Fractint Version 15.0 Page 1 New Features in Version 15.0...............................3 Introduction...............................................5 1. Fractint Commands.......................................7 Plotting Commands......................................7 Color Cycling Commands................................10 Palette Editing Commands..............................11 Save/Restore/Print Commands...........................14 "3D" Commands.........................................15 Interrupting and Resuming.............................15 View Window...........................................16 More Zoom Box Commands................................17 Hints.................................................19 2. Fractal Types..........................................20 The Mandelbrot Set (type=mandel)......................20 Julia Sets (type=julia)...............................21 Newton domains of attraction (type=newtbasin).........22 Newton (type=newton)..................................23 Complex Newton (type=complexnewton/complexbasin)......23 Lambda Sets (type=lambda).............................24 Mandellambda Sets (type=mandellambda).................24 Plasma Clouds (type=plasma)...........................24 Lambdafn (type=lambdafn)..............................25 Mandelfn (type=mandelfn)..............................26 Barnsley Mandelbrot/Julia Sets (barnsleym1/.../j3)....26 Barnsley IFS Fractals (type=ifs, ifs3d)...............27 Sierpinski Gasket (type=sierpinski)...................28 Quartic Mandelbrot/Julia (type=mandel4/julia4)........28 Distance Estimator (distest=nnn)......................29 Pickover Mandelbrot/Julia Types.......................29 Pickover Popcorn (type=popcorn/popcornjul)............29 Peterson Variations (xxmarksxx types).................30 Unity (type=unity)....................................30 Scott Taylor / Lee Skinner Variations.................31 Kam Torus (type=kamtorus, kamtorus3d).................31 Bifurcation (type=bifxxx).............................31 Lorenz Attractors (type=lorenz/lorenz3d)..............33 Rossler Attractors (type=rossler3D)...................34 Henon Attractors (type=henon).........................34 Pickover Attractors (type=pickover)...................35 Gingerbreadman (type=gingerbreadman)..................35 Test (type=test)......................................35 Formula (type=formula)................................36 Julibrots (type=julibrot).............................37 Diffusion Limited Aggregation (type=diffusion)........39 Magnetic Fractals (type=magnet1m/.../magnet2j)........39 L-Systems (type=lsystem)..............................41 3. Doodads, Bells, and Whistles...........................43 Distance Estimator Method.............................43 Inversion.............................................43 Decomposition.........................................44 Fractint Version 15.0 Page 2 Logarithmic Palettes..................................45 Biomorphs.............................................45 Starfields............................................46 4. "3D" Images............................................47 3D Mode Selection.....................................47 Select Fill Type Screen...............................49 Stereo 3D Viewing.....................................50 Rectangular Coordinate Transformation.................51 Color Parameters......................................52 Light Source Parameters...............................52 Spherical Projection..................................53 3D Overlay Mode.......................................54 Special Note for CGA or Hercules Users................54 Making Terrains.......................................55 Making 3D Slides......................................56 5. Command-Line Parameters, Setting Defaults, Batch Mode..58 General Parameters....................................58 Video Parameters......................................63 Printer Parameters....................................65 3D Parameters.........................................66 Setting Defaults (SSTOOLS.INI)........................67 Batch Mode............................................68 6. File Formats, Color Maps and All That..................70 Fractint and .GIF Files...............................70 Continuous Potential..................................71 Palette Maps..........................................73 7. Hardware Support.......................................74 Notes on Video Modes, "Standard" and Otherwise........74 FRACTINT.CFG (Customized Video Mode List).............77 8. Common Problems........................................79 9. Fractals and the PC....................................81 A Little History......................................81 Before Mandelbrot ...................................81 Who Is This Guy, Anyway? ............................82 A Little Code.........................................83 Periodicity Logic ...................................83 Limitations of Integer Math (And How We Cope) .......84 The Fractint "Fractal Engine" Architecture ..........85 Appendix A Mathematics of the Fractal Types..............87 Appendix B Stone Soup With Pixels: The Authors...........92 Appendix C Revision History..............................97 Appendix D Bibliography.................................103 Appendix E Other Programs...............................104 Appendix F Version13 to Version 14 Type Mapping.........105 Fractint Version 15.0 Page 3 New Features in Version 15.0 New user interface! Enjoy! Some key assignments have changed and some have been removed. If you're familiar with a prior version of Fractint, a quick exploration of the new main menu will show what is different. New palette editing from Ethan Nagel. Reduced memory requirements - Fractint now uses overlays and will run on a 512K machine. The bad news is that the executable can no longer be compressed with LZEXE, and has bloated up to almost 400K! New iew command: use to get small window for fast preview, or to setup an image which will eventually be rendered on hard copy with different aspect ratio L-System fractal type from Adrian Mariano Postscript printer support from Scott Taylor Better Tandy video support and faster CGA video from Joseph A Albrecht 16 bit continuous potential files have changed considerably; see the Continuous Potential section for details. Continuous potential is now resumable. Mandelbrot calculation is faster again (thanks to Mike Gelvin) - double speed in 8086 32 bit case Compressed log palette and sqrt palette from Chuck Ebbert Calculation automatically resumes whenever current image is resumable and is not paused for a visible reason. Auto increment of savename changed to be more predictable New video modes: trident 1024x768x256 mode 320x480x256 tweak mode (good for reduced 640x480 viewing) changed NEC GB-1, hopefully it works now Integer mandelbrot and julia now work with periodicitycheck Initial zoombox color auto-picked for better contrast (usually) New adapter=cga|ega|mcga|vga for systems having trouble with auto-detect New textsafe=no|yes for systems having trouble with garbled text mode and <3> commands now present list of video modes to pick from; can reduce a non-standard or unviewable image size. Diffusion fractal type is now resumable after interrupt/save Fractint Version 15.0 Page 4 Exitmode=n parameter, sets video mode to n when exiting from fractint When savetime is used with 1 and 2 pass and solid guessing, saves are deferred till the beginning of a new row, so that no calculation time is lost. 3d photographer's mode now allows the first image to be saved to disk textcolors=mono|12/34/56/... -- allows setting user interface colors Code (again!) compilable under TC++ (we think!) bug fixes: multiple restores (msc 6.0, fixed in 14.0r) repeating 3d loads problem; slow 3d loads of images with float=yes map= is now a real substitute for default colors starfield and julibrot no longer cause permanent color map replacement starfield parameters bug fix - if you couldn't get the starfield parameters to do anything interesting before, try again with this release Newton and newtbasin orbit display fixed .TIW files (from v9.3) are no longer supported as input to 3D transformations Fractint Version 15.0 Page 5 Introduction FRACTINT plots and manipulates images of "objects" -- actually, sets of mathematical points -- that have fractal dimension. See chapter 9 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 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 recognizes and automatically uses an 80x87 chip if it's present. It's even faster on systems using Intel's 80386 and 80486 microprocessors, where the integer math can be executed in their native 32-bit mode. Fractint works with many adapters and graphics modes from CGA to the 1024x768, 256-color 8514/A mode. Even "larger" images, up to 2048x2048x256, can be plotted to expanded memory, extended memory, or disk: this bypasses the screen and allows you to create images with higher resolution than your current display can handle, and to run in "background" under multi-tasking control programs such as DESQview and Windows 3. Fractint is an experiment in collaboration. Many volunteers have joined Bert Tyler, the program's first author, in improving successive versions. Through electronic mail messages, first on CompuServe's PICS forum and now on 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. Fractint Version 15.0 Page 6 Fractint is freeware. The copyright is retained by the Stone Soup Group. Conditions on use: Fractint may be freely copied and distributed but may not be sold. It may be used personally or in a business - if you can do your job better by using Fractint, or use images from it, that's great! It may be given away with commercial products under the following conditions: o It must be clearly stated that Fractint does not belong to the vendor and is included as a free give-away. o It must be a complete unmodified release of Fractint, with documentation, unless other arrangements are made with the Stone Soup Group There is no warranty of Fractint's suitability for any purpose, nor any acceptance of liability, express or implied. Source code for Fractint is also freely available. See the FRACTSRC.DOC file included with it for conditions on use. (In most cases we just want credit.) ********************************************************************** * Contribution policy: Don't want money. Got money. Want admiration. * ********************************************************************** Fractint Version 15.0 Page 7 1. Fractint Commands To start the program, enter FRACTINT at the DOS prompt. The program displays an initial "credits" screen. Hitting gets you to the main menu. You can either select options from the menu by moving the highlight with the cursor, or you can enter commands directly. 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. Help is available at any time by pressing the key. AT ANY TIME, you can hit one of the keys described in the following sections to select a function. You do not need to wait for a calculation to finish, nor do you have to return to the main menu. Note that for the "typewriter" keys, upper and lower case are equivalent, e.g. and , have the same result. Many commands and parameters can be passed to FRACTINT.EXE as command- line arguments or read from a configuration file; see Chapter 5 for details. Plotting Commands Function keys & various combinations are used to select a video mode and redraw the screen. For a quick start, try one of the following: If you have MCGA, VGA, or better: If you have EGA: If you have CGA: Otherwise, monochrome: When you choose "select video mode" from the main menu, you will see a list of possible video modes. Display a help screen. The function keys available in help mode are displayed at the bottom of the help screen. or Return from a displayed image to the main menu. From the main menu, is used to exit from Fractint. 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. Display and expand the zoom box ("zoom out"). Pulling the mouse towards you with the left button down has the same effect. Fractint Version 15.0 Page 8 Cursor ("arrow") keys Move ("pan") the zoom box. Moving the mouse has the same effect. With an enhanced-keyboard BIOS, plus a cursor key moves the box 5 times faster. ("Return") Redraw the area inside the zoom box as a full-screen image. Or double click the left mouse button. If there's no zoom box on the screen, and the current image is incomplete (e.g. if you interrupted it to adjust the colors) Enter is also used to resume calculation after a pause. It is only necessary to do this when there is a message on the screen waiting to be acknowledged, such as the message shown after you save an image to disk. "Zoom out," so that your screen would fill the current zoom box, and redraw. Or double click the right mouse button. <\> (previously ) Redraw the previous image. The program tracks 25 sets of pervious coordinates and fractal types, but does not remember other options which were different for those past images. 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. The Tab function is non- destructive - if you press it while in the midst of generating an image, you will continue generating it when you return. The Tab function tells you if your image is still being generated or has finished - a handy feature for those overnight, 1024x768 resolution fractal images. If the image is incomplete, it also tells you whether it can be interrupted and resumed. (Any function other than and counts as an "interrupt".) Select a fractal type. Move the cursor to your choice and hit . Next you will be prompted for any parameters used by the selected type - hit Enter to use the defaults. Note that any values you change on this parameter screen are remembered for the next time you select the same type. Select a number of eXtended options. Brings up a full-screen menu of options, any of which you can change at will. These options are: Image generation algorithm ("passes=" option) Floating point algorithm toggle "maxiter=" iteration maximum "inside=" and "outside=" colors "logmap=" logarithmic palette value "biomorph=" option "decomp=" decomposition value "sound=" option "overwrite=" option "distest=" value Fractint Version 15.0 Page 9 "savename=" filename The "passes option" (which used to be selected by <1>, <2>, or ) selects single-pass, dual-pass, or solid-guessing (default) mode, then redraws 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. Floating point is discussed below, under the key. The other extended options are detailed elsewhere in the manual. More options which we couldn't fit under the command. Finite attractor option, continuous potential parameters, and inversion options. These options are discussed elsewhere in the manual. Toggle between Mandelbrot set images and their corresponding Julia-set images. Read the notes in chapter 2 before trying this option if you want to see anything interesting. 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. Toggles the use of floating-point algorithms. Whether floating point is in use is shown on the status screen. The floating point option can also be turned on and off using the "X" options screen. 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. (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 chapter 9.) Fractint Version 15.0 Page 10 Restart at the "credits" screen and reset most variables to their initial state. Variables which are not reset are: savename, lightname, ifs parms, video, startup filename. Shell to DOS. Return to Fractint by entering "exit" at a DOS prompt. Color Cycling Commands The color-cycling commands are available ONLY for VGA adapters and EGA adapters in 640x350x16 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-cycling mode and begin cycling the palette by shifting each color to the next "contour." Switch to color-cycling mode but do not start cycling. The normally black "overscan" border of the screen changes to white. When you are in color-cycling mode, you will either see the screen colors cycling, or will see a white "overscan" border when paused, as a reminder that you are still in this mode. The keyboard commands available once you've entered color-cycling. are described below. Bring up a HELP screen with commands specific to color cycling mode. Leave color-cycling mode. <+> 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. through Switches from simple rotation to color selection using randomly generated 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. Fractint Version 15.0 Page 11 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). Pause cycling with white overscan area. Cycling restarts with any command key (including another spacebar). - Pause cycling and reset the palette to a preset two color "straight" assignment, such as a spread from black to white. (Not for EGA) - Pause and use a two-color cyclical assignment, e.g. red->yellow->red (not for EGA). - Pause and use a 3-color cyclical assignment, e.g. green->white->blue (not for EGA). , , 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: , , Pause and decrease the red, green, or blue component of all colors by a small amount (not for EGA). or Pause and load an external color map from the files DEFAULT.MAP or ALTERN.MAP, supplied with the program. Pause and load an external color map (.MAP file). Several .MAP files are supplied with Fractint. (See chapter 6 for details on creating your own map files manually, or save customized palettes in .MAP files using the command in color-cycling or palette-editing mode.) Pause, prompt for a filename, and save the current palette to the named file (.MAP assumed). Palette Editing Commands Palette-editing mode provides a number of tools for modifying the colors in an image. It can be used only with 256 color images. Use the key to enter palette-editing mode from a displayed image or from the main menu. When this mode is entered, an empty palette frame is displayed. You can use the cursor keys to position the frame outline, and and Fractint Version 15.0 Page 12 to change its size. (The upper and lower limits on the size depend on the current video mode.) When the frame is positioned where you want it, hit Enter to display the current palette in the frame. Note that the palette frame shows R(ed) G(reen) and B(lue) values for two color registers at the top. The active color register has a solid frame, the inactive register's frame is dotted. Within the active register, the active color component is framed. Using the commands described below, you can assign particular colors to the registers and manipulate them. Note that color zero (top left) cannot be edited. Color 255 (bottom right) is also special - it can only be edited by rotating the palette to move it to another slot, then editing that slot and rotating the palette back again. Once the palette frame is displayed and filled in, the following commands are available: Bring up a HELP screen with commands specific to palette-editing mode. Leave palette-editing mode Hide the palette frame to see full image; the cross-hair remains visible and all functions remain enabled; hit again to restore the palette display. Cursor keys Move the cross-hair cursor around. In 'auto' mode (the default) the color under the center of the cross-hair is automatically assigned to the active color register. Control-Cursor keys move the cross-hair faster. A mouse can also be used to move around. Select the Red, Green, or Blue component of the active color register for subsequent commands Select previous or next color component in active register <+> <-> Increase or decrease the active color component value by 1 Numeric keypad (gray) + and - keys do the same. Increase or decrease the active color component value by 5; Moving the mouse up/down with left button held is the same <0> <1> <2> <3> <4> <5> Set the active color component's value to 0 10 20 ... 60 Select the other color register as the active one. In the default Fractint Version 15.0 Page 13 'auto' mode this results in the now-inactive register being set to remember the color under the cursor, and the now-active register changing from whatever it had previously remembered to now follow the color. <,> <.> Rotate the palette one step. The rotation matches the color-cycling mode rotation method - only colors 1 through 255 inclusive are involved. "<" ">" Rotate the palette continuously (until next keystroke) Enter Color-Cycling Mode. When you invoke color-cycling from here, it will subsequently return to palette-editing when you from it. <=> Create a smoothly shaded range of colors between the colors selected by the two color registers. Duplicate the inactive color register's values to the active color register. Stripe-shade - create a smoothly shaded range of colors between the two color registers, setting only every Nth register. After hitting , hit a numeric key from 2 to 9 to specify N. For example, if you press <3>, smooth shading is done between the two color registers, affecting only every 3rd color between them. The other colors between them remain unchanged. ... Store the current palette in a temporary save area associated with the function key. The temporary save palettes are useful for quickly comparing different palettes or the effect of some changes - see next command. The temporary palettes are only remembered until you exit from palette-editing mode. ... Restore the palette from a temporary save area. If you haven't previously saved a palette for the function key, you'll get a simple grey scale. <\> Move or resize the palette frame. The frame outline is drawn - it can then be repositioned and sized with the cursor keys, and , just as was done when first entering palette-editing mode. Hit Enter when done moving/sizing Invert frame colors. With some colors the palette is easier to see when the frame colors are interchanged. Fractint Version 15.0 Page 14 Pause and load an external color map (.MAP file). Pause, prompt for a filename, and save the current palette to the named file (.MAP assumed). Toggle 'auto' mode on or off. When on (the default), the active color register follows the cursor; when off, must be pressed to set the active register to the color under the cursor. Only useful when 'auto' is off, as described above; double clicking the left mouse button is the same as Enter Toggle 'exclude' mode on or off - when toggled on, only those image pixels which match the active color are displayed. Toggle 'exclude' range on or off - similar to , but all pixels matching colors in the range of the two color registers are displayed. Save/Restore/Print Commands Save the screen contents to disk. Progress is marked by colored lines moving down the screen's edges. The default filename is FRACT001.GIF; multiple saves in a session are automatically incremented 002, 003... Use the "savename=" parameter or options screen to change the name. By default, files left over from previous sessions are not overwritten - the first unused FRACTnnn name is used. Use the "overwrite=yes" parameter or options screen) to overwrite existing files. A save operation can be interrupted by pressing any key. If you interrupt, you'll be asked whether to keep or discard the partial file. Restore an image previously saved with , or an ordinary GIF file. After pressing you are shown the file names in the current directory which match the current file mask. To select a file to restore, move the cursor to it and press . Directories are shown in the file list with a "\" at the end of the name. When you select a directory, the contents of that directory are shown. Or, you can type the name of a different directory (and optionally a different drive) and press for a new display. You can also type a mask such as "*.XYZ" and press to display files whose name ends with the matching suffix (XYZ). Once you have selected a file to restore, a summary description of the file is shown, with a video mode selection list. Usually you can just press to go past this screen and load the image. Other choices available at this point are: Fractint Version 15.0 Page 15 Cursor keys: select a different video mode : display more information about the fractal : for help about the "err" column in displayed video modes If you restore a file into a video mode which does not have the same pixel dimensions as the file, Fractint will make some adjustments: The view window parameters (see command) will automatically be set to an appropriate size, and if the image is larger than the screen dimensions, it will be reduced by using only every Nth pixel during the restore.

Print the current fractal image on your (Laserjet, Paintjet, Epson- compatible, or PostScript) printer. See chapter 5 for how to let Fractint know about your printer setup. "Disk video" modes can be used to generate images for printing at higher resolutions than your screen supports. "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 the "3D Images" chapter for details. Restore in 3D and overlay the result on the current screen. (This only works when there's a completed image on screen, so it can't be confused with the -for-orbits toggle that ONLY operates WHILE a fractal is being generated.) Interrupting and Resuming Fractint command keys can be loosely grouped as: o Keys which suspend calculation of the current image (if one is being calculated) and automatically resume after the function. (display status information) and (display help), are the only keys in this group. o Keys which automatically trigger calculation of a new image. Examples: selecting a video mode (e.g. ); selecting a fractal type using ; using the screen to change an option such as maximum iterations. o Keys which do something, then wait for you to indicate what to do next. Examples: to go to main menu; to enter color cycling mode; to bring up a zoom box. After using a command in this group, calculation automatically resumes when you return from the function (e.g. from color cycling, to clear zoom box). There are a few fractal types which cannot resume calculation, they are noted below. Note that after saving an image with , you must press to clear the "saved" message from the screen and resume. Fractint Version 15.0 Page 16 An image which is aved before it completes can later be estored and continued. The calculation is automatically resumed when you restore such an image. When a slow fractal type resumes after an interruption in the third category above, there may be a lag while nothing visible happens. This is because most cases of resume restart at the beginning of a screen line. If unsure, you can check whether calculation has resumed with the key. The following fractal types cannot (currently) be resumed: plasma, 3d transformations, julibrot, and 3d orbital types like lorenz3d. To check whether resuming an image is possible, use the key while it is calculating. It is resumable unless there is a note under the fractal type saying it is not. The "Batch Mode" section discusses how to resume in batch mode. To estore and resume a "formula" type fractal (described later) your "formulafile" must contain the required formula. View Window The command is used to set the view window parameters described below. These parameters can be used to: o Define a small window on the screen which is to contain the generated images. Using a small window speeds up calculation time (there are less pixels to generate). You can use a small window to explore quickly, then turn the view window off to recalculate the image at full screen size. o Generate an image with a different "aspect ratio"; e.g. in a square window or in a tall skinny rectangle. o View saved GIF images which have pixel dimensions different from any mode supported by your hardware. This use of view windows occurs automatically when you restore such an image. Note that there are currently no command-line parameters for view windows - they can only be set interactively. "Preview display" Set this to "yes" to turn on view window, "no" for full screen display. While this is "no", the only view parameter which has any affect is "final media aspect ratio". When a view window is being used, all other Fractint functions continue to operate normally - you can zoom, color- cycle, and all the rest. "Reduction factor" When an explicit size is not given, this determines the view window size, as a factor of the screen size. E.g. a reduction factor of 2 makes the window 1/2 as big as the screen in both dimensions. "Final media aspect ratio" This is the height of the final image you want, divided by the width. The default is 0.75 because standard PC monitors have a height:width Fractint Version 15.0 Page 17 ratio of 3:4. E.g. set this to 2.0 for an image twice as high as it is wide. The effect of this parameter is visible only when "preview display" is enabled. "Crop starting coordinates" This parameter affects what happens when you change the aspect ratio. If set to "no", then when you change aspect ratio, the prior image will be squeezed or stretched to fit into the new shape. If set to "yes", the prior image is "cropped" to avoid squeezing or stretching. "Explicit size" Setting these to non-zero values over-rides the "reduction factor" with explicit sizes in pixels. If only the "x pixels" size is specified, the "y pixels" size is calculated automatically based on x and the aspect ratio. More about final aspect ratio: If you want to produce a high quality hard-copy image which is say 8" high by 5" down, based on a vertical "slice" of an existing image, you could use a procedure like the following. You'll need some method of converting a GIF image to your final media (slide or whatever) - Fractint can only do the whole job with a PostScript printer, it does not preserve aspect ratio with other printers. o restore the existing image o set view parameters: preview to yes, reduction to anything (say 2), aspect ratio to 1.6, and crop to yes o zoom, rotate, whatever, till you get the desired final image o set preview display back to no o trigger final calculation in some high res disk video mode, using the appropriate video mode function key o print directly to a PostScript printer, or save the result as a GIF file and use external utilities to convert to hard copy. More Zoom Box Commands In addition to resizing the zoom box and moving it around, you can do some rather warped things with it. The general approach is always the same: Frame an area, then to expand what's in the frame to fill the whole screen (zoom in); or to shrink the current image into the framed area (zoom out). With a mouse, double-click the left button to zoom in, double click the right button to zoom out. , Use to initially bring up the zoom box. It starts at full screen size. Subsequent use of these keys makes the zoom box smaller or larger. Using to enlarge the zoom box when it is already at maximum size removes the zoom box from the display. Moving the mouse away from you or toward you while holding the left button down performs the same functions as these keys. The cursor "arrow" keys, holding with the "arrow" keys, and moving the mouse without holding any buttons down, moves the zoom box. Fractint Version 15.0 Page 18 Panning: If you move a fullsize zoombox and don't change anything else before performing the zoom, Fractint just moves what's already on the screen and then fills in the new edges, to reduce drawing time. This feature applies to most fractal types but not all. A side effect is that while an image is incomplete, a full size zoom box moves in steps larger than one pixel. Fractint keeps the box on multiple pixel boundaries, to make panning possible. As a multi-pass (e.g. solid guessing) image approaches completion, the zoom box can move in smaller increments. , Holding and pressing the numeric keypad's + or - keys rotates the zoom box. Moving the mouse left or right while holding the right button down performs the same function. , These commands change the zoom box's "aspect ratio", stretching or shrinking it vertically. Moving the mouse away from you or toward you while holding both buttons (or the middle button on a 3-button mouse) down performs the same function. There are no commands to directly stretch or shrink the zoom box horizontally - the same effect can be achieved by combining vertical stretching and resizing. , These commands "skew" the zoom box, moving the top and bottom edges in opposite directions. Moving the mouse left or right while holding both buttons (or the middle button on a 3-button mouse) down performs the same function. There are no commands to directly skew the left and right edges - the same effect can be achieved by using these functions combined with rotation. , These commands change the zoom box color. This is useful when you're having trouble seeing the zoom box against the colors around it. Moving the mouse away from you or toward you while holding the right button down performs the same function. You may find it difficult to figure out what combination of size, position, rotation, stretch, and skew to use to get a particular result. (We do.) A good way to get a feel for all these functions is to play with the Gingerbreadman fractal type. Gingerbreadman's shape makes it easy to see what you're doing to him. A warning though: Gingerbreadman will run forever, he's never quite done! So, pre-empt with your next zoom when he's baked enough. If you accidentally change the shape of your zoom box, or rotate and forget which way is up, just use to make it bigger until it is removed from the display, then to get a fresh one. With the mouse, after removing the old zoom box from the display, release and re- press the left button to get a fresh one. If your screen does not have a 4:3 "aspect ratio" (i.e. if the visible display area on it is not 1.333 times as wide as it is high), rotating and zooming will have some odd effects - angles will change, including the zoom box's shape itself, circles (if you are so lucky as to see any Fractint Version 15.0 Page 19 with a non-standard aspect ratio) become non-circular, and so on. The vast majority of PC screens *do* have a 4:3 aspect ratio. Zooming is not implemented for the plasma and diffusion fractal types, nor for verlayed and <3>d images. A few fractal types support zooming but do not support rotation and skewing - nothing happens when you try it. 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. Or use the uessing mode and pre- empt with a zoom before it finishes. Plotting time also varies with the maximum iteration setting, the fractal type, and your choice of drawing mode. Solid-guessing (the default) is fastest, but it can be wrong: perfectionists will want to use dual-pass mode (its first-pass preview is handy if you might zoom pre-emptively) or single-pass mode. When you start systematically exploring, you can save time (and hey, every little bit helps -- these "objects" are INFINITE, remember!) by aving your last screen in a session to a file, and then going straight to it the next time with FRACTINT FRACTxxx (the .GIF 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 chapter 6 for more on batch mode and the command-line arguments the command writes. Fractint Version 15.0 Page 20 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 chapter 9. 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 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 (type=mandel) This set is the classic: the only one implemented in many plotting programs, and the source of most of the printed fractal images published in recent years. Like most of the other types in Fractint, it is simply a graph: the x (horizontal) and y (vertical) coordinate axes represent ranges of two independent quantities, with various colors used to symbolize levels of a third quantity which depends on the first two. So far, so good: basic analytic geometry. Now things get a bit hairier. The x axis is ordinary, vanilla real numbers. The y axis is an imaginary number, i.e. a real number times i, where i is the square root of -1. Every point on the plane -- in this case, your PC's display screen -- represents a complex number of the form: x-coordinate + i * y-coordinate If your math training stopped before you got to imaginary and complex numbers, this is not the place to catch up. Suffice it to say that they are just as "real" as the numbers you count fingers with (they're used every day by electrical engineers) and they can undergo the same kinds of algebraic operations. OK, now pick any complex number -- any point on the complex plane -- and call it C, a constant. Pick another, this time one which can vary, and call it Z. Starting with Z=0 (i.e., at the origin, where the real and imaginary axes cross), calculate the value of the expression Z^2 + C Take the result, make it the new value of the variable Z, and calculate again. Take that result, make it Z, and do it again, and so on: in mathematical terms, iterate the function Z(n+1) = Z(n)^2 + C. For certain values of C, the result "levels off" after a while. For all others, it grows without limit. The Mandelbrot set you see at the start -- the solid-colored lake (blue by default), the blue circles sprouting Fractint Version 15.0 Page 21 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 options screen 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 (type=julia) 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 outside, and it breaks up into scattered points. In fact, all Julia sets for C within the M-set Fractint Version 15.0 Page 22 share the "connected" property of the M-set, and all those for C outside lack it. Fractint's spacebar toggle lets you "flip" between any view of the M-set and the Julia set for the point C at the center of that screen. You can then toggle back, or zoom your way into the Julia set for a while and then return to the M-set. So if the infinite complexity of the M-set palls, remember: each of its infinite points opens up a whole new Julia set. Historically, the Julia sets came first: it was while looking at the M- set as an "index" of all the Julia sets' origins that Mandelbrot noticed its properties. The relationship between the Mandelbrot set and Julia set can hold between other sets as well. Many of Fractint's types are "Mandelbrot/Julia" pairs (sometimes called "M-sets" or "J-sets". All these are generated by equations that are of the form z(k+1) = f(z(k),c), where the function orbit is the sequence z(0), z(1), ..., and the variable c is a complex parameter of the equation. The value c is fixed for "Julia" sets and is equal to the first two parameters entered with the "params=Creal/Cimag" command. The initial orbit value z(0) is the complex number corresponding to the screen pixel. For Mandelbrot sets, the parameter c is the complex number corresponding to the screen pixel. The value z(0) is c plus a perturbation equal to the values of the first two parameters. See discussion of "type=mandellambda". This approach may or may not be the "standard" way to create "Mandelbrot" sets out of "Julia" sets. Some equations have additional parameters. These values is entered as the third for fourth params= value for both Julia and Mandelbrot sets. The variables x and y refer to the real and imaginary parts of z; similarly, cx and cy are the real and imaginary parts of the parameter c and fx(z) and fy(z) are the real and imaginary parts of f(z). The variable c is sometimes called lambda for historical reasons. NOTE: if you use the "PARAMS=" argument to warp the M-set by starting with an initial value of Z other than 0, the M-set/J-sets correspondence breaks down and the spacebar toggle no longer works. 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 ype command and enter "newtbasin" in response to the prompt. You will be asked for a parameter, the "order" of the equation (an integer from 3 through 10 -- Fractint Version 15.0 Page 23 3 for x^3-1, 7 for x^7-1, etc.). A second parameter is a flag to turn on alternating shades showing changes in the number of iterations needed to attract an orbit. Some people like stripes and some don't, as always, Fractint gives you a choice! The coloring of the plot shows the "basins of attraction" for each root of the polynomial -- i.e., an initial guess within any area of a given color would lead you to one of the roots. As you can see, things get a bit weird along certain radial lines or "spokes," those being the lines between actual roots. By "weird," we mean infinitely complex in the good old fractal sense. Zoom in and see for yourself. This fractal type is symmetric about the origin, with the number of "spokes" depending on the order you select. It uses floating-point math if you have an FPU, or a somewhat slower integer algorithm if you don't have one. 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/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! Fractint Version 15.0 Page 24 Lambda Sets (type=lambda) This type calculates the Julia set of the formula lambda*Z*(1-Z). That is, the value Z[0] is initialized with the value corresponding to each pixel position, and the formula iterated. The pixel is colored according to the iteration when the sum of the squares of the real and imaginary parts exceeds 4. Two parameters, the real and imaginary parts of lambda, are required. Try 0 and 1 to see the classical fractal "dragon". Then try 0.2 and 1 for a lot more detail to zoom in on. It turns out that all quadratic Julia-type sets can be calculated using just the formula z^2+c (the "classic" Julia"), so that this type is redundant, but we include it for reason of it's prominence in the history of fractals. Mandellambda Sets (type=mandellambda) This type is the "Mandelbrot equivalent" of the lambda set. A comment is in order here. Almost all the Fractint "Mandelbrot" sets are created from orbits generated using formulas like z(n+1) = f(z(n),C), with z(0) and C initialized to the complex value corresponding to the current pixel. Our reasoning was that "Mandelbrots" are maps of the corresponding "Julias". Using this scheme each pixel of a "Mandelbrot" is colored the same as the Julia set corresponding to that pixel. However, Kevin Allen informs us that the MANDELLAMBDA set appears in the literature with z(0) initialized to a critical point (a point where the derivative of the formula is zero), which in this case happens to be the point (.5,0). Since Kevin knows more about Dr. Mandelbrot than we do, and Dr. Mandelbrot knows more about fractals than we do, we defer! Starting with version 14 Fractint calculates MANDELAMBDA Dr. Mandelbrot's way instead of our way. But ALL THE OTHER "Mandelbrot" sets in Fractint are still calculated OUR way! (Fortunately for us, for the classic Mandelbrot Set these two methods are the same!) Well now, folks, apart from questions of faithfulness to fractals named in the literature (which we DO take seriously!), if a formula makes a beautiful fractal, it is not wrong. In fact some of the best fractals in Fractint are the results of mistakes! Nevertheless, thanks to Kevin for keeping us accurate! (See description of "initorbit=" command for a way to experiment with different orbit intializations). 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 Fractint Version 15.0 Page 25 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. Lambdafn (type=lambdafn) Function=[sin|cos|sinh|cosh|exp|log|sqr]) is specified with this type. Prior to version 14, these types were lambdasine, lambdacos, lambdasinh, lambdacos, and lambdaexp. Where we say "lambdasine" or some such below, the good reader knows we mean "lambdafn with function=sin".) These types calculate the Julia set of the formula lambda*fn(Z), for various values of the function "fn", where lambda and Z are both complex. Two values, the real and imaginary parts of lambda, should be given in the "params=" option. For the feathery, nested spirals of LambdaSines and the frost-on-glass patterns of LambdaCosines, make the real part = 1, and try values for the imaginary part ranging from 0.1 to 0.4 (hint: values near 0.4 have the best patterns). In these ranges the Julia set "explodes". For the tongues and blobs of LambdaExponents, try a real part of 0.379 and an imaginary part of 0.479. A 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! Fractint Version 15.0 Page 26 Mandelfn (type=mandelfn) Function=[sin|cos|sinh|cosh|exp|log|sqr]) is specified with this type. Prior to version 14, these types were mandelsine, mandelcos, mandelsinh, mandelcos, and mandelexp. Same comment about our lapses into the old terminology as above! These are "pseudo-Mandelbrot" mappings for the LambdaFn Julia functions. They map to their corresponding Julia sets via the spacebar command in exactly the same fashion as the original M/J sets. In general, they are interesting mainly because of that property (the function=exp set in particular is rather boring). Generate the appropriate "Mandelfn" set, zoom on a likely spot where the colors are changing rapidly, and hit the spacebar key to plot the Julia set for that particular point. Try "FRACTINT TYPE=MANDELFN CORNERS=4.68/4.76/-.03/.03 FUNCTION=COS" for a graphic demonstration that we're not taking Mandelbrot's name in vain here. We didn't even know these little buggers were here until Mark Peterson found this a few hours before the version incorporating Mandelfns was released. Note: If you created images using the lambda or mandel "fn" types prior to version 14, and you wish to update the fractal information in the "*.fra" file, simply read the files and save again. You can do this in batch mode via a command line like: "fractint oldfile.fra savename=newfile.gif batch=yes" For example, this procedure can convert a version 13 "type=lambdasine" image to a version 14 "type=lambdafn function=sin" GIF89a image. We do not promise to keep this "backward compatibility" past version 14 - if you want to keep the fractal information in your *.fra files accurate, we recommend conversion. Barnsley Mandelbrot/Julia Sets (barnsleym1/.../j3) Michael Barnsley has written a fascinating college-level text, "Fractals Everywhere," on fractal geometry and its graphic applications. (See the bibliography, Appendix. 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 spacebar. The parameters have the same meaning as they do for the "regular" Mandelbrot and Julia. For types M1, M2, and M3, they are used to "warp" the image by setting the initial value of Z. For the types J1 through J3, they are the values of C in the generating formulas, found in Appendix A. Be sure to try the rbit function while plotting these types. Fractint Version 15.0 Page 27 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 -- parameters for a set of functions that map a region of two-dimensional space onto itself. In principle (and increasingly in practice), a scene of any level of complexity and detail can be stored as a handful of numbers, achieving amazing "compression" ratios... how about a super-VGA image of a forest, more than 300,000 pixels at eight bits apiece, from a 1-KB "seed" file? Again, Michael Barnsley and his co-workers at the Georgia Institute of Technology are to be thanked for pushing the development of these iterated function systems (IFS). Invoking Fractint with "TYPE=ifs" defaults to an IFS "fern". Other IFS images can be created via the *.IFS files included with Fractint (FERN, TRIANGLE, TREE, and more), 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 (the files supplied with Fractint are almost all 2D). You will then be asked to pick which function to edit. Hitting 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 presumably 2D, we have added rotation, translation, and perspective capabilities. These share values with the same variables used in Fractint's other 3D facilities; for their meaning see "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) Fractint Version 15.0 Page 28 Alternatively, entering from main screen and then selecting "3D Transform Parameters" will allow you to modify 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 USED to be dedicated to Bruce Goren, as a bribe to get him to send us MORE knockout stereo slides of 3D ferns, now that we have made it so easy! Bruce, what have you done for us *LATELY* ?? (Just kidding, really!) Sierpinski Gasket (type=sierpinski) Another pre-Mandelbrot classic, this one found by W. Sierpinski around World War I. It is generated by dividing a triangle into four congruent smaller triangles, doing the same to each of them, and so on, yea, even unto infinity. (Notice how hard we try to avoid reiterating "iterating"?) 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/Julia (type=mandel4/julia4) These fractal types are the moral equivalent of the original M and J sets, except that they use the formula Z(n+1) = Z(n)^4 + C, which adds additional pseudo-symmetries to the plots. The "Mandel4" set maps to the "Julia4" set via -- surprise! -- the spacebar toggle. The M4 set is kind of boring at first (the area between the "inside" and the "outside" of the set is pretty thin, and it tends to take a few zooms to get to any interesting sections), but it looks nice once you get there. The Julia sets look nice right from the start. Other powers, like Z(n)^3 or Z(n)^7, work in exactly the same fashion. We used this one only because we're lazy, and Z(n)^4 = (Z(n)^2)^2. Fractint Version 15.0 Page 29 Distance Estimator (distest=nnn) This used to be type=demm and type=demj. These types have not died, but are only hiding! They are equivalent to the mandel and julia types with the "distest=" option selected with a predetermined value. The "Distance Estimator Method" (described under "Doodads, Bells and Whistles") can be used to produce higher quality images of M and J sets, especially suitable for printing in black and white. If you have some *.fra files made with the old types demm/demj, you may want to convert them to the new form. See "MANDELFN" section for directions to carry out the conversion. Pickover Mandelbrot/Julia Types (type=manfn+zsqrd/julfn+zsqrd, manzpowr/julzpowr, manzzpwr/julzzpwr, manfn+exp/julfn+exp - formerly included man/julsinzsqrd and man/julsinexp which have now been generalized) These types have been explored by Clifford A. Pickover, of the IBM Thomas J. Watson Research center. As implemented in Fractint, they are regular Mandelbrot/Julia set pairs that may be plotted with or without the "biomorph" 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=manfn+sqrd biomorph=0 corners=-8/8/-6/6 function=sin to see a big biomorph digesting little biomorphs! Pickover Popcorn (type=popcorn/popcornjul) Here is another Pickover idea. This one computes and plots the orbits of the dynamic system defined by: x(n+1) = x(n) - h*sin(y(n)+tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n)+tan(3*x(n)) with the initializers x(0) and y(0) equal to ALL the complex values within the "corners" values, and h=.01. ALL these orbits are superimposed, resulting in "popcorn" effect. You may want to use a maxiter value less than normal - Pickover recommends a value of 50. As a bonus, type=popcornjul shows the Julia set generated by these same equations with the usual escape-time coloring. Turn on orbit viewing with the "O" command, and as you watch the orbit pattern you may get some insight as to where the popcorn comes from. Although you can zoom and rotate popcorn, the results may not be what you'd expect, due to the Fractint Version 15.0 Page 30 superimposing of orbits and arbitrary use of color. Just for fun we added type popcornjul, which is the plain old Julia set calculated from the same formula. Peterson Variations (xxmarksxx types) (type=marksmandel, marksjulia, 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. 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! Fractint Version 15.0 Page 31 Scott Taylor / Lee Skinner Variations (type=fn(z*z), fn*fn, fn*z+z, fn+fn, sqr(1/fn), sqr(fn), spider, tetrate, manowar) Two of Fractint's faithful users went bonkers when we introduced the "formula" type, and came up with all kinds of variations on escape-time fractals using trig functions. We decided to put them in as regular types, but there were just too many! So we defined the types with variable functions and let you, the, overwhelmed user, specify what the functions should be! Thus Scott Taylor's "z = sin(z) + z^2" formula type is now the "fn+fn" regular type, and EITHER function can be one of sin, cos, sinh, cosh, exp, log, or sqr. Plus we give you 4 parameters to set, the complex coefficients of the two functions! Thus the innocent- looking "fn+fn" type is really at least 36 different types in disguise, not counting the damage done by the parameters! Lee informs us that you should not judge fractals by their "outer" appearance. For example, the images produced by z = sin(z) + z^2 and z = sin(z) - z^2 look very similar, but are different when you zoom in. Kam Torus (type=kamtorus, kamtorus3d) This type is created by superimposing orbits generated by a set of equations, with a variable incremented each time. x(0) = y(0) = orbit/3; x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a) y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a) After each orbit, 'orbit' is incremented by a step size. The parameters are angle "a", step size for incrementing 'orbit', stop value for 'orbit', and points per orbit. Try this with a stop value of 5 with sound=x for some weird fractal music (ok, ok, fractal noise)! You will also see the KAM Torus head into some chaotic territory that Scott Taylor wanted to hide from you by setting the defaults the way he did, but now we have revealed all! The 3D variant is created by treating 'orbit' as the z coordinate. Bifurcation (type=bifxxx) The wonder of fractal geometry is that such complex forms can arise from such simple generating processes. A parallel surprise has emerged in the study of dynamical systems: that simple, deterministic equations can yield chaotic behavior, in which the system never settles down to a steady state or even a periodic loop. Often such systems behave normally up to a certain level of some controlling parameter, then go through a transition in which there are two possible solutions, then four, and finally a chaotic array of possibilities. This emerged many years ago in biological models of population growth. Consider a (highly over-simplified) model in which the rate of growth is Fractint Version 15.0 Page 32 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 population values, from 0 to 4/3. Notice that within the chaotic region, there are narrow bands where there is a small, odd number of stable values. It turns out that the geometry of this branching is fractal; zoom in where changing pixel colors look suspicious, and see for yourself. 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 (smaller) one as the first "PARAMS=" value. "PARAMS=1" produces an unfiltered map. Many formulae can be used to produce bifurcations. Mitchel Feigenbaum studied lots of bifurcations in the mid-70's, using a HP-65 calculator (IBM PCs, Fractals, and Fractint, were all Sci-Fi then !). He studied where bifurcations occurred, for the formula r*p*(1-p), the one described above. He found that the ratios of lengths of adjacent areas of bifurcation were four and a bit. These ratios vary, but, as the growth rate increases, they tend to a limit of 4.669+. This helped him guess where bifurcation points would be, and saved lots of time. When he studied bifurcations of r*sin(PI*p) he found a similar pattern, which is not surprising in itself. However, 4.669+ popped out, again. Different formulae, same number ? Now, THAT's surprising ! He tried many other formulae and ALWAYS got 4.669+ - Hot Damn !!! So hot, in fact, that he phoned home and told his Mom it would make him Famous ! He also went on to tell other scientists. The rest is History... (It has been conjectured that if Feigenbaum had a copy of Fractint, and used it to study bifurcations, he may never have found his Number, as it only became obvious from long perusal of hand-written lists of values, without the distraction of wild color-cycling effects !). We now know that this number is as universal as PI or E. It appears in situations ranging from fluid-flow turbulence, electronic oscillators, chemical reactions, and even the Mandelbrot Set - yup, fraid so: "budding" of the Mandelbrot Set along the negative real axis occurs at intervals determined by Feigenbaum's Number, 4.669201660910..... Fractint Version 15.0 Page 33 Fractint does not make direct use of the Feigenbaum Number (YET !). However, it does now reflect the fact that there is a whole sub-species of Bifurcation-type fractals. Those implemented to date, and the related formulae, (writing P for pop[n+1] and p for pop[n]) are : bifurcation P = p + r*p*(1-p) Verhulst Bifurcations. biflambda P = r*p*(1-p) Real equivalent of Lambda Sets. bif+sinpi P = p + r*sin(PI*p) Population scenario based on... bif=sinpi P = r*sin(PI*p) ...Feigenbaum's second formula. It took a while for bifurcations to appear here, despite them being over a century old, and intimately related to chaotic systems. However, they are now truly alive and well in Fractint! Lorenz Attractors (type=lorenz/lorenz3d) The "Lorenz Attractor" is a "simple" set of three deterministic equations developed by Edward Lorenz while studying the non- repeatability of weather patterns. The weather forecaster's basic problem is that even very tiny changes in initial patterns ("the beating of a butterfly's wings" - the official term is "sensitive dependence on initial conditions") eventually reduces the best weather forecast to rubble. The lorenz attractor is the plot of the orbit of a dynamic system consisting of three first order non-linear differential equations. The solution to the differential equation is vector-valued function of one variable. If you think of the variable as time, the solution traces an orbit. The orbit is made up of two spirals at an angle to each other in three dimensions. We change the orbit color as time goes on to add a little dazzle to the image. The equations are: dx/dt = -a*x + a*y dy/dt = b*x - y -z*x dz/dt = -c*z + x*y We solve these differential equations approximately using a method known as the first order taylor series. Calculus teachers everywhere will kill us for saying this, but you treat the notation for the derivative dx/dt as though it really is a fraction, with "dx" the small change in x that happens when the time changes "dt". So multiply through the above equations by dt, and you will have the change in the orbit for a small time step. We add these changes to the old vector to get the new vector after one step. This gives us: xnew = x + (-a*x*dt) + (a*y*dt) ynew = y + (b*x*dt) - (y*dt) - (z*x*dt) znew = z + (-c*z*dt) + (x*y*dt) (default values: dt = .02, a = 5, b = 15, c = 1) We connect the successive points with a line, project the resulting 3D orbit onto the screen, and voila! The Lorenz Attractor! Fractint Version 15.0 Page 34 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 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 coefficients used in the differential equation (a, b, and c). The default values are 5, 15, and 1. Try changing these a little at a time to see the result. 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 Fractint Version 15.0 Page 35 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 (type=gingerbreadman) 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. Fractint Version 15.0 Page 36 After you get it working, send your code to one of the authors and we might just add it to the next release of Fractint, with full credit to you. Our criteria are: 1) an interesting image and 2) a formula significantly different from types already supported. (Bribery may also work. THIS author is completely honest, but I don't trust those other guys.) Be sure to include an explanation of your algorithm and the 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 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 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(), Fractint Version 15.0 Page 37 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 modulus squared operator |z| is also parenthetic and always sets the imaginary component to zero. This means 'c * |z - 4|' first subtracts 4 from z, calculates the modulus squared then multiplies times 'c'. Nested modulus squared operators require overriding parenthesis: c * |z + (|pixel|)| The 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 lieu 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) = . . . . Fractint Version 15.0 Page 38 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 spatial dimensions of height and width, and the Mandelbrot 'y' dimension as the third spatial 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 spatial 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. Fractint Version 15.0 Page 39 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 gray. If it appears bluish lower the ratio. Julibrots can only be shown in 256 red/blue colors for viewing in either stereo-graphic (red/blue funny glasses) or gray-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. Magnetic Fractals (type=magnet1m/.../magnet2j) These fractals use formulae derived from the study of hierarchical lattices, in the context of magnetic renormalisation transformations. This kinda stuff is useful in an area of theoretical physics that deals with magnetic phase-transitions (predicting at which temperatures a given substance will be magnetic, or non-magnetic). In an attempt to clarify the results obtained for Real temperatures (the kind that you and I can feel), the study moved into the realm of Complex Numbers, aiming to spot Real phase-transitions by finding the intersections of lines representing Complex phase-transitions with the Real Axis. The first people to try this were two physicists called Yang and Lee, who Fractint Version 15.0 Page 40 found the situation a bit more complex than first expected, as the phase boundaries for Complex temperatures are (surprise!) fractals. And that's all the technical (?) background you're getting here! For more details (are you SERIOUS ?!) read "The Beauty of Fractals". When you understand it all, you might like to re-write this section, before you start your new job as a professor of theoretical physics... In Fractint terms, the important bits of the above are "Fractals", "Complex Numbers", "Formulae", and "The Beauty of Fractals". Lifting the Formulae straight out of the Book and iterating them over the Complex plane (just like the Mandelbrot set) produces Fractals. The formulae are a bit more complicated than the Z^2+C used for the Mandelbrot Set, that's all. They are : / Z^2 + (C-1) \ MAGNET1 : | ------------- | ^ 2 \ 2*Z + (C-2) / / Z^3 + 3*(C-1)*Z + (C-1)*(C-2) \ MAGNET2 : | --------------------------------------- | ^ 2 \ 3*(Z^2) + 3*(C-2)*Z + (C-1)*(C-2) - 1 / These aren't quite as horrific as they look (oh yeah ?!) as they only involve two variables (Z and C), but cubing things, doing division, and eventually squaring the result (all in Complex Numbers) don't exactly spell S-p-e-e-d ! These are NOT the fastest fractals in Fractint ! As you might expect, for both formulae there is a single related Mandelbrot-type set (magnet1m, magnet2m) and an infinite number of related Julia-type sets (magnet1j, magnet2j), with the usual toggle between the corresponding Ms and Js via the spacebar. If you fancy delving into the Julia-types by hand, you will be prompted for the Real and Imaginary parts of the parameter denoted by C. The result is symmetrical about the Real axis (and therefore the initial image gets drawn in half the usual time) if you specify a value of Zero for the Imaginary part of C. Fractint Historical Note: Another complication (besides the formulae) in implementing these fractal types was that they all have a finite attractor (1.0 + 0.0i), as well as the usual one (Infinity). This fact spurred the development of Finite Attractor logic in Fractint. Without this code you can still generate these fractals, but you usually end up with a pretty boring image that is mostly deep blue "lake", courtesy of Fractint's standard Periodicity Checking. See the Finite Attractors section for more information on this aspect of Fractint internals. (Thanks to Kevin Allen for Magnetic type documentation above). Fractint Version 15.0 Page 41 L-Systems (type=lsystem) These fractals are constructed from line segments using rules specified in drawing commands. Starting with an initial string, the axiom, transformation rules are applied a specified number of times, to produce the final command string which is used to draw the image. Like the type=formula fractals, this type requires a separate data file. A sample file, FRACTINT.L, is included with this distribution. When you select type lsystem, you are asked for the filename. Next you are asked for the lsystem name you wish to run. Last, you are asked for one parameter - the "order", or number of times to execute all the transformation rules. It is wise to start with small orders, because the size of the substituted command string grows exponentially and it is very easy to exceed your resolution. (Higher orders take longer to generate too.) The command line options "lname=" and "lfile=" can be used to over-ride the default file name and lsystem name. Each L-System entry in the file contains a specification of the angle, the axiom, and the transformation rules. Each item must appear on its own line and each line must be less than 160 characters long. The statement "angle n" sets the angle to 360/n degrees; n must be an integer greater than two and less than fifty. "Axiom string" defines the axiom. Transformation rules are specified as "a=string" and convert the single character 'a' into "string." If more than one rule is specified for a single character all of the strings will be added together. This allows specifying transformations longer than the 160 chararacter limit. Transformation rules may operate on any characters except space, tab or '}'. A line which starts with ;; is a comment line. Here is a sample lsystem: | This comment section would NOT be legal. | Comments inside an L-System start with ;; at | the start of the line. Dragon { | Name of lsystem, { indicates start ;; Comment! | This is a comment Angle 8 | Specify the angle increment to 45 degrees Axiom FX | Starting character string F= | First rule: Delete 'F' y=+FX--FY+ | Change 'y' into "+fx--fy+" x=-FX++FY- | Similar transformation on 'x' } | final } indicates end The standard drawing commands are: F Draw forward G Move forward (without drawing) + Increase angle - Decrease angle Fractint Version 15.0 Page 42 | Try to turn 180 degrees. (If angle is odd, the turn will be the largest possible turn less than 180 degrees.) These commands increment angle by the user specified angle value. They should be used when possible because they are fast. If greater flexibility is needed, use the following commands which keep a completely separate angle pointer which is specified in degrees. D Draw forward M Move forward \nn Increase angle nn degrees /nn Decrease angle nn degrees Color control: Cnn Select color nn nn decrement color by nn Advanced commands: ! Reverse directions (Switch meanings of +, - and \, /) @nnn Multiply line segment size by nnn nnn may be a plain number, or may be preceeded by I for inverse, or Q for sqare root. (e.g. @IQ2 divides size by the square root of 2) [ Push. Stores current angle and position on a stack ] Pop. Return to location of last push Other characters are perfectly legal in command strings. They are ignored for drawing purposes, but can be used to achieve complex translations. Fractint Version 15.0 Page 43 3. Doodads, Bells, and Whistles Distance Estimator Method This is Phil Wilson's implementation of an alternate method for the M and J sets, based on work by mathematician John Milnor and described in "The Science of Fractal Images", p. 198. While it takes full advantage of your color palette, one of the best uses is in preparing monochrome images for a printer. Using the 1600x1200x2 disk-video mode and an HP LaserJet, we have produced pictures of quality equivalent to the black and white illustrations of the M-set in "The Beauty of Fractals." The distance estimator method widens very thin "strands" which are part of the "inside" of the set. Instead of hiding invisibly between pixels, these strands are made one pixel wide. To turn on the distance estimator method with any escape time type fractal, set the "distest" value on the options screen to a nonzero value. You should use 1 or 2 pass mode to avoid missing some of the thin strands made visible by distance estimator. For the highest quality, "maxiter" should also be set to a high value, say 1000 or so. You'll probably also want "inside" set to zero, to get a black interior. In color modes, this method also produces more evenly spaced contours. Set "distest" to a higher value for narrower color bands, a lower value for wider ones. 1000 is a good value to start with. Setting "distest" automatically toggles to floating point mode. When you reset distest back to zero, remember to also turn off floating point mode if you want it off. Unfortunately, images using the distance estimator method can take many hours to calculate even on a fast machine with a coprocessor, especially if a high "maxiter" value is used. One way of dealing with this is to leave distest turned off while you find and frame an image. Then hit to save the coordinates to FRABATCH.BAT. Edit the batch file, adding "distest=1", "video=something" to select a high-resolution monochrome disk-video mode, "maxiter=1000", "inside=0" and "batch=yes". (Watch out for the line exceeding the DOS limit of 120 characters - if it does, use "fractint @filename" in your batch and put the parameters in another file "filename".) Run the batch file when you won't be needing your machine (over the weekend?) 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.) Fractint Version 15.0 Page 44 The Inversion option performs a related transformation on most of the fractal types. You define the center point and radius of a circle; Fractint maps each point inside the circle to a corresponding point outside, and vice-versa. This is known to mathematicians as inverting (or if you want to get precise, "everting") the plane, and is something they can contemplate without getting a headache. John Milnor, 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 inversion parameters on the options screen allow entry of the radius and center coordinates of the inversion circle. A default choice of -1 sets the radius at 1/6 the smaller dimension of the image currently on the screen. The default values for Xcenter and Ycenter use the coordinates currently mapped to the center of the screen. Try this one out with a Newton plot, so its radial "spokes" will give you something to hang on to. Plot a Newton-method image, then set the inversion radius to 1, with default center coordinates. The center "explodes" to the periphery. Inverting through a circle not centered on the origin produces bizarre effects that we're not even going to try to describe. Aren't computers wonderful? Decomposition You'll remember that most fractal types are calculated by iterating a simple function of a complex number, producing another complex number, until either the number exceeds some pre-defined "bailout" value, or the iteration limit is reached. The pixel corresponding to the starting point is then colored based on the result of that calculation. The decomposition option ("decomp=", on the screen) toggles to another coloring protocol. Here the points are colored according to which quadrant of the complex plane (negative real/positive imaginary, positive real/positive imaginary, etc.) the final value is in. If you use 4 as the parameter, points ending up in each quadrant are given their own color; if 2 (binary decomposition), points in alternating quadrants are given 2 alternating colors. The result is a kind of warped checkerboard coloring, even in areas that would ordinarily be part of a single contour. Remember, for the M-set all points whose final values exceed 2 (by any amount) after, say, 80 iterations are normally the same color; under decomposition, Fractint runs [bailout-value] iterations and then colors according to where the actual final value falls on the complex plane. Fractint Version 15.0 Page 45 When using decomposition, a higher bailout value will give a more accurate plot, at some expense in speed. You might want to set the bailout value (in the parameters prompt following selection of a new fractal type; present for most but not all types) to a higher value than the default. A value of about 50 is a good compromise for M/J sets. Logarithmic Palettes By default, Fractint maps iterations to colors 1-1. 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, the entire range of iteration values is compressed to map to one span of the color range. This results in spectacularly different images if you are using a high iteration limit near the current iteration maximum of 32000 and are zooming in on an area near a "lakelet". Fractint has a number of types of compressed palette, selected by the "Log Palette" line on the screen, or by the "logmap=" command line parameter. logmap=1: for standard logarithmic palette. logmap=-1: "old" logarithmic palette. This variant was the only one used before Fractint 14.0. It differs from logmap=1 in that some colors are not used - logmap=1 "spreads" low color numbers which are unused by logmap=-1's pure logarithmic mapping so that all colors are assigned. logmap=N (>1): Same as logmap=1, but starting from iteration count N. Pixels with iteration counts less than N are mapped to color 1. This is useful when zooming in an area near the lake where no points in the image have low iteration counts - it makes use of the low colors which would otherwise be unused. logmap=-N (<-1): Similar to logmap=N, but uses a square root distribution of the colors instead of a logarithmic one. When using a compressed palette in a 256 color mode, we suggest changing your colors from the usual defaults. The last few colors in the default IBM VGA color map are black. This results in points nearest the "lake" smearing into a single dark band, with little contrast from the blue (by default) lake. 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 bailout value. Normally the pixel corresponding to that orbit is colored according to the iteration when bailout happened. To create biomorphs, this is modified so that if Fractint Version 15.0 Page 46 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. Starfields Once you have generated your favorite fractal image, you can convert it into a fractal starfield with the 'a' transformation (for 'astronomy'? - once again, all of the good letters were gone already). Stars are generated on a pixel-by-pixel basis - the odds that a particular pixel will coalesce into a star are based (partially) on the color index of that pixel. (The following was supplied by Mark Peterson, the starfield author). If the screen were entirely black and the 'Star Density per Pixel' were set to 30 then a starfield transformation would create an evenly distributed starfield with an average of one star for every 30 pixels. If you're on a 320x200 screen then you have 64000 pixels and would end up with about 2100 stars. By introducing the variable of 'Clumpiness' we can create more stars in areas that have higher color values. At 100% Clumpiness a color value of 255 will change the average of finding a star at that location to 50:50. A lower clumpiness values will lower the amount of probability weighting. To create a spiral galaxy draw your favorite spiral fractal (IFS, Julia, or Mandelbrot) and perform a starfield transformation. For general starfields I'd recommend transforming a plasma fractal. Real starfields have many more dim stars than bright ones because very few stars are close enough to appear bright. To achieve this effect the program will create a bell curve based on the value of ratio of Dim stars to bright stars. After calculating the bell curve the curve is folded in half and the peak used to represent the number of dim stars. Starfields can only be shown in 256 colors. Fractint will automatically try to load ALTERN.MAP and abort if the map file cannot be found. Fractint Version 15.0 Page 47 4. "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 .GIF images as a 3D PROJECTION or STEREO IMAGE PAIR. The iteration values you've come to know and love, the ones that determine pixel colors, are translated into "height" so that your saved screen becomes a landscape viewed in perspective. You can even wrap the landscape onto a sphere for realistic-looking planets and moons that never existed outside your PC! We suggest starting with a saved plasma-cloud screen. Hit <3> in main command mode to begin the process. Next, select the file to be transformed, and the video mode. (Usually you want the same video mode the file was generated in; other choices may or may not work.) 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 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 will take you back to the previous screen. Once you're familiar with the effects of the 3D option values you have a variety of options on how to specify them. You can specify them all on the command line (there ARE a lot of them so they may not all fit within the DOS command line limits), with an SSTOOLS.INI file, or with a command file. Here's an example for you power FRACTINTers, the command FRACTINT MYFILE SAVENAME=MY3D 3D=YES BATCH=YES would make Fractint load MYFILE.GIF, re-plot it as a 3D landscape (taking all of the defaults), save the result as MY3D.GIF, and exit to DOS. By the time you've come back with that cup of coffee, you'll have a new world to view, if not conquer. Note that the image created by 3D transformation is treated as if it were a plasma cloud - We have NO idea how to retain the ability to zoom and pan around a 3D image that has been twisted, stretched, perspective- ized, and water-leveled. Actually, we do, but it involves the kind of hardware that Industrial Light & Magic, Pixar et al. use for feature films. So if you'd like to send us a check equivalent to George Lucas' net from the "Star Wars" series... 3D Mode Selection After hitting <3> and getting past the filename prompt and video mode selection, you're presented with a "3d Mode Selection" screen. 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 . Fractint Version 15.0 Page 48 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: 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. Fractint Version 15.0 Page 49 Option "2" gives you full spatial resolution but with only 16 shades of gray. If the red and blue images overlap, the colors are mixed. Good for wire-frame images (we call them surface grids), lorenz3d and 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. Fractint Version 15.0 Page 50 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 these numbers: 1 - surface grid 2 - (default) - no fill at all - just draw the dots. 3 - wire frame - joins points with lines 4 - surface fill - (colors interpolated) 5 - surface fill - (interpolation turned off) 6 - solid fill - draws lines from the "ground" up to the point 7 - surface fill with light model - calculated before 3D transforms 8 - surface fill with light model - calculated after 3D transforms Types 4, 7, and 8 interpolate colors when filling, making a very smooth fill if the palette is continuous. This may not be desirable if the palette is not continuous. Type 5 is the same as type 4 with interpolation turned off. You might want to use fill type 5, for example, to project a .GIF photograph onto a sphere. With type 4, you might see the filled-in points, since chances are the palette is not continuous; type 5 fills those same points in with the colors of adjacent pixels. However, for most fractal images, fill type 4 works better. 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! Fractint Version 15.0 Page 51 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 percent. Initially, leave the X and Y axes alone and play with Z, now the vertical axis, which translates into surface "roughness." High values of Z make spiky, on- beyond-Alpine mountains and improbably deep valleys; low values make gentle, rolling terrain. Negative roughness is legal: if you're doing an 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 Fractint Version 15.0 Page 52 back is, etc. Try about 150% for reasonable results. Much larger values put you far away for even less distortion, while values smaller than 100% put you "inside" the box. Try larger values first, and work your way in. Next, you are prompted for two types of X and Y shifts (now back in the plane of your screen) that let you move the final image around if you'd like to re-center it. The first set, x and y shift with perspective, move the image and the effect changes the perspective you see. The second set, "x and y adjust without perspective", move the image but do not change perspective. They are used just for positioning the final image on the screen. Color Parameters Now, you are asked for a range of "transparent" colors, if any. This option is most useful when using the verlay command below. Enter the color range (minimum and maximum value) for which you do not want to overwrite whatever may already be on the screen. The default is no transparency (overwrite everything). Now, for the final option. This one will smooth the transition between colors by randomizing them and reduce the banding that occurs with some maps. Select the value of randomize to between 0 (for no effect) and 7 (to randomize your colors almost beyond use). 3 is a good starting point. Well, OK, we lied. If you selected one of the light source fill options, there are still MORE options. If you have selected a fill type which uses light source, you can select the final image to be either in monochrome or full color. 0 selects monochrome and 1 selects glorious full color. That's all for this screen. Press enter and the next and final screen will appear (honestly!). Light Source Parameters This one deals with all the aspects of light source. 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 i.e.. it will be low in the sky, say, afternoon or morning. Fractint Version 15.0 Page 53 Then you are asked for a smoothing factor. Unless you used continuous potential (see chapter 6) when generating the starting image, the illumination when using light source fills may appear "sparkly", like a sandy beach in bright sun. A smoothing factor of 2 or 3 will allow you to see the large-scale shapes better. This is primarily useful when doing light source fill types with plasma clouds. 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 software company here], used by permission.) The ambient option sets the minimum light value a surface has if it has no direct lighting at all. All light values are scaled from this value to white. This effectively adjusts the depth of the shadows and sets the overall contrast of the image. If you selected the full color option, you have a few more choices. The next is the haze factor. Set this to make distant objects more hazy. Close up objects will have little effect, distant objects will have most. 0 disables the function. 100 is the maximum effect, the farthest objects will be lost in the mist. Currently, this does not really use distance from the viewer, we cheat and use the y value of the original image. So the effect really only works if the y-rotation (set earlier) is between +/- 30. Finally, absolutely the last option (this time we mean it): you can chose the name under which to save your light file. If you have a RAM disk handy, you might want to create the file on it for speed, so include its full path name in this option. If you have not set "overwrite=yes" then the file name will be incremented to avoid over- writing previous files. By the way, the background on the full color file is sky blue instead of black. You'll probably want to adjust the final colors for monochrome fill types using light source via color cycling. Try one of the more continuous palettes ( through ), or load the GRAY palette with the lternate-map command. Now, lie down for a while in a quiet room with a damp washcloth on your forehead. Feeling better? Good -- because it's time to go back almost to the top of the 3D options and just say yes to: 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. Fractint Version 15.0 Page 54 The next entry is for a radius factor that controls the over-all size of the globe. All the rest of the entries are the same as in the landscape projection. You may want less surface roughness for a plausible look, unless you prefer small worlds with big topography, a la "The Little Prince." WARNING: When the "construction" process begins at the edge of the globe (default) or behind it, it's plotting points that will be hidden by subsequent points as the process sweeps around the sphere toward you. Our nifty hidden-point algorithms "know" this, and the first few dozen lines may be invisible unless a high mountain happens to poke over the horizon. If you start a spherical projection and the screen stays black, wait for a while (a longer while for higher resolution or fill type 6) to see if points start to appear. Would we lie to you? If you're still waiting hours later, first check that the power's still on, then consider a faster system. 3D Overlay Mode While the <3> command creates its image on a blank screen, the command draws a second image over an existing displayed image. This image can be any restored image from a command or the result of a just executed <3> command. So you can do a landscape, then hit and choose spherical projection to re-plot that image or another as a moon in the sky above the landscape. can be repeated as many times as you like. It's worth noting that not all that many years ago, one of us watched Benoit Mandelbrot and fractal-graphics wizard Dick Voss creating just such a moon-over-landscape image at IBM's research center in Yorktown Heights, NY. The system was a large and impressive mainframe with floating-point facilities bigger than the average minicomputer, running LBLGRAPH -- what Mandelbrot calls "an independent-minded and often very ill-mannered heap of graphics programs that originated in work by Alex Hurwitz and Jack Wright of IBM Los Angeles." We'd like to salute LBLGRAPH, its successors, and their creators, because it was their graphic output (like "Planetrise over Labelgraph Hill," plate C9 in Mandelbrot's "Fractal Geometry of Nature") that helped turn fractal geometry from a mathematical curiosity into a phenomenon. We'd also like to point out that it wasn't as fast, flexible or pretty as Fractint on a 386/16 PC with S-VGA graphics. Now, a lot of the difference has to do with the incredible progress of micro-processor power since then, so a lot of the credit should go to Intel rather than to our highly tuned code. OK, twist our arms -- it IS awfully good code. 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. Fractint Version 15.0 Page 55 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! Making Terrains If you enjoy using Fractint for making landscapes, we now have several new features for you to work with. When doing 3d transformations banding tends to occur because all pixels of a given height end up the same color. Now, colors can be randomized to make the transitions between different colors at different altitudes smoother. Use the new "RANDOMIZE= " variable to accomplish this. If your light source images all look like lunar landscapes since they are all monochrome and have very dark shadows, we now allow you to set the ambient light for adjusting the contrast of the final image. Use the "Ambient= " variable. In addition to being able to create scenes with light sources in monochrome, you can now do it in full color as well. Setting fullcolor=1 will generate a Targa-24 file with a full color image which will be a combination of the original colors of the source image (or map file if you select map=something) and the amount of light which reflects off a given point on the surface. Since there can be 256 different colors in the original image and 256 levels of light, you can now generate an image with *lots* of colors. To convert it to a GIF if you can't view Targa files directly, you can use PICLAB (see Appendix E for information about getting PICLAB), and the following commands: SET PALETTE 256 SET CREZ 8 TLOAD yourfile.tga MAKEPAL MAP GSAVE yourfile.gif EXIT Using the full color option allows you to also set a haze factor with the "haze= " variable to make more distant objects more hazy. As a default, full color files also have the background set to sky blue. Warning, the files which are created with the full color option are very large, 3 bytes per pixel. So make sure to use a disk with enough space. The file is created using Fractint's disk-video caching, but is always created on real disk (expanded or extended memory is not used.) Try the following settings of the new variables in sequence to get a feel for the effect of each one: ;use this with any filltype map=topo randomize=3; adjusting this smooths color transitions Fractint Version 15.0 Page 56 ;now add this using filltype 5 or 6 ambient=20; adjusting this changes the contrast filltype=6 smoothing=2; makes the light not quite as granular as the terrain ;now add the following, and this is where it gets slow fullcolor=1; use PICLAB to reduce resulting lightfile to a GIF ;and finally this haze=20; sets the amount of haze for distant objects When full color is being used, the image you see on the screen will represent the amount of light being reflected, not the colors in the final image. Don't be disturbed if the colors look weird, they are an artifact of the process being used. The image being created in the lightfile won't look like the screen. However, if you are worried, hit ESC several times and when Fractint gets to the end of the current line it will abort. Your partial image will be there as LIGHT001.TGA or with whatever file name you selected with the lightname option. Convert it as described above and adjust any parameters you are not happy with. Its a little awkward, but all we haven't figured out a better way yet. Making 3D Slides Bruce Goren, CIS's resident stereoscopic maven, contributed these tips on what to do with your 3D images (Bruce inspired and prodded us so much we automated much of what follows, allowing both this and actual on screen stereo viewing, but we included it here for reference and a brief tutorial.) "I use a Targa 32 video card and TOPAS graphic software, moving the viewport or imaginary camera left and right to create two separate views of the stationary object in x,y,z, space. The distance between the two views, known as the inter-ocular distance, toe-in or convergence angle, is critical. It makes the difference between good 3-D and headache- generating bad 3-D. "For a 3D fractal landscape, I created and photographed the left and right eye views as if flying by in an imaginary airplane and mounted the film chips for stereo viewing. To make my image, first I generated a plasma cloud based on a color map I calculated to resemble a geological survey map (available on CIS as TARGA.MAP). In the 3D reconstruction, I used a perspective value of 150 and shifted the camera -15 and +15 on the X-axis for the left and right views. All other values were left to the defaults. "The images are captured on a Matrix 3000 film recorder -- basically a box with a high-resolution (1400 lines) black and white TV and a 35mm camera (Konica FS-1) looking at the TV screen through a filter wheel. The Matrix 3000 can be calibrated for 8 different film types, but so far I have only used Kodak Ektachrome 64 daylight for slides and a few print films. I glass mount the film chips myself. Fractint Version 15.0 Page 57 "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." Fractint Version 15.0 Page 58 5. Command-Line Parameters, Setting Defaults, Batch Mode 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 from the command line using a command like "FRACTINT @MYFILE". Fractint 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 which will over-ride those from the .INI file. General 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. In an SSTOOLS.INI file 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 choose one of "yes", "no", and "whatever" COMMAND=1st[/2nd[/3rd]] - the slash-separated parameters "2nd" and "3rd" are optional General Parameters @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. 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 the starting point, bypassing the initial information screens. The filetype is optional and defaults to .GIF. Non-Fractint GIF files are restored as fractal type "plasma". On the command line you may omit FILENAME= and just give the name; the full argument is required in SSTOOLS.INI. 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 chapter 2. Fractint Version 15.0 Page 59 (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[/x3rd/y3rd] Example: corners=-0.739/-0.736/0.288/0.291 Begin with these coordinates as the range of x and y coordinates, rather than the default values of (for type=mandel) -2.0/2.0/-1.5/1.5. When you specify four values (the usual case), this defines a rectangle: x- coordinates are mapped to the screen, left to right, from xmin to xmax, y-coordinates are mapped to the screen, bottom to top, from ymin to ymax. Six parameters can be used to describe any rotated or stretched parallelogram: (xmin,ymax) are the coordinates used for the top-left corner of the screen, (xmax,ymin) for the bottom-right corner, and (x3rd,y3rd) for the bottom-left. CENTER-MAG=[Xctr/Yctr/Mag] This is an alternative way to enter corners as a center point and a magnification that is popular with some fractal programs and publications. Entering just "CENTER-MAG=" tells Fractint whether to use this form rather than corners when creating a batch file with the command. The status display shows the "corners" in both forms. Note that an aspect ratio of 1.3333 is assumed; if you have altered the zoom box proportions or rotated the zoom box, this form can no longer be used. PASSES=1|2|guess|btm Selects single-pass, dual-pass, solid-Guessing mode, or the Boundary Tracing algorithm. The first two take the same total time to finish the display. Solid-guessing is faster at the risk of errors for a few pixels. Boundary tracing is sometimes faster, sometimes slower, and isn't correct for some fractal types at all (like the Newton fractals). 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. 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). INSIDE=nnn|bof60|bof61|attractor 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. Two more options reveal hidden structure inside the lake. These are inside=bof61, and inside=bof62, named after the figures on pages 61 and Fractint Version 15.0 Page 60 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. BAILOUT=nnn Over-rides the default bailout criterion for escape-time fractals. Can also be set from the parameters screen after selecting a fractal type. INITORBIT=pixel INITORBIT=nnn/nnn Allows control over the value used to begin each Mandelbrot-type orbit. "initorbit=pixel" is the default for most types; this command initializes the orbit to the complex number corresponding to the screen pixel. The command "initorbit=nnn/nnn" uses the entered value as the initializer. See the discussion of the MANDELLAMBDA type for more on this topic. POTENTIAL=maxcolor[/slope[/modulus[/16bit]]] Enables 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 whether 16 bit values are to be calculated. Example: "POTENTIAL=240/2000/40/16bit". The Mandelbrot and Julia types ignore the modulus bailout value and use their own hardwired value of 4.0 instead. See chapter 6 on the continuous-potential algorithm for details. LOGMAP=yes|old|n Selects a compressed relationship between escape-time iterations and palette colors. See "Logarithmic Palette" section for details. DISTEST=nnn Nonzero values enable the distance estimator method. Any nonzero value has the same effect in monochrome modes. Higher values result in more and narrower color bands in color modes; 1000 is a good value to try. DECOMP=2|4|8|16|32|64|128|256 Invokes the corresponding decomposition coloring scheme. BIOMORPH=nnn Turn on biomorph option; set affected pixels to color nnn. INVERT=nn/nn/nn Turns on inversion. The parameters are radius of inversion, x-coordinate of center, and y-coordinate of center. -1 as the first parameter sets the radius to 1/6 the smaller screen dimension; no x/y parameters defaults to center of screen. The values are displayed with the command. Fractint Version 15.0 Page 61 FINATTRACT=no|yes Another option to show coloring inside some Julia "lakes" to show escape time to finite attractors. Works with lambda, magnet types, and possibly others. See appendix for more information. FUNCTION=[fn1[/fn2[/fn3[/fn4]]]] Allows setting variable functions found in some fractal type formulae. Possible values are sin, cos, sinh, cosh, exp, log, and sqr. FORMULAFILE=[formulafilename] Specifies the default formula file for type=formula fractals (if not FRACTINT.FRM). Handy if you want to generate one of these fractal types in batch mode. FORMULANAME=[formulaname] Specifies 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. IFS=[filename] Use the IFS coded values in [filename] to generate Barnsley IFS fractal images. The default extension is .IFS. See chapter 2 for details. IFS3D=[filename] Use the IFS coded values in [filename] to generate Barnsley IFS3D fractals. LFILE=[lsystemfile] Specifies the default L-System file for type=lsystem fractals (if not FRACTINT.L). LNAME=[lsystemname] Specifies the default L-System name. Required if you want to generate one of these fractal types in batch mode, as this is the only way to specify an L-System name in that case. PERIODICITY=no|show|nnn Allows control of periodicity checking. "no" turns it off, "show" lets you see which pixels were painted the "inside" color due to being caught by periodicity. Specifying a number causes a more conservative periodicity test (each increase of 1 divides the test tolerance by 2). Entering a negative number lets you turn on "show" with that number. Type lambdafn function=exp needs periodicity turned off to be accurate - - there may be other cases. SOUND=off|x|y|z We're all MUCH too busy to waste time with Fractint at work, and no doubt you are too, so "sound=off" is included only for use at home, to avoid waking the kids or your Significant Other, late at night. (By the way, didn't you tell yourself "just one more zoom on LambdaSine" an hour ago?) Suggestions for a "boss" hot-key will be cheerfully ignored, as this sucker is getting big enough without including a spreadsheet screen too. The "sound=x/y/x" options are for the "attractor" fractals, like the Lorenz fractals - they play with the sound on your PC speaker as Fractint Version 15.0 Page 62 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. SAVENAME=[name] Set the filename to use when you ave a screen. The default filename is FRACT001. The .GIF extension is optional (Example: SAVENAME=myfile) OVERWRITE=no|yes Sets the savename overwrite flag (default is 'no'). If 'yes', saved files will over-write existing files from previous sessions; otherwise the automatic incrementing of FRACTnnn.GIF will find the first unused filename. GIF87a=YES Backward-compatibility switch to force creation of GIF files in the GIF87a format. As of version 14, Fractint now defaults to the new GIF89a format which permits storage of fractal information within the format. This switch is only needed if you wish to view Fractint images with a GIF decoder that cannot accept the newer format. BATCH=yes See "Batch Mode notes later in this chapter. SAVETIME=nnn Tells Fractint to automatically do a save every nnn minutes while a calculation is in progress. This is mainly useful with long batches - see "Batch Mode". 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 chapter 6. The difference between this argument and an alternate map read in via in color-command mode is that this one is permanent. CYCLELIMIT=nnn Sets the speed of color cycling. Technically, the number of DAC registers updated during a single vertical refresh cycle. Legal values are 1 - 256, default is 55. 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 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." Fractint Version 15.0 Page 63 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 chapter 7). SYMMETRY=xxx Forces symmetry to None, Xaxis, Yaxis, XYaxis, Origin, or Pi symmetry. Useful for debugging. ; Indicates the rest of the command line (or the rest of the individual line inside command files) is a comment. Video Parameters VIDEO=xxx Set the initial video mode (and bypass the informational screens). Handy for batch runs. (Example: VIDEO=F4 for IBM 16-color VGA.) (You can obtain the current VIDEO= values 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 this arguments in a command file to be used with future versions of Fractint, particularly for the super-VGA modes. ADAPTER=cga|ega|mcga|vga Skips the video autodetect logic, assumes the specified kind of adapter is present. Use this only if you have trouble running up. TEXTSAFE=no If return from menu, prompt, or help to an image garbles the image, try specifying this. TEXTSAFE=yes To get rid of the irritating flicker on EGA/VGA during Fractint startup, try specifying this. Remove it if you find that images are scrambled when returning to them from the menu. EXITMODE=nn Sets the bios-supported videomode to use upon exit to the specified value. nn is in hexadecimal. The default is 3, which resets to 80x25 color text mode on exit. TEXTCOLORS=mono Set text screen colors to simple black and white. TEXTCOLORS=aa/bb/cc/... Set text screen colors. Omit any value to use the default (e.g. Fractint Version 15.0 Page 64 textcolors=////50 to set just the 5th value). Each value is a 2 digit hexadecimal value; 1st digit is background color (from 0 to 7), 2nd digit is foreground color (from 0 to F). Color values are: 0 black 8 gray 1 blue 9 light blue 2 green A light green 3 cyan B light cyan 4 red C light red 5 magenta D light magenta 6 brown E yellow 7 white F bright white 27 colors can be specified, their meanings are as follows: heading: 1 Fractint version info 2 heading line development info (not used in released version) help: 3 sub-heading 4 main text 5 instructions at bottom of screen menu, selection boxes, parameter input boxes: 6 background around box and instructions at bottom 7 low intensity information 8 medium intensity information 9 high intensity information (e.g. heading) 10 current keyin field 11 current choice in multiple choice list 12 speed key prompt in multiple choice list 13 speed key keyin in multiple choice list general (tab key display, IFS parameters, "thinking" display): 14 high intensity information 15 medium intensity information 16 low intensity information 17 current keyin field disk video: 18 background around box 19 high intensity information 20 low intensity information diagnostic messages: 21 error 22 information credits screen: 23 bottom lines 24 high intensity divider line 25 low intensity divider line 26 primary authors 27 contributing authors The default is textcolors=1F/1A/2E/70/28/78/17/1F/1E/2F/ 5F/07/0D/71/70/78/0F/70/0E/0F/ 4F/20/17/20/28/0F/07 (In a real command file, all values must be on one line.) Fractint Version 15.0 Page 65 Printer Parameters PRINTER=type[/resolution[/port#]] Defines your printer setup. The SSTOOLS.INI file is a REAL handy place to put this option, so that it's available whenever you have that sudden, irresistible urge for hard copy. Printer types: IB IBM-compatible (default) EP Epson-compatible HP LaserJet CO Star Micronics Color printer, supposedly Epson-color-compatible PA Paintjet PS PostScript PSL Postscript, landscape mode Resolution: In dots per inch. Currently available values are 60, 120, 240 for Epson/IBM; 75, 150, 300 for LaserJet; 90, 180 for Paintjet; 10 through 600 for PostScript. Port: 1, 2, 3 for LPT1-3; 11, 12, 13, 14 for COM1-4. With PostScript, a negative port number can be used to redirect printing to a file. Default print-to-filename is FRACT001.PRN and is incremented after each print to file operation. COMPORT=port/baud/options Serial printer port initialization. Port=1,2,3,etc. Baud=115,150,300,600,1200,2400,4800,9600 Options: 7,8 | 1,2 | e,n,o (any order). Example: comport=1/9600/n81 for COM1 set to 9600, no parity, 8 bits per character, 1 stop bit. TITLE=yes If specified, title information is added to printouts. Additional options for PostScript printers follow. PRINTFILE=filename Forces print-to-file (like negative port number) and specifies filename. EPSF=1|2|3 Forces print-to-file and PostScript, but the default filename is FRACT001.EPS. The number determines how 'well-behaved' a .EPS file is. 1 means by-the-book. 2 allows some EPS 'no-nos' like settransfer and setscreen - BUT includes code that should make the code still work without affecting the rest of the non-EPS document. 3 is a free-for-all. TRANSLATE=yes|-n|n Translate=yes prints the negative image of the fractal. Translate=n reduces the image to that many colors. A negative value causes a color reduction as well as a negative image. HALFTONE=frq/ang/sty Tells the PostScript printer how to define its halftone screen. The first value, frequency, defines the number of halftone lines per inch. Fractint Version 15.0 Page 66 The second chooses the angle (in degrees) that the screen lies at. The third option chooses the halftone 'spot' style. Good default frequencies are between 60 and 80; Good default angles are 45 and 0; the default style is 0. If the halftone= option is not in SSTOOLS, Fractint will print using the printer's default halftone screen, which should have been already set to do a fine job on the printer. This option is presented here for the advanced PostScript owners and those who want to play around. Current halftone styles (no warranty on these!): 0 - common circle screen (probably same as hardware default) 2 - horizontal line screen 4 - vertical line screen others - some have worked, some haven't. They SHOULD produce screens such as: boxes, crosses, rounded crosses, but aren't working as expected yet - explore if you like but don't expect wonders. 3D Parameters 3D=Yes Resets all 3d parameters to default values. 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. If 3D=Yes follows the setting of any 3d parameters on the command line or in a command file, they will be reset to default values. (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 SCALEZYZ=nn/nn/nn X,y,and z scale factors ROUGHNESS=nn Same as z scale factor WATERLINE=nn Colors nn and below will be "inside" color FILLTYPE=nn 3D filltype PERSPECTIVE=nn Perspective distance XYSHIFT=nn/nn Shift image in x and y directions with perspective LIGHTSOURCE=nn/nn/nn Coordinates for light-source vector SMOOTHING=nn Smooths images in light-source fill modes TRANSPARENT=min/max Defines a range of colors to be treated as "transparent" when 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 funny glasses filter parameters Fractint Version 15.0 Page 67 Below are new commands as of version 14 that support Marc Reinig's terrain features. RANDOMIZE=nnn (0 - 100) This feature randomly varies the color of a pixel to near by colors. Useful to minimize map banding in 3d transformations. Usable with all FILLTYPES. 0 disables, max values is 7. Try 3 - 5. AMBIENT=nnn (0 - 100) Set the depth of the shadows when using full color and light source filltypes. "0" disables the function, higher values lower the contrast. FULLCOLOR=yes Valid with any light source FILLTYPE. Allows you to create a Targa-24 file which uses the color of the image being transformed or the map you select and shades it as you would see it in real life. Well, its better than B&W. A good map file to use is topo HAZE=nnn (0 - 100) Gives more realistic terrains by setting the amount of haze for distant objects when using full color in light source FILLTYPES. Works only in the "y" direction currently, so don't use it with much y rotation. Try "rotation=85/0/0". 0 disables. LIGHTNAME= The name of the Targa-24 file to be created when using full color with light source. Default is light001.tga. If overwrite=no (the default), the file name will be incremented until an unused filename is found. Background in this file will be sky blue. To stay out of trouble, use the complete 3D 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. Setting Defaults (SSTOOLS.INI) If you are familiar with Microsoft's TOOLS.INI or WINDOWS.INI configuration 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 Version 15.0 Page 68 [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 of the file. (Why use a convention like that when Fractint is the only program you know of that uses an SSTOOLS.INI file? Because there are other programs (such as Lee Crocker's PICLAB) that now use the same file, and 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..!) Batch Mode It IS possible, believe it or not, to become so jaded with the screen drawing process, so familiar with the types and options, that you just want to hit a key and do something else until the final images are safe on disk. To do this, start Fractint with the BATCH=yes parameter added to the command line. To create a batch file with the parameters required for a particular image you might: o Find an interesting area. Note the parameters from the display. Then use an editor to write a batch file. o Find an interesting area. Set all the parameters you'll want in the batch run. Use the command to store the parameters in FRABATCH.BAT. Then use an editor to add the required batch mode parameters to the generated FRABATCH.BAT line. Another approach to batch mode calculations, using "FILENAME=" and resume, is described later. When modifying a command line generated by the command, the only parameters you must add for a batch mode run are "BATCH=yes", and "VIDEO=xxx" to select a video mode. You might want to also add "SAVENAME=[name]" to name the result as something other than the default FRACT001.GIF. Watch out for the line exceeding the DOS limit of 120 characters - if it does, use "fractint @filename" in your batch and put the parameters in another file "filename". "BATCH=yes" tells Fractint to run in batch mode -- that is, Fractint draws the image using whatever other parameters you specified, then acts as if you had hit to save the image, then exits to DOS. "FILENAME=" can be used with "BATCH=yes" to resume calculation of an incomplete image. For instance, you might interactively find an image you like; then select some slow options (a high resolution disk video mode, distance estimator method, high maxiter, or whatever); start the calculation; then interrupt immediately with a ave. Rename the save file (fract001.gif if it is the first in the session and you didn't name it with the options or "savename=") to xxx.gif. Later you can run a batch file to finish the job: fractint batch=yes filename=xxx savename=xxx Fractint Version 15.0 Page 69 "SAVETIME=nnn" is useful with long batch calculations, to store a checkpoint every nnn minutes. If you start a many hour calculation with say "savetime=60", and a power failure occurs during the calculation, you'll have lost at most an hour of work on the image. You can resume calculation from the save file as above. Automatic saves triggered by SAVETIME do not increment the save file name. The same file is overwritten by each auto save until the image completes. But note that Fractint does not directly over-write save files. Instead, each save operation writes a temporary file FRACTINT.TMP, then deletes the prior save file, then renames FRACTINT.TMP to be the new save file. This protects against power failures which occur during a save operation - if such a power failure occurs, the prior save file is intact and there's a harmless incomplete FRACTINT.TMP on your disk. If you want to spread a many-hour image over multiple bits of free machine time you could use a batch like: fractint batch=yes filename=xxx savename=xxx savetime=60 While this batch is running, hit (almost any key actually) to tell fractint to save what it has done so far and give your machine back. Kick off the batch again when you have another time slice for it. The SAVETIME= parameter, and batch resumes of partial calculations, only work with fractal types which can be resumed. See "Interrupting and Resuming" for information about non-resumable types. Fractint Version 15.0 Page 70 6. File Formats, Color Maps and All That Fractint and .GIF Files Since version 5.0, Fractint has had the 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 estore-from-disk capability. Unfortunately, this created a problem. The then-current GIF87a specification did not offer a place to store the small amount of extra information that Fractint needs in order to implement the 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 got around this restriction in a non-standard manner, saving its application-specific information AFTER the official GIF terminator character. This trick worked with all of the popular GIF decoders that we had tested (although some of them required renaming the file to xxxx.GIF first). We did not claim that these files are true GIF files. For one thing, information after the GIF terminator has the potential to confuse the online GIF viewers used on Compuserve. For another, it is the opinion of some GIF developers that the addition of this extra information violates the GIF87a spec. That's why we used the default filetype .FRA instead. On August 1, 1990, Compuserve released an official upwardly-compatible extension of the GIF87a spec, called GIF89a. This new spec allows the placement of application data within "extension blocks". Fractint version 14 supports the new spec. We have changed default savename extension from *.FRA to *.GIF. Version 14 can still read *.FRA files generated by earlier versions. If for some reason you wish to save files in the older GIF87a format, because, for example, your favorite GIF decoder has not yet been upgraded to GIF89a, just add the command- line parameter "GIF87a=yes". Then any saved files will use the original GIF87a format without any application-specific information. An easy way to convert an older .FRA file into true .GIF format suitable for uploading is something like this at the DOS prompt: FRACTINT MYFILE.FRA SAVENAME=MYFILE.GIF BATCH=YES Fractint will load MYFILE.FRA, save it in true .GIF format as MYFILE.GIF, and return to DOS. There is one significant advantage to the new GIF89a format compared to the old GIF87a-based .FRA format for Fractint purposes: the new .GIF files may be uploaded to the Compuserve graphics forums (such as Fractint's home forum, COMART) with fractal information intact. Therefore anyone downloading a Fractint image from Compuserve will also be downloading all the information needed to regenerate the image. GIF and "Graphics Interchange Format" are trademarks of Compuserve Incorporated, an H&R Block Company. Fractint Version 15.0 Page 71 Continuous Potential Note: This option can only be used with 256 color modes. Fractint's images are usually calculated by the "level set" method, producing bands of color corresponding to regions where the calculation gives the same value. When 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. Continuous potential is approximated by calculating potential = log(modulus)/2^iterations where "modulus" is the orbit value (magnitude of the complex number) when the modulus bailout was exceeded, at the "iterations" iteration. Clear as mud, right? Fortunately, you don't have to understand all the details. However, there ARE a few points to understand. First, Fractint's criterion for halting a fractal calculation, the "modulus bailout value", is generally set to 4. Continuous potential is inaccurate at such a low value. The bad news is that the integer math which makes the "mandel" and "julia" types so fast imposes a hard-wired maximum value of 127. You can still make interesting images from those types, though, so don't avoid them. You will see "ridges" in the "hillsides." Some folks like the effect. The good news is that the other fractal types, particularly the (generally slower) floating point algorithms, have no such limitation. The even better news is that there is a floating-point algorithm for the "mandel" and "julia" types. To force the use of a floating-point algorithm, use Fractint with the "FLOAT=YES" command-line toggle. Only a few fractal types like plasma clouds, the Barnsley IFS/IFS3d types, and "test" are unaffected by this toggle. The parameters for continuous potential are: potential=maxcolor[/slope[/modulus[/16bit]]] These parameters are present on the options screen. "Maxcolor" is the color corresponding to zero potential, which plots as the TOP of the mountain. Generally this should be set to one less than the number of colors, i.e. usually 255. Remember that the last few colors of the default IBM VGA palette are BLACK, so you won't see what you are really getting unless you change to a different palette. "Slope" affects how rapidly the colors change -- the slope of the "mountains" created in 3D. If this is too low, the palette will not Fractint Version 15.0 Page 72 cover all the potential values and large areas will be black. If it is too high, the range of colors in the picture will be much less than those available. There is no easy way to predict in advance what this value should be. "Modulus" is the bailout value used to determine when an orbit has "escaped". Larger values give more accurate and smoother potential. A value of 500 gives excellent results. As noted, this value must be <128 for the integer fractal types (if you select a higher number, they will use 127). "16bit": If you transform a continuous potential image to 3D, the illumination modes 5 and 6 will work fine, but the colors will look a bit granular. This is because even with 256 colors, the continuous potential is being truncated to integers. The "16bit" option can be used to add an extra 8 bits of goodness to each stored pixel, for a much smoother result when transforming to 3D. Fractint's visible behavior is unchanged when 16bit is enabled, except that solid guessing and boundary tracing are not used. But when you save an image generated with 16bit continuous potential: o The saved file is a fair bit larger. o Fractint names the file with a .POT extension instead of .GIF, if you didn't specify an extension in "savename". o The image can be used as input to a subsequent <3> command to get the promised smoother effect. o If you happen to view the saved image with a GIF viewer other than Fractint, you'll find that it is twice as wide as it is supposed to be. (Guess where the extra goodness was stored!) Though these files are structurally legal GIF files the double-width business made us think they should perhaps not be called GIF - hence the .POT filename extension. A 16bit (.POT) file can be converted to an ordinary 8 bit GIF by estoring it, changing "16bit" to "no" on the options screen, and aving. You might find with 16bit continuous potential that there's a long delay at the start of an image, and disk activity during calculation. Fractint uses its disk-video cache area to store the extra 8 bits per pixel - if there isn't sufficient memory available, the cache will page to disk. The following commands can be used to recreate the image that Mark Peterson first prototyped for us, and named "MtMand": TYPE=mandel CORNERS=-0.19920/-0.11/1.0/1.06707 INSIDE=255 MAXITER=255 POTENTIAL=255/2000/1000/16bit PASSES=1 FLOAT=yes Fractint Version 15.0 Page 73 Note that prior to version 15.0, Fractint: o Produced "16 bit TGA potfiles" This format is no longer generated, but you can still (for a release or two) use and <3> with those files. o Assumed "inside=maxit" for continuous potential. It now uses the current "inside=" value - to recreate prior results you must be explicit about this parameter. 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 "L" and "S" commands in color-cycling and palette-editing modes. 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 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), LANDSCAP.MAP (Guruka Singh Khalsa's favorite color-map for plasma "landscapes"), TOPO.MAP (Monte Davis's contribution to full color terrain), and GRID.MAP (great for stereo surface grid images). Fractint Version 15.0 Page 74 7. 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 holographic 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 Video 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 displayed as part of the help screens and in the video mode selection list. 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 successfully with a PS/2 model 80. These entries, coupled with the descriptive comments in the table definition and the information supplied (or that should have been supplied!) with your video adapter, should be all you need to add your own entries. 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 (help) and commands still work, because they temporarily switch the screen to an alternate video mode. Fractint Version 15.0 Page 75 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 excruciatingly 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 (chapter 5), however, works with both TARGA and VGA boards and enables you to redefine the default color maps with either board. "DISK-VIDEO" MODES These "video modes" do not involve a video adapter at all. They use (in order or preference) your expanded memory, your extended memory, or your disk drive (as file FRACTINT.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. While you are in a disk-video mode, your screen will display text information indicating whether memory or your disk drive is being used, and what portion of the "screen" is being read from or written to. A "Cache size" figure is also displayed. 24K is the maximum cache size. If you see a number less than this, it means that you don't have a lot of memory free, and that performance will be less than optimum. With a very low cache size such as 4 or 6k, performance gets considerably worse Fractint Version 15.0 Page 76 in cases using solid guessing, boundary tracing, plasma, or anything else which paints the screen non-linearly. If you have this problem, all we can suggest is having less TSR utilities loaded before starting Fractint, or changing in your config.sys file, such as reducing a very high BUFFERS value. The zoom box is disabled during disk-video modes (you couldn't see where it is anyway). So is the orbit display feature. When using real disk for your disk-video, Fractint will not generate some "attractor" types (e.g. lorenz) nor "IFS" images. These would kill your disk drive. Boundary tracing is allowed - it may give your drive a bit of a workout, but is generally tolerable. When using a real disk, and you are not directing the file to a RAM disk, and you aren't using a disk caching program on your machine, specifying BUFFERS=10 (or more) in your config.sys file is best for performance. BUFFERS=10,2 or even BUFFERS=10,4 is also good. It is also best to keep your disk relatively "compressed" (or "defragmented") if you have a utility to do this. In order to use extended memory, you must have HIMEM.SYS or an equivalent that supports the XMS 2.0 standard or higher. Also, you can't have a VDISK installed in extended memory. Himem.sys is distributed with Microsoft Windows 286/386 and 3.0. If you have problems using the extended memory, try rebooting with just himem.sys loaded and see if that clears up the problem. If you are running background disk-video fractals under Windows 3, and you don't have a lot of real memory (over 2Mb), you might find it best to force Fractint to use real disk for disk-video modes. (Force this by using a .pif file with extended memory and expanded memory set to zero.) Try this if your disk goes crazy when generating background images, which are supposedly using extended or expanded memory. This problem can occur because, to multi-task, sometimes Windows must page an application's expanded or extended memory to disk, in big chunks. Fractint's own cached disk access may be faster in such cases. "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: o vertical or horizontal overscan (displaying dots beyond the edges of your visible CRT area) o flickering (caused by a too-slow refresh rate) o vertical roll or total garbage on the screen (your monitor simply can't keep up, or is attempting to "force" the image into a pre-set mode that doesn't fit). Fractint Version 15.0 Page 77 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 (Customized Video Mode List) 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 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, i.e. "disk-video" 12) 8514/A video modes 13) CGA 320x200x4-color and 640x200x2-color modes 14) Reserved for Tandy 1000 video modes 15) SuperVGA 256-Color mode using the Trident chipset 16) SuperVGA 256-Color mode using the Chips & Tech chipset 17) SuperVGA 256-Color mode using the ATI VGA Wonder chipset 18) SuperVGA 256-Color mode using the EVEREX chipset 19) Roll-your-own video mode (as you've defined it in YOURVID.C) Fractint Version 15.0 Page 78 20) SuperVGA 1024x768x16 mode for the ATI VGA Wonder chipset 21) SuperVGA 1024x768x16 mode for the Tseng Labs chipset 22) SuperVGA 1024x768x16 mode for the Trident chipset 23) SuperVGA 1024x768x16 mode for the Video 7 chipset 24) SuperVGA 1024x768x16 mode for the Paradise chipset 25) SuperVGA 1024x768x16 mode for the Chips & Tech chipset 26) SuperVGA 1024x768x16 mode for the Everex Chipset 27) SuperVGA Auto-Detect mode (we poke around looking for your adapter) 28) VESA modes 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! Fractint Version 15.0 Page 79 8. Common Problems Of course, Fractint would never stoop to having a "common" problem. These notes describe some, ahem, "special situations" which come up occasionally and which even we haven't the gall to label as "features". Hang during startup: There might be a problem with Fractint's video detection logic and your particular video adapter. Try running with "fractint adapter=xxx" where xxx is cga, ega, mcga, or vga. Note that when Fractint is started this way, you may not be able to use the highest resolution modes supported by your adapter. If this solves the problem, create an SSTOOLS.INI file with the "adapter=xxx" command in it so that the fix will apply to every run. Another possible cause: If you install the latest Fractint in say directory "newfrac", then run it from another directory with the command "\newfrac\fractint", *and* you have an older version of fractint.exe somewhere in your DOS PATH, a silent hang is all you'll get. See the notes under "Cannot find FRACTINT.EXE message" for the reason. Scrambled image when returning from a text mode display: If an image which has been partly or completely generated gets partly destroyed when you return to it from the menu, help, or the information display, try starting with "fractint textsafe=no". If this cures the problem, create an SSTOOLS.INI file with the "textsafe=no" command so that the fix will apply to every run. There are a number of known cases where "textsafe=no" will not cure the problem. This happens mostly with SVGA boards in higher resolution modes. Sorry, but we don't have a fix for this yet - we're working on it, perhaps for a near-future version 15.1. The problem appears to be bios-related; unfortunately the fix will probably be an option which will slow down switches to and from text mode quite a bit. For now, if you have this problem and "textsafe=no" doesn't fix it , all we can suggest is avoiding invoking the menu, help, and displays while generating or viewing an image in the video modes which have this problem. Coarse monochrome text: You may find that the menu, help screens, and fractal information screen sometimes display in color, sometimes in coarse monochrome characters. The monochrome text occurs if you specify "textsafe=no", or if Fractint finds during initialization that your video adapter does not behave in a standard way. In these cases you may prefer to run with the "textcolors=mono" command for a more consistent appearance in Fractint's text screens. Other video problems: We've had one case where a video driver for Windows does not work properly with Fractint. If running under Windows, DesqView, or some other layered environment, try running Fractint directly from DOS to see if that avoids the problem. We've also had one case of a problem co-existing with "386 to the Max". Fractint Version 15.0 Page 80 Also, see the "Hardware Support" section of the manual for information about enhanced video modes - Fractint makes only limited attempts to verify that a video mode you request is actually supported by your adapter. Insufficient memory: Fractint requires a fair bit of memory to run. Most machines with at least 640k (ok sticklers, make that "PC-compatible" machines) will have no problem. Machines with 512k and machines with many TSR utilities and/or a LAN interface may have problems. Some Fractint features allocate memory when required during a run. If you get a message about insufficient memory, or suspect that some problem is due to a memory shortage, you could try commenting out some TSR utilities in your AUTOEXEC.BAT file, some non-critical drivers in your CONFIG.SYS file, or reducing the BUFFERS parameter in your CONFIG.SYS. "Cannot find FRACTINT.EXE" message: Fractint is an overlayed program - some parts of it are brought from disk into memory only when used. The overlay manager needs to know where to find the program. It must be named FRACTINT.EXE (which it is unless somebody renamed it), and you should either be in the directory containing it when you start Fractint, or that directory should be in your DOS PATH. Disk video mode performance: This won't be blindingly fast at the best of times, but there are things which can slow it down and can be tuned. See "Disk Video Modes" in "Hardware Support" for details. Fractint Version 15.0 Page 81 9. Fractals and the PC A Little History Before Mandelbrot Like new forms of life, new branches of mathematics and science don't appear from nowhere. The ideas of fractal geometry can be traced to the late nineteenth century, when mathematicians created shapes -- sets of points -- that seemed to have no counterpart in nature. By a wonderful irony, the "abstract" mathematics descended from that work has now turned out to be MORE appropriate than any other for describing many natural shapes and processes. Perhaps we shouldn't be surprised. The Greek geometers worked out the mathematics of the conic sections for its formal beauty; it was two thousand years before Copernicus and Brahe, Kepler and Newton overcame the preconception that all heavenly motions must be circular, and found the ellipse, parabola, and hyperbola in the paths of planets, comets, and projectiles. In the 17th century Newton and Leibniz created calculus, with its techniques for "differentiating" or finding the derivative of functions -- in geometric terms, finding the tangent of a curve at any given point. True, some functions were discontinuous, with no tangent at a gap or an isolated point. Some had singularities: abrupt changes in direction at which the idea of a tangent becomes meaningless. But these were seen as exceptional, and attention was focused on the "well- behaved" functions that worked well in modeling nature. Beginning in the early 1870s, though, a 50-year crisis transformed mathematical thinking. Weierstrass described a function that was continuous but nondifferentiable -- no tangent could be described at any point. Cantor showed how a simple, repeated procedure could turn a line into a dust of scattered points, and Peano generated a convoluted curve that eventually touches every point on a plane. These shapes seemed to fall "between" the usual categories of one-dimensional lines, two- dimensional planes and three-dimensional volumes. Most still saw them as "pathological" cases, but here and there they began to find applications. In other areas of mathematics, too, strange shapes began to crop up. Poincare attempted to analyze the stability of the solar system in the 1880s and found that the many-body dynamical problem resisted traditional methods. Instead, he developed a qualitative approach, a "state space" in which each point represented a different planetary orbit, and studied what we would now call the topology -- the "connectedness" -- of whole families of orbits. This approach revealed that while many initial motions quickly settled into the familiar curves, there were also strange, "chaotic" orbits that never became periodic and predictable. Fractint Version 15.0 Page 82 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 provocative essay titled "How Long Is the Coast of Britain?" Mandelbrot noted that the answer depends on the scale at which one measures: it grows longer and longer as one takes into account every bay and inlet, every stone, every grain of sand. And he codified the "self-similarity" characteristic of many fractal shapes -- the reappearance of geometrically similar features at all scales. First in isolated papers and lectures, then in two editions of his seminal book, he argued that many of science's traditional mathematical models are ill-suited to natural forms and processes: in fact, that many of the "pathological" shapes mathematicians had discovered generations before are useful approximations of tree bark and lung tissue, clouds and galaxies. Fractint Version 15.0 Page 83 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 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. Fractint Version 15.0 Page 84 Limitations of Integer Math (And How We Cope) By default, Fractint uses 16-bit and/or 32-bit integer math to generate nearly all its fractal types. The advantage of integer math is speed: this is by far the fastest such plotter that we have ever seen on any PC. The disadvantage is an accuracy limit. Integer math represents numbers like 1.00 as 32-bit integers of the form [1.00 * (2^29)] (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 noticeable. If it has no floating-point algorithm, Fractint does the best it can: it switches to its minimal drawing mode, with adjacent pixels having initial values differing by 1 (really 0.000000002). Attempts to zoom further may result in moving the image around a bit, but won't actually zoom. If you are stuck with an integer algorithm, you can reach minimal mode with your fifth consecutive "maximum zoom", each of which covers about 0.25% of the previous screen. By then your full-screen image is an area less than 1/(10^13)th [~0.0000000000001] the area of the initial screen. (If your image is rotated or stretched very slightly, you can run into the wall of jello as early as the fourth consecutive maximum zoom. Rotating or stretching by larger amounts has less impact on how soon you run into it.) Think of it this way: at minimal drawing mode, your VGA display would have to have a surface area of over one million square miles just to be able to display the entire M-set using the integer algorithms. Using the floating-point algorithms, your display would have to be big enough to fit the entire solar system out to the orbit of Saturn inside it. So there's a considerable saving on hardware, electricity and desk space involved here. Also, you don't have to take out asteroid insurance. 32 bit integers also limit the largest number which can be stored. This doesn't matter much since numbers outside the supported range (which is between -4 and +4) produce a boring single color. If you try to zoom-out to reduce the entire Mandelbrot set to a featureless speck, or to squeeze it to a pancake, you'll find you can't do so in integer math mode. Fractint Version 15.0 Page 85 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 straightforward algebra-like language, and Fractint will compile them and bolt them in for you! Fractint Version 15.0 Page 86 There is a terrible down side to this flexibility. Fractint users everywhere are going berserk. Fractal-inventing creativity is running rampant. Proposals for new fractal types are clogging the mail and the telephones. All we can say is that non-productivity software has never been so potent, and we're sorry, it's our fault! Fractint was compiled using Microsoft C 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 or /ML switch to link with the "C" code. Because it has become too large to distribute comfortably as a single compressed file, and because many downloaders have no intention of ever modifying it, it is now distributed as two files: one containing FRACTINT.EXE, auxiliary files and this document, and another containing complete source code (including a .MAK file and MAKEFRAC.BAT). Fractint Version 15.0 Page 87 Appendix A Mathematics of the Fractal Types For functions below, "pixel" is the complex number corresponding to the current pixel. "Fn" can be replace by sin, cos, sinh, cosh, exp, log or sqr. "^" is the exponential operator, z(n) denotes the nth complex orbit value in the sequence z(0), z(1), ..., z(n), z(n+1),... Fractal Type(s) Formula(s) used ----------------------- ------------------------------------------- Mandel, Julia z(n+1) = z(n)^2 + C Newton, Newtbasin (roots of) z^n - 1, where n is an integer ComplexNewton, ComplexBasin (roots of) z^a - b, where a,b are complex plasma (see the Plasma section for details) Mandelfn, Lambdafn z(n+1) = Lambda*fn(z(n))+C (function=sin, cos, sinh, cosh, exp, log, or sqr) 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) Manfn+zsqrd, Julfn+zsqrd z(n+1) = fn(z(n)) + z(n)^2 + C Manzpower, Julzpower z(n+1) = z(n)^M + C (M is a parameter) Mandel4/Julia4 Special case of manzpower/julzpower (M=4) Manzzpwr, Julzzpwr z(n+1) = z(n)^z(n) + z(n)^M + C Manfn+exp, Julfn+exp z(n+1) = fn(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)) popcornjul Julia calculation from the popcorn formula. Bifurcation (see the Bifurcation section for details) Bif+sinpi Bif=sinpi Biflambda Fractint Version 15.0 Page 88 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 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 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) Defaults: 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 See Julibrot type description fn(z*z) z(0)=pixel; z(n+1) = fn(z(n)*z(n)) fn*fn z(0)=pixel; z(n+1) = fn(n)*fn(n) fn*z+z z(0)=pixel; z(n+1) = p1*fn(z(n))+p2*z(n) fn+fn z(0)=pixel; z(n+1)=p1*fn1(z(n))+p2*fn2(z(n)) sqr(1/fn) z(0)=pixel; z(n+1) = (1/fn(z(n))*(1/fn(z(n)) sqr(fn) z(0)=pixel; z(n+1) = fn(z(n))*fn(z(n)) spider c(0)=z(0)=pixel; z(n+1) = z(n)*z(n) + c(n); c(n+1) = c(n)/2 + z(n+1) manowar c = z1(0) = z(0) = Xcoord + i*Ycoord; z(n+1) = z(n)^2 + z1(n) + c; z1(n+1) = z(n); tetrate z(0) = c = pixel; z(n+1) = c^z(n) kamtorus, kamtorus3d (See kamtorus type description for details) magnetj1,j2,m1,m2 (See type descriptions for details) Fractint Version 15.0 Page 89 INSIDE=bof60|bof61 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! FINITE ATTRACTORS (inside=attractor) Many of Fractint's fractals involve the iteration of functions of complex numbers until some "bailout" value is exceeded, then coloring the associated pixel according to the number of iterations performed. This process identifies which values tend to infinity when iterated, and gives us a rough measure of how "quickly" they get there. In dynamical terms, we say that "Infinity is an Attractor", as many initial values get "attracted" to it when iterated. The set of all points that are attracted to infinity is termed The Basin of Attraction of Infinity. The coloring algorithm used divides this Basin of Attraction into many distinct sets, each a single band of one color, representing all the points that are "attracted" to Infinity at the same "rate". These sets (bands of color) are termed "Level Sets" - all points in such a set are at the same "Level" away from the attractor, in terms of numbers of iterations required to exceed the bailout value. Thus, Fractint produces colored images of the Level Sets of the Basin of Attraction of Infinity, for all fractals that iterate functions of Complex numbers, at least. Now we have a sound mathematical definition of what Fractint's "bailout" processing generates, and we have formally introduced the terms Attractor, Basin of Attraction, and Level Set, so you should have little trouble following the rest of this section! Fractint Version 15.0 Page 90 For certain Julia-type fractals, Fractint can also display the Level Sets of Basins of Attraction of Finite Attractors. This capability is a by-product of the implementation of the MAGNETic fractal types, which always have at least one Finite Attractor. Most Julia-types that have a "lake" (normally colored blue by default) have a Finite Attractor within this lake, and the lake turns out to be, quite appropriately, the Basin of Attraction of this Attractor. The "inside=attractor" option (command-line or options screen) instructs Fractint to seek out and identify a possible Finite Attractor and, if found, to display the Level Sets of its Basin of Attraction, in addition to those of the Basin of Attraction of Infinity. In many cases this results in a "lake" with colored "waves" in it; in other cases there may be little change in the lake's appearance. For a quick demonstration, select a fractal type of LAMBDA, with a parameter of 0.5 + 0.5i. You will obtain an image with a large blue lake. Now set "inside=attractors" with the "X" menu. The image will be re-drawn with a much more colorful lake. A Finite Attractor lives in the center of one of the resulting "ripple" patterns in the lake - turn the rbits display on to see where it is - the orbits of all initial points that are in the lake converge there. Fractint tests for the presence of a Finite Attractor by iterating a Critical Value of the fractal's function. If the iteration doesn't bail out before exceeding twice the iteration limit, it is almost certain that we have a Finite Attractor - we assume that we have. Next we define a small circle around it and, after each iteration, as well as testing for the usual bailout value being exceeded, we test to see if we've hit the circle. If so, we bail out and color our pixels according to the number of iterations performed. Result - a nicely colored-in lake that displays the Level Sets of the Basin of Attraction of the Finite Attractor. Sometimes ! First exception : This does not work for the lakes of Mandel-types. Every point in a Mandel-type is, in effect, a single point plucked from one of its related Julia-types. A Mandel-type's lake has an infinite number of points, and thus an infinite number of related Julia-type sets, and consequently an infinite number of finite attractors too. It *MAY* be possible to color in such a lake, by determining the attractor for EVERY pixel, but this would probably treble (at least) the number of iterations needed to draw the image. Due to this overhead, Finite Attractor logic has not been implemented for Mandel-types. Secondly, certain Julia-types with lakes may not respond to this treatment, depending on the parameter value used. E.g., the Lambda Set for 0.5 + 0.5i responds well; the Lambda Set for 0.0 + 1.0i does not - its lake stays blue. Attractors that consist of single points, or a cycle of a finite number of points are ok. Others are not. If you're into fractal technospeke, the implemented approach fails if the Julia- type is a Parabolic case, or has Siegel Disks, or has Herman Rings. Fractint Version 15.0 Page 91 However, all the difficult cases have one thing in common - they all have a parameter value that falls exactly on the edge of the related Mandel-type's lake. You can avoid them by intelligent use of the Mandel-Julia Space-Bar toggle: Pick a view of the related Mandel-type where the center of the screen is inside the lake, but not too close to its edge, then use the space-bar toggle. You should obtain a usable Julia-type with a lake, if you follow this guideline. Thirdly, the initial implementation only works for Julia-types that use the "Standard" fractal engine in Fractint. Fractals with their own special algorithms are not affected by Finite Attractor logic, as yet. Despite these restrictions, the Finite Attractor logic can produce interesting results. Just bear in mind that it is principally a bonus off-shoot from the development of the MAGNETic fractal types, and is not specifically tuned for optimal performance for other Julia types. (Thanks to Kevin Allen for the above). 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)) Fractint Version 15.0 Page 92 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 questions as if he planned to stay for the night. [No! No! It was three Russian Soldiers! - Lee Crocker] [Wait! I heard it was a Wandering Confessor! - Doug Quinn] [Well *my* kids have a book that uses Russian Soldiers! - Bert] [Look, who's writing this documentation, anyway? - Monte] [Ah, but who gets it *last* and gets to upload it? - Bert] "There's not a bite to eat in the whole province," he was told. "Better keep moving on." "Oh, I have everything I need," he said. "In fact, I was thinking of making some stone soup to share with all of you." He pulled an iron cauldron from his wagon, filled it with water, and built a fire under it. Then, with great ceremony, he drew an ordinary-looking stone from a velvet bag and dropped it into the water. By now, hearing the rumor of food, most of the villagers had come to the square or watched from their windows. As the peddler sniffed the "broth" and licked his lips in anticipation, hunger began to overcome their skepticism. "Ahh," the peddler said to himself rather loudly, "I do like a tasty stone soup. Of course, stone soup with CABBAGE -- that's hard to beat." Soon a villager approached hesitantly, holding a cabbage he'd retrieved from its hiding place, and added it to the pot. "Capital!" cried the peddler. "You know, I once had stone soup with cabbage and a bit of salt beef as well, and it was fit for a king." The village butcher managed to find some salt beef...and so it went, through potatoes, onions, carrots, mushrooms, and so on, until there was indeed a delicious meal for all. The villagers offered the peddler a great deal of money for the magic stone, but he refused to sell and traveled on the next day. And from that time on, long after the famine had ended, they reminisced about the finest soup they'd ever had. *** That's the way Fractint has grown, with quite a bit of magic, although without the element of deception. (You don't have to deceive programmers to make them think that hours of painstaking, often frustrating work is fun... they do it to themselves.) It wouldn't have happened, of course, without Benoit Mandelbrot and the explosion of interest in fractal graphics that has grown from his work at IBM. Or without the example of other Mandelplotters for the PC. Or without those wizards who first realized you could perform Mandelbrot calculations using integer math (it wasn't us - we just recognize good Fractint Version 15.0 Page 93 algorithms when we steal--uhh--see them). Or those graphics experts who hang around the Compuserve PICS forum and keep adding video modes to the program. Or... A WORD ABOUT THE AUTHORS Fractint is the result of a synergy between the main authors, many contributors, and published sources. All four of the main authors have had a hand in many aspects of the code. However, each author has certain areas of greater contribution and creativity. Since there is not room in the 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 sophomore year at college - in other words, he hasn't done an honest day's work in his life. He has been known to pass himself off as a PC expert, a UNIX expert, a statistician, and even a financial modeling expert. He is currently masquerading as an independent PC consultant, supporting the PC-to-Mainframe communications environment at NIH. If you sent mail from the Internet to an NIH staffer on his 3+Mail system, it was probably Bert's code that mangled it during the Internet-to-3+Mail conversion. He also claims to support the MS-Kermit environment at NIH. Fractint is Bert's first effort at building a graphics program. Tim Wegner contributed the original implementation of palette animation, and is responsible for most of the 3D mechanisms. He provided the main outlines of the "StandardFractal" engine and data structures, and is accused by his cohorts of being "obsessed with options". Tim is quite proud of having originally integrated the 256 color super VGA modes in Fractint, especially since he knows almost nothing about it! Tim has BA and MA degrees in mathematics from Carleton College and the University of California Berkeley. He worked for 7 years overseas as a volunteer, doing things like working with Egyptian villagers building water systems. Since returning to the US in 1982, he has written shuttle navigation software, a software support environment prototype, and supported strategic information planning, all at NASA's Johnson Space Center. Mark Peterson invented the periodicity detection logic, several original fractal types, transcendental function libraries, alternate math implementations, the formula compiler, and the "Julibrot" intrinsic 3D fractals - in other words, most of the truly original ideas in Fractint! Fractint Version 15.0 Page 94 Mark's knowledge of higher mathematics and programming 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. Pieter Branderhorst is a late-comer to the group who likes to distract the other authors with enhancements impacting at least 1/2 of the source files at once. His contributions include super solid guessing, image rotation, resume, and fast disk caching. Pieter left high school to work with computers, back when huge machines had 64k of core. He's been happily computing since, mostly programming and designing software from comms firmware to database and o/s, and anything between, and large scale online transaction processing applications. He has worked as a free-lance computer consultant (whatever that means) since 1983. 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 addresses. Usenet/Internet/Bitnet/Whatevernet users can reach CIS users directly if they know the user ID (i.e., Bert Tyler can be reached as 73477.433@compuserve.com). Just remember that CIS charges by the minute, so it costs us a little bit to read a message -- don't kill us with kindness. And don't send all your mail to Bert -- spread it around a little! Main authors: 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 Fractint Version 15.0 Page 95 Mark Peterson [70441,3353] on CIS The Yankee Programmer 405-C Queen St., Suite #181 Southington, CT 06489 (203) 276-9721 Pieter Branderhorst [72611,2257] on CIS Amthor Computer Consultants 3915 Bedford Rd. Victoria, B.C., Canada V8N 4K6 (604) 477-1197 Contributing authors (list alphabetically); Joseph A Albrecht 9250 Old Cedar Ave #215 Bloomington, Mn 55425 (612) 884-3286 Kevin Allen PO Box 335 Seven Hills NSW 2147 Australia Rob Beyer [71021,2074] on CIS 23 Briarwood Lane Laguna Hills, CA, 92656 (714) 831-7665 (7-12pm PST & weekends) Michael D. Burkey burkey@sun9.math.utk.edu on Internet 6600 Crossgate Rd. Knoxville, TN 37912 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 Fractint Version 15.0 Page 96 Michael L. Kaufman [71610,431] on CIS 2247 Ridge Ave, #2K (also accessible via EXEC-PC bbs) Evanston, IL, 60201 (708) 864-7916 Adrian Mariano theorem@blake.acs.washington.edu on INTERNET 2729 72nd AVE SE Mercer Island, WA 98040 Joe McLain [75066,1257] on CIS McLain Imaging 2417 Venier Costa Mesa, CA 92627 (714) 642-5219 Bob Montgomery [73357,3140] on CIS (Author of VPIC) 132 Parsons Road Longwood, Fl 32779 Ethan Nagel [70022,2552] on CIS PO Box 7 Yarnell, Arizona 85362 (602)427-3233 Marc Reinig [72410,77] on CIS 3415 Merrill Rd. marco@sun.com!daver!cypress on Usenet Aptos, CA. 95003 abu on BIX (408) 475-2132 Lee H. Skinner P.O. Box 14944 Albuquerque, NM 87191 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 DGWM18A on Prodigy 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 Fractint Version 15.0 Page 97 Appendix C Revision History Features first appearing in: =========================== Version 14 8/90 LAST MINUTE NEWS FLASH! Compuserve announces the GIF89a on August 1, 1990, and Fractint supports it on August 2! GIF files can now contain fractal information! Fractint now saves its files in the new GIF89a format by default, and uses .GIF rather than .FRA as a default filetype. Note that Fractint still *looks* for a .FRA file on file restores if it can't find a .GIF file, and can be coerced into using the old GIF87a format with the new 'gif87a=yes' command-line option. Pieter Branderhorst mounted a major campaign to get his name in lights: Mouse interface: Diagonals, faster movement, improved feel. Mouse button assignments have changed - see the online help. Zoom box enhancements: The zoom box can be rotated, stretched, skewed, and panned partially offscreen. See "More Zoom Box Commands". FINALLY!! You asked for it and we (eventually, by talking Pieter into it [actually he grabbed it]) did it! Images can be saved before completion, for a subsequent restore and continue. See "Interrupting and Resuming" and "Batch Mode". Off-center symmetry: Fractint now takes advantage of x or y axis symmetry anywhere on the screen to reduce drawing time. Panning: If you move an image up, down, left, or right, and don't change anything else, only the new edges are calculated. Disk-video caching - it is now possible, reasonable even, to do most things with disk video, including solid guessing, 3d, and plasma. Logarithmic palette changed to use all colors. It now matches regular palette except near the "lake". "logmap=old" gets the old way. New "savetime=nnn" parameter to save checkpoints during long calculations. Calculation time is shown in display. Kevin Allen, of Antipodean origin, did the following: Made Bifurcation/Verhulst into a generalized Fractal Engine (like StandardFractal, but for Bifurcation types), and implemented periodicity checking for Bifurcation types to speed them up. Added Integer version of Verhulst Bifurcation (lots faster now). Integer is the default. The Floating-Point toggle works, too. Added NEW Fractal types BIFLAMBDA, BIF+SINPI, and BIF=SINPI. These are Bifurcation types that make use of the new Engine. Floating- point/Integer toggle is available for BIFLAMBDA. The SINPI types are Floating-Point only, at this time. Corrected the generation of the MandelLambda Set. Sorry, but it's always been wrong (up to v 12, at least). Ask Mandelbrot ! Added NEW Fractal types MAGNET1M, MAGNET1J, MAGNET2M, MAGNET2J from "The Beauty of Fractals". Floating-Point only, so far, but what do you expect with THESE formulae ?! Added new symmetry types XAXIS NOIMAG and XAXIS NOREAL, required by the new MAGNETic Fractal types. Fractint Version 15.0 Page 98 Added Finite Attractor Bailout (FAB) logic to detect when iterations are approaching a known finite attractor. This is required by the new MAGNETic Fractal types. Added Finite Attractor Detection (FAD) logic which can be used by *SOME* Julia types prior to generating an image, to test for finite attractors, and find their values, for use by FAB logic. Can be used by the new MAGNETic Fractal Types, Lambda Sets, and some other Julia types too. Mike Burkey sent us new tweaked video modes: VGA - 400x600x256 376x564x256 400x564x256 ATI VGA - 832x612x256 New HP Paintjet support from Chris Martin New "FUNCTION=" command to allow substition of different transcendental functions for variables in types (allows one type with four of these variables to represent 7*7*7*7 different types! ALL KINDS of new fractal types, some using "FUNCTION=": fn(z*z), fn*fn, fn*z+z, fn+fn, sqr(1/fn), sqr(fn), spider, tetrate, and Manowar. Most of these are generalizations of formula fractal types contributed by Scott Taylor and Lee Skinner. Distance Estimator logic can now be applied to many fractal types using distest= option. The types "demm" and "demj" have been replaced by "type=mandel distest=nnn" and "type=julia distest=nnn" Added extended memory support for diskvideo thanks to Paul Varner Added support for "center and magnification" format for corners. Color 0 is no longer generated except when specifically requested with inside= or outside=. Formula name is now included in display and in aved images. Bug fixes - formula type and diskvideo, batch file outside=-1 problem. Now you can produce your favorite fractal terrains in full color instead of boring old monochrome! Use the fullcolor option in 3d! Along with a few new 3D options. New "INITORBIT=" command to allow alternate Mandelbrot set orbit initialization. Version 13.0, 5/90 F1 was made the help key. Use F1 for help Use F9 for EGA 320x200x16 video mode Use CF4 for EGA 640x200x16 mode (if anybody uses that mode) Super-Solid-guessing (three or more passes) from Pieter Branderhorst (replaces the old solid-guessing mode) Boundary Tracing option from David Guenther ("fractint passes=btm", or use the new 'x' options screen) "outside=nnn" option sets all points not "inside" the fractal to color "nnn" (and generates a two-color image). 'x' option from the main menu brings up a full-screen menu of many popular options and toggle switches "Speed Key" feature for fractal type selection (either use the cursor keys for point-and-shoot, or just start typing the name of your favorite fractal type) "Attractor" fractals (Henon, Rossler, Pickover, Gingerbread) Diffusion fractal type by Adrian Mariano Fractint Version 15.0 Page 99 "type=formula" formulas from Scott Taylor and Lee H. Skinner. "sound=" options for attractor fractals. Sound=x plays speaker tones according to the 'x' attractor value Sound=y plays speaker tones according to the 'y' attractor value. Sound=z plays speaker tones according to the 'z' attractor value (These options are best invoked with the floating-point algorithm flag set.) "hertz=" option for adjusting the "sound=x/y/z" output. Printer support for color printers (printer=color) from Kurt Sowa Trident 4000 and Oak Technologies SuperVGA support from John Bridges Improved 8514/A support (the zoom-box keeps up with the cursor keys now!) Tandy 1000 640x200x16 mode from Brian Corbino (which does not, as yet, work with the F1(help) and TAB functions) The Julibrot fractal type and the Starmap option now automatically verify that they have been selected with a 256-color palette, and search for, and use, the appropriate GLASSESn.MAP or ALTERN.MAP palette map when invoked. *You* were supposed to be doing that manually all along, but *you* probably never read the docs, huh? Bug Fixes: TAB key now works after R(estore) commands PS/2 Model 30 (MCGA) adapters should be able to select 320x200x256 mode again (we think) Everex video adapters should work with the Autodetect modes again (we think) Version 12.0, 3/90 New SuperVGA Autodetecting and VESA Video modes (you tell us the resolution you want, and we'll figure out how to do it) New Full-Screen Entry for most prompting New Fractal formula interpreter ('type=formula') - roll your own 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 - 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. Fractint Version 15.0 Page 100 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 Most fractal types given an integer and a floating point algorithm. "Float=yes" option now determines whether integer or floating-point algorithms are used for most fractal types. "F" command toggles the use of floating-point algorithms, flagged in the 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 uaternary decomposition toggle and "DECOMP=" argument 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 ave and Restore () commands ogarithmic and 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 Fractint Version 15.0 Page 101 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" () command 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 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 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 ( to ) "Disk-video, RAM-video, EMS-video" modes Lambda sets now use integer math (with 80386 speedups) "WARN=yes" argument to prevent over-writing old .GIF files Version 7.0, 4/89 Restore from disk (from prior save-to-disk using v. 7.0 or later) New types: Newton, Lambda, Mandelfp, Juliafp, Plasma, Lambdasine Many new color-cycling options (for VGA adapters only) New periodicity logic (Mark Peterson) Initial displays recognize (and use) symmetry Solid-guessing option (now the default) Context-sensitive elp Customizable video mode configuration file (FRACTINT.CFG) Fractint Version 15.0 Page 102 "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 Fractint Version 15.0 Page 103 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*!!! Fractint Version 15.0 Page 104 Appendix E Other Programs WINFRACT. Bert Tyler has ported Fractint to run under Windows 3! The same underlying code is used, with a Windows user interface. Winfract has almost all the functionality of Fractint - the biggest difference is the absence of a zillion weird video modes. Fractint for DOS will continue to be the definitive version. Winfract is available from CompuServe in COMART Lib 15, as WINFRA.ZIP (executable) and WINSRC.ZIP (source). PICLAB, by Lee Crocker - a freeware image manipulation utility available from Compuserve in PICS Lib 14, as PICLAB.EXE. PICLAB can do very sophisticated resizing and color manipulation of GIF and TGA files. It can be used to reduce 24 bit TGA files generated with the Fractint "lightname" option to GIF files. FDESIGN, by Doug Nelson (CIS ID 70431,3374) - a freeware IFS fractal generator available from Compuserve in COMART Lib 15, and probably on your local BBS. This program requires a VGA adapter and a Microsoft- compatible mouse, and a floating point coprocessor is highly recommended. It generates IFS fractals in a *much* more intuitive fashion than Fractint. It can also (beginning with version 3.0) save its IFS formulas in Fractint-style .IFS files. Fractint Version 15.0 Page 105 Appendix F Version13 to Version 14 Type Mapping A number of types in version 13 and earlier have been generalized in version 14. We have added a "backward compatibility" hook that (hopefully) automatically translates these to the new form when the old files are read. Files may be converted via: FRACTINT OLDFILE.FRA SAVENAME=NEWFILE.GIF BATCH=YES In a few cases the biomorph flag was incorrectly set in older files. In that case, add "biomorph=no" to the command line. This procedure can also be used to convert any *.fra file to the new GIF89a spec, which now allows storage of fractal information. TYPES CHANGED FROM VERSION 13 - V13 NAME V14 NAME + PARAMETERS -------- -------------------------------------- LOGMAP=YES LOGMAP=OLD for identical Logmap type DEMJ JULIA DISTEST=nnn DEMM MANDEL DISTEST=nnn Note: DISTEST also available on many other fractals MANSINEXP MANFN+EXP FUNCTION=SIN Note: New functions for this type are cos sinh cosh exp log sqr JULSINEXP JULFN+EXP FUNCTION=SIN Note: New functions for this type are cos sinh cosh exp log sqr MANSINZSQRD MANFN+ZSQRD FUNCTION=SQR/SIN Note: New functions for this type are cos sinh cosh exp log sqr JULSINZSQRD JULFN+ZSQRD FUNCTION=SQR/SIN Note: New functions for this type are cos sinh cosh exp log sqr LAMBDACOS LAMBDAFN FUNCTION=COS LAMBDACOSH LAMBDAFN FUNCTION=COSH Fractint Version 15.0 Page 106 LAMBDAEXP LAMBDAFN FUNCTION=EXP LAMBDASINE LAMBDAFN FUNCTION=SIN LAMBDASINH LAMBDAFN FUNCTION=SINH Note: New functions for this type are log sqr MANDELCOS MANDELFN FUNCTION=COS MANDELCOSH MANDELFN FUNCTION=COSH MANDELEXP MANDELFN FUNCTION=EXP MANDELSINE MANDELFN FUNCTION=SIN MANDELSINH MANDELFN FUNCTION=SINH Note: New functions for this type are log sqr MANDELLAMBDA MANDELLAMBDA INITORBIT=PIXEL POPCORN SYMMETRY=NONE POPCORNJUL ------------------------------------------------------------- Formulas from FRACTINT.FRM in version 13 MANDELGLASS MANDELLAMBDA INITORBIT=.5/0 INVMANDEL V13 divide bug may cause some image differences. NEWTON4 V13 divide bug may cause some image differences. SPIDER V13 divide bug may cause some image differences. MANDELSINE MANDELFN FUNCTION=SIN BAILOUT=50 MANDELCOSINE MANDELFN FUNCTION=COS BAILOUT=50 MANDELHYPSINE MANDELFN FUNCTION=SINH BAILOUT=50 MANDELHYPCOSINE MANDELFN FUNCTION=COSH BAILOUT=50 SCOTTSIN PARAMS=nnn FN+FN FUNCTION=SIN/SQR BAILOUT=nnn+3 SCOTTSINH PARAMS=nnn FN+FN FUNCTION=SINH/SQR BAILOUT=nnn+3 SCOTTCOS PARAMS=nnn FN+FN FUNCTION=COS/SQR BAILOUT=nnn+3 SCOTTCOSH PARAMS=nnn FN+FN FUNCTION=COSH/SQR BAILOUT=nnn+3 Fractint Version 15.0 Page 107 SCOTTLPC PARAMS=nnn FN+FN FUNCTION=LOG/COS BAILOUT=nnn+3 SCOTTLPS PARAMS=nnn FN+FN FUNCTION=LOG/SIN BAILOUT=nnn+3 Note: New functions for this type are sin/sin sin/cos sin/sinh sin/cosh sin/exp cos/cos cos/sinh cos/cosh cos/exp sinh/sinh sinh/cosh sinh/exp sinh/log cosh/cosh cosh/exp cosh/log exp/exp exp/log exp/sqr log/log log/sqr sqr/sqr SCOTTSZSA PARAMS=nnn FN(Z*Z) FUNCTION=SIN BAILOUT=nnn+3 SCOTTCZSA PARAMS=nnn FN(Z*Z) FUNCTION=COS BAILOUT=nnn+3 Note: New functions for this type are sinh cosh exp log sqr SCOTTZSZZ PARAMS=nnn FN*Z+Z FUNCTION=SIN BAILOUT=nnn+3 SCOTTZCZZ PARAMS=nnn FN*Z+Z FUNCTION=COS BAILOUT=nnn+3 Note: New functions for this type are sinh cosh exp log sqr SCOTTSZSB PARAMS=nnn FN*FN FUNCTION=SIN/SIN BAILOUT=nnn+3 SCOTTCZSB PARAMS=nnn FN*FN FUNCTION=COS/COS BAILOUT=nnn+3 SCOTTLTS PARAMS=nnn FN*FN FUNCTION=LOG/SIN BAILOUT=nnn+3 SCOTTLTC PARAMS=nnn FN*FN FUNCTION=LOG/COS BAILOUT=nnn+3 Note: New functions for this type are sin/cos sin/sinh sin/cosh sin/exp sin/sqr cos/sinh cos/cosh cos/exp cos/sqr sinh/sinh sinh/cosh sinh/exp sinh/log sinh/sqr cosh/cosh cosh/exp cosh/log cosh/sqr exp/exp exp/log exp/sqr log/log log/sqr sqr/sqr SCOTTSIC PARAMS=nnn SQR(1/FN) FUNCTION=COS BAILOUT=nnn+3 SCOTTSIS PARAMS=nnn SQR(1/FN) FUNCTION=SIN BAILOUT=nnn+3 TETRATE PARAMS=nnn TETRATE BAILOUT=nnn+3 Note: New function type sqr(1/fn) with sin cos sinh cosh exp log sqr Note: New function type sqr(fn) with sin cos sinh cosh exp log sqr