 41,935 pixels (of the 307,200 pixels on the entire screen), every one of which is actually diverging to infinity. Thus, every pixel center in this figure falls in a hole in the lace. Nevertheless, this region of the plane contains infinitely many interior points of the MSet and, in fact, infinitely many small imbedded copies of the MSet, itself. Successive further magnifications, of a detail in this figure, yields figures that look like MSHVL1, MSHVL2, and MSHVL3. It is this second fact that prevented early computer figures from being correct (See References [5], [6]), and [8]. In fact, the production of realistic figures had to await the discovery of the Distance Estimate Formula (See Reference [5]), by William Thurston. Thurston's formula yields an estimate of the distance from the center of a pixel to the boundary of the set. The Distance Estimate Method involves the iteration of both the function and its derivative(s) until either (i) a good distance estimate can be made, (ii) the iterates escape, or (iii) a certain maxiter number of iterations has been made. YABMP uses Thurston's Distance Estimate Formula in the following way. If the distance from a pixel to the boundary of the set is very small, say a pixel side or so, the pixel is colored bright yellow to indicate that the pixel is "near the boundary." If a pixel is inside the set, it is colored bright red. Additionally, for MSet figures, an attempt is made to estimate the distance from the pixel, corresponding to an interior point of the set, to the boundary and this is used to color a disk red. If a pixel is found to lie outside the set, it is colored according to one of six selectable Complement Coloring Schemes. With Scheme 1, Solid, the distance estimate is used to color a solid disc consisting entirely of blue points that are escaping to infinity. This yields the best run-time performance and, also, the most mathematically realistic figures. There are, after all, only two kinds of points: those that belong to the set and those that belong to its complement. Consequently, only two colors should be needed. However, by distinguishing between the yellow points near the boundary and the red interior points of the set, we can print good monochrome figures by printing the yellow points. The implementation of the Distance Estimate Method, used by YABMP, is an adaptation of the recursive algorithm of Juval Fisher, given by him in his paper "Exploring the Mandelbrot Set," see [4], pp. 287-296. The MSet iteration process is always started with the initial value z = 0. The function f(z) = z² + c has exactly one critical point at z = 0 (where the derivative f'(z) = 2·z is zero). Consequently, if we know what happens in the neighbor- hood of z = 0, we know what happens everywhere and, therefore, nothing further is to be learned by making figures with non-zero initial values of z. REFERENCES ---------- [1] Robert Brooks and J. Peter Matelski, "The Dynamics of 2-generator subgroups of PSL(2,C)," pp. 65-71, in "Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference" (Irwin Kra and Bernard Maskit, Eds.), Annals of Math. Studies vol. 97, Princeton University Press, 1981. Contains the first figure of the MSet and the JSet figure for c = 0.1 + 0.6·i. [2] Adrien Duady et John Hamal Hubbard, "Itération des polynômes quadratique complexes," C. R. Acad. Sc. Paris, Série I, t. 294 (18 janvier 1982), pp. 123-126. [3] H.-O. Peitgen, P. H. Richter, et al., "The Beauty of Fractals," Springer Verlag, 1986. Contains many good colored figures, including interesting details of the MSet and their locations (coordinates). [4] H.-O. Peitgen, Dietmar Saupe, et al., "The Science of Fractal Images," Springer Verlag, 1988. Contains more good figures, mathematical exposition, and pseudo-code for computational algorithms. [5] John Milnor, "Self-Similarity and Hairiness in the Mandelbrot Set," pp. 211-257 of "Computers in Geometry and Topology", Edited by Tangora, Lecture Notes in Pure and Applied Mathematics, vol. 114, Marcel Dekker Inc., 1989. Contains the derivation of the Thurston Distance Estimate Formula, many good technical references, and some very interesting figures. [6] Steven G. Krantz, "Fractal Geometry," pp. 12-19, The Mathematical Intelligencer, vol. 11, No. 4, 1989 (Springer Verlag). An interesting book review of [3] and [4], above. See also the letters to the editor, from Robert Brooks and Benoit Mandelbrot, in the following issue of The Mathematical Intelligencer, vol. 11, No. 5, 1989 (Springer Verlag). [7] John Horgan, "Mandelbrot Set-To --- Did the father of fractals 'discover' his namesake set?", Scientific American, April 1990, pp. 30-34. [8] John Milnor, "Dynamics in One Complex Variable: Introductory Lectures", April 1990, SUNY Stony Brook, Institute for Mathematical Sciences, Preprint # 1990/5. Elegant exposition, with some outstanding figures and a fine treatment of the problems of computer graphics. Copyright, Disclaimer, Licenses, and All That Stuff --------------------------------------------------- The author of YABMP disclaims all warranties, either express or implied, and makes no claim as to the suitability of this software for any purpose. In no event shall either he or his Institution be liable for any damages whatsoever. You may freely make copies of YABMP and give them to others, subject only to the condition that they be accompanied by the full documentation. The author would appreciate your comments, suggestions, and bug reports. He can be reached at: Prof. Eugene Zaustinsky Department of Mathematics State University of New York Stony Brook, New York 11794-3561 email: ezaust@math.sunysb.edu