═══ 1. Copyright ═══ Copyright (C) 1992 The Stone Soup Group. FRACTINT for OS/2 2.0 may be freely copied and distributed, but may not be sold. GIF and "Graphics Interchange Format" are trademarks of Compuserve Incorporated, an H&R Block Company. ═══ 2. What's New ═══ Release 17.2 of Fractint for OS/2 2.0 The following is what I remember adding. o This program is now full 32-bit 386 protected mode. o The fractal calculation engine is from Fractint for DOS version 17.2 (the "portable" source). All formulas available in that release are available here. This version appears externally identical to Fractint for DOS version 17.1. o Palette Manager support has been added. This means that, on proper hardware and OS/2 2.0 driver levels, the exact color RGB values used in the program will be displayed (limited to the mapping done by the Palette Manager. The foreground program gets first crack at setting its colors, then background programs). At least I think so. At the present, there are NO drivers delivered with Palette Manager support. The calls exist, I am not receiving errors, But The Color's Don't Change! Note: It is my best information that a basic VGA (256K memory, 640x480 16-color) Is Not and NEVER Will Be Proper Hardware and Drivers for Palette Manager functions. Sorry! o The fractal calculation engine has finally recieved a reprieve from the limits of the 64K segments of DOS and Windows. As such, the fractal image can now be 4096 by 4096 pixels. Warning: Don't over do this new-found freedom. Do a little reality check first. The fact is PMFRACT needs a pixel memory array of the size you ask, and OS/2 NEEDS ONE, TOO. A little calculation: A 2-color image needs 1 bit per pixel, a 16-color image needs a half byte (4 bits), and a 256-color image needs 1 full byte (8 bits). An image 4096 by 4096 by 256 colors needs 4096x4096/1 = 16,777,216 bytes or 16 megabytes -- TIMES 2; or 32 MEGABYTES of memory. OS/2 2.0 can handle it. Can your system? Remember, what doesn't fit in your real memory comes out of your hard disk's SWAPPER.DAT file. Do you have that much space free to begin with? Can the drive live long enough to handle all the swapping? A test (well, really a finger check) during development revealed that you can easily end up with a situation that could only be described as disk-video at its worst. (And no, OS/2 2.0 Did Not Crash.) o Support for reading and writing in the PC Paintbrush . PCX format has been added. o Printer support has been enhanced to include a Printer Setup dialog, allowing a Presentation Manager printer to be selected and Job Properties to be set. The following was added to version 3.0 of Fractint for Presentation Manager - the 16-bit predecessor and base for Fractint for OS/2 2.0: o This entire help system is new. Much of the on-line help system from Fractint for DOS has been adapted to the OS/2 PM Help manager. Try hitting F1 at any point in the dialogs. The result is supposed to be in context and helpful. The entire help file is available both in context with the running program as well as in an on-line book format. To view the on-line book, issue the command "VIEW PMFRACT.INF". o A simple keyboard interface of some of the useful Fractint for DOS command keys has been added. See the selection "Keys Help" from the main Help menu selection. o Support has been added for Bitmap files in the Windows 3.0 compressed formats (RLE4 and RLE8), and OS/2 2.0 standard and RLE4 and RLE8 compressed formats. This is in addition to the original support for OS/2 1.x and Windows 3.0 uncompressed bitmap formats. o Print support has been extended to whatever type printer (Color or Black and White) the Presentation Manager supports. This complicates printing on a Black and White printer. To print Black and White, select one of the Black and White palettes from the Settings/Palette menu before printing. o I have found a few speed-ups for the PM interface code. ═══ 3. 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. ═══ 3.1. History of this program ═══ 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 for OS/2 2.0 was adapted from Fractint-for-DOS by Donald P. Egen, CIS ID 73507,3143. This program was a training exercise in Presentation Manager and SAA programming, which goes a long way towards explaining a lot of the bugs. My task was made a lot easier by Pieter Branderhorst, who separated the DOS-specific code from Fractint-for-DOS's fractal generator modules, and the efforts of Bert Tyler in porting Fractint-for-DOS to Windows. By noting what Bert had to do to get the fractal generator running under Windows, and the user interface functionality needed for the Windows environment, I was able to create a Presentation Manager user interface that could adequately drive the fractal generator. Besides, I like looking at the pretty pictures. Fractint for OS/2 2.0 is based heavily on (and uses the fractal generator engines straight out of) Fractint-for-DOS. A partial list of the authors of Fractint-for-DOS includes: ------------------ Primary Authors (this changes over time) ----------------- Bert Tyler CompuServe (CIS) ID: [73477,433] Timothy Wegner CIS ID: [71320,675] Internet: twegner@mwunix.mitre.org Mark Peterson CIS ID: [70441,3353] Pieter Branderhorst CIS ID: [72611,2257] --------- Contributing Authors ---------- Michael Abrash 360x480x256, 320x400x256 VGA video modes Joseph Albrecht Tandy video, CGA video speedup Kevin Allen Finite attractor and bifurcation engine Steve Bennett restore-from-disk logic Rob Beyer [71021,2074] Barnsley IFS, Lorenz fractals Mike Burkey 376x564x256, 400x564x256, and 832x612x256 VGA video modes John Bridges [75300,2137] superVGA support, 360x480x256 mode Brian Corbino [71611,702] Tandy 1000 640x200x16 video mode Lee Crocker [73407,2030] Fast Newton, Inversion, Decomposition.. Monte Davis [71450,3542] Documentation Chuck Ebbert [76306,1226] cmprsd & sqrt logmap, fpu speedups Richard Finegold [76701,153] 8/16/../256-Way Decomposition option Frank Fussenegger Mandelbrot speedups Mike Gelvin [73337,520] Mandelbrot speedups Lawrence Gozum [73437,2372] Tseng 640x400x256 Video Mode David Guenther [70531,3525] Boundary Tracing algorithm Norman Hills [71621,1352] Ranges option Richard Hughes [70461,3272] "inside=", "outside=" coloring options Mike Kaufman [kaufman@eecs.nwu.edu] mouse support, other features Wesley Loewer fast floating-point Mandelbrot/Julia logic Adrian Mariano [adrian@u.washington.edu] Diffusion & L-Systems Charles Marslett [75300,1636] VESA video and IIT math chip support Joe McLain [75066,1257] TARGA Support, color-map files Bob Montgomery [73357,3140] (Author of VPIC) Fast text I/O routines Bret Mulvey plasma clouds Roy Murphy [76376,721] Lyapunov Fractals Ethan Nagel [70022,2552] Palette editor, integrated help/doc system Jonathan Osuch [73277,1432] IIT detect Marc Reinig [72410,77] Lots of 3D options Kyle Powell [76704,12] 8514/A Support Matt Saucier [72371,3101] Printer Support Herb Savage [71640,455] 'inside=bof60', 'inside=bof61' options Lee Skinner Tetrate, Spider, Mandelglass fractal types and more Dean Souleles [75115,1671] Hercules Support Kurt Sowa [73467,2013] Color Printer Support Hugh Steele cyclerange feature Chris Taylor Floating&Fixed-point algorithm speedups, Tesseral Option Scott Taylor [72401,410] (DGWM18A) PostScript, Kam Torus, many fn types. Bill Townsend Mandelbrot Speedups Paul Varner [73237,441] Extended Memory support for Disk Video Dave Warker Integer Mandelbrot Fractals concept Phil Wilson [76247,3145] Distance Estimator, Bifurcation fractals Nicholas Wilt Lsystem speedups Richard Wilton Tweaked VGA Video modes ... Byte Magazine Tweaked VGA Modes MS-Kermit Keyboard Routines PC Magazine Sound Routines PC Tech Journal CPU, FPU Detectors ═══ 3.2. Distribution Policy ═══ 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. **** Warning **** Warning **** Warning **** No Warranties are either Expressed or Implied! **** Warning **** Warning **** Warning **** So, that's it. Please let me know what you think. I will be checking the COMART forum on CompuServe periodically. ═══ 3.3. Contacting the Author ═══ You may contact me as follows: Donald P. Egen 409 Cameron Circle, Apt. 1204 Chattanooga, TN 37402 CIS 73507,3143 ═══ 4. The Fractal Formulas ═══ These panels give details of the fractal formulas and parameters. Select them from the fractal descriptions or from the table of contents. ═══ 4.1. barnsleyj1 formula ═══ z(0) = pixel; z(n+1) = (z-1)*c if real(z) >= 0, else z(n+1) = (z+1)*modulus(c)/c Two parameters: real and imaginary parts of c Select here for details. ═══ 4.2. barnsleyj2 formula ═══ z(0) = pixel; if real(z(n)) * imag(c) + real(c) * imag(z((n)) >= 0 z(n+1) = (z(n)-1)*c else z(n+1) = (z(n)+1)*c Two parameters: real and imaginary parts of c Select here for details. ═══ 4.3. barnsleyj3 formula ═══ z(0) = pixel; if real(z(n) > 0 then z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1) + i * (2*real(z((n)) * imag(z((n))) else z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1 + real(c) * real(z(n)) + i * (2*real(z((n)) * imag(z((n)) + imag(c) * real(z(n)) Two parameters: real and imaginary parts of c. Select here for details. ═══ 4.4. barnsleym1 formula ═══ z(0) = c = pixel; if real(z) >= 0 then z(n+1) = (z-1)*c else z(n+1) = (z+1)*modulus(c)/c. Parameters are perturbations of z(0) Select here for details. ═══ 4.5. barnsleym2 formula ═══ z(0) = c = pixel; if real(z)*imag(c) + real(c)*imag(z) >= 0 z(n+1) = (z-1)*c else z(n+1) = (z+1)*c Parameters are perturbations of z(0) Select here for details. ═══ 4.6. barnsleym3 formula ═══ z(0) = c = pixel; if real(z(n) > 0 then z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1) + i * (2*real(z((n)) * imag(z((n))) else z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1 + real(c) * real(z(n)) + i * (2*real(z((n)) * imag(z((n)) + imag(c) * real(z(n)) Parameters are pertubations of z(0) Select here for details. ═══ 4.7. bifurcation formula ═══ Pictorial representation of a population growth model. Let P = new population, p = oldpopulation, r = growth rate The model is: P = p + r*p*(1-p). No parameters. Select here for details. ═══ 4.8. bif+sinpi formula ═══ Bifurcation variation: model is: P = p + r*sin(PI*p). No parameters. Select here for details. ═══ 4.9. bif=sinpi formula ═══ Bifurcation variation: model is: P = r*sin(PI*p). No parameters. Select here for details. ═══ 4.10. biflambda formula ═══ Bifurcation variation: model is: P = r*p*(1-p)P. No parameters. Select here for details. ═══ 4.11. bifstewart formula ═══ Bifurcation variation: model is: P = (r*p*p) - 1. Two parameters: Filter Cycles and Seed Population. Select here for details. ═══ 4.12. Circle formula ═══ Circle pattern by John Connett x + iy = pixel z = a*(x^2 + y^2) c = integer part of z color = c modulo(number of colors) Select here for details. ═══ 4.13. cmplxmarksjul formula ═══ A generalization of the marksjulia fractal. z(0) = pixel; z(n+1) = (c^exp)*z(n) + c. Four parameters: real and imaginary parts of c and exp. Select here for details. ═══ 4.14. cmplxmarksmand formula ═══ A generalization of the marksmandel fractal. z(0) = c = pixel; z(n+1) = (c^exp)*z(n) + c. Four parameters: real and imaginary parts of perturbation of z(0) and exp. Select here for details. ═══ 4.15. complexnewton and complexbasin formula ═══ Newton fractal types extended to complex degrees. Complexnewton colors pixels according to the number of iterations required to escape to a root. Complexbasin colors pixels according to which root captures the orbit. The equation is based on the newton formula for solving the equation z^p = r z(0) = pixel; z(n+1) = ((p - 1) * z(n)^p + r)/(p * z(n)^(p - 1)). Four parameters: real & imaginary parts of degree p and root r Select here for details. ═══ 4.16. diffusion formula ═══ Diffusion Limited Aggregation. Randomly moving points accumulate. One parameter: border width (default 10) Select here for details. ═══ 4.17. fn+fn(pix) formula ═══ c = z(0) = pixel; z(n+1) = fn1(z) + p*fn2(c) Six parameters: real and imaginary parts of the perturbation of z(0) and factor p, and the functions fn1, and fn2. Select here for details. ═══ 4.18. fn(z*z*) formula ═══ z(0) = pixel; z(n+1) = fn(z(n)*z(n)) One parameter: the function fn. Select here for details. ═══ 4.19. fn*fn formula ═══ z(0) = pixel; z(n+1) = fn1(n)*fn2(n) Two parameters: the functions fn1 and fn2. Select here for details. ═══ 4.20. fn*z+z formula ═══ z(0) = pixel; z(n+1) = p1*fn(z(n))*z(n) + p2*z(n) Six parameters: the real and imaginary components of p1 and p2, and the functions fn1 and fn2. Select here for details. ═══ 4.21. fn+fn ═══ z(0) = pixel; z(n+1) = p1*fn1(z(n))+p2*fn2(z(n)) Six parameters: The real and imaginary components of p1 and p2, and the functions fn1 and fn2. Select here for details. ═══ 4.22. gingerbread man formula ═══ Orbit in two dimensions defined by: x(n+1) = 1 - y(n) + |x(n)| y(n+1) = x(n) Two parameters: initial values of x(0) and y(0). Select here for details. ═══ 4.23. henon ═══ Orbit in two dimensions defined by: x(n+1) = 1 + y(n) - a*x(n)*x(n) y(n+1) = b*x(n) Two parameters: a and b Select here for details. ═══ 4.24. Hopalong formula ═══ Hopalong attractor by Barry Martin - orbit in two dimensions. z(0) = y(0) = 0; x(n+1) = y(n) - sign(x(n))*sqrt(abs(b*x(n)-c)) y(n+1) = a - x(n) Parameters are a, b, and c. Select here for details. ═══ 4.25. julfn+exp formula ═══ A generalized Clifford Pickover fractal. z(0) = pixel; z(n+1) = fn(z(n)) + e^z(n) + c. Three parameters: real & imaginary parts of c, and fn Select here for details. ═══ 4.26. julfn+zsqrd formula ═══ z(0) = pixel; z(n+1) = fn(z(n)) + z(n)^2 + c Three parameters: real & imaginary parts of c, and fn Select here for details. ═══ 4.27. julia formula ═══ Classic Julia set fractal. z(0) = pixel; z(n+1) = z(n)^2 + c. Two parameters: real and imaginary parts of c. Select here for details. ═══ 4.28. julia4 formula ═══ Fourth-power Julia set fractals, a special case of julzpower kept for speed. z(0) = pixel; z(n+1) = z(n)^4 + c. Two parameters: real and imaginary parts of c. Select here for details. ═══ 4.29. julzpower formula ═══ z(0) = pixel; z(n+1) = z(n)^m + c. Three parameters: real & imaginary parts of c, exponent m Select here for details. ═══ 4.30. julzzpwr formula ═══ z(0) = pixel; z(n+1) = z(n)^z(n) + z(n)^m + c. Three parameters: real & imaginary parts of c, exponent m Select here for details. ═══ 4.31. kamtorus, kamtorus3d formulas ═══ Series of orbits superimposed. 3d version has 'orbit' the z dimension. x(0) = y(0) = orbit/3; x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a) y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a) After each orbit, 'orbit' is incremented by a step size. Parameters: a, step size, stop value for 'orbit', and points per orbit. Select here for details. ═══ 4.32. lambda formula ═══ Classic Lambda fractal. 'Julia' variant of Mandellambda. z(0) = pixel; z(n+1) = lambda*z(n)*(1 - z(n)^2). Two parameters: real and imaginary parts of lambda. Select here for details. ═══ 4.33. lambdafn formula ═══ z(0) = pixel; z(n+1) = lambda * fn(z(n)). Three parameters: real, imag portions of lambda, and fn Select here for details. ═══ 4.34. lorenz, lorenz3d forumla ═══ Lorenz attractor - orbit in three dimensions. In 2d the x and y components are projected to form the image. z(0) = y(0) = z(0) = 1; x(n+1) = x(n) + (-a*x(n)*dt) + ( a*y(n)*dt) y(n+1) = y(n) + ( b*x(n)*dt) - ( y(n)*dt) - (z(n)*x(n)*dt) z(n+1) = z(n) + (-c*z(n)*dt) + (x(n)*y(n)*dt) Parameters are dt, a, b, and c. Select here for details. ═══ 4.35. lorenz3d1 formula ═══ Lorenz one lobe attractor - orbit in three dimensions. The original formulas were developed by Rick Miranda and Emily Stone. z(0) = y(0) = z(0) = 1; norm = sqrt(x(n)^2 + y(n)^2) x(n+1) = x(n) + (-a*dt-dt)*x(n) + (a*dt-b*dt)*y(n) + (dt-a*dt)*norm + y(n)*dt*z(n) y(n+1) = y(n) + (b*dt-a*dt)*x(n) - (a*dt+dt)*y(n) + (b*dt+a*dt)*norm - x(n)*dt*z(n) - norm*z(n)*dt z(n+1) = z(n) +(y(n)*dt/2) - c*dt*z(n) Parameters are dt, a, b, and c. Select here for details. ═══ 4.36. lorenz3d3 ═══ Lorenz three lobe attractor - orbit in three dimensions. The original formulas were developed by Rick Miranda and Emily Stone. z(0) = y(0) = z(0) = 1; norm = sqrt(x(n)^2 + y(n)^2) x(n+1) = x(n) +(-(a*dt+dt)*x(n) + (a*dt-b*dt+z(n)*dt)*y(n))/3 + ((dt-a*dt)*(x(n)^2-y(n)^2) + 2*(b*dt+a*dt-z(n)*dt)*x(n)*y(n))/(3*norm) y(n+1) = y(n) +((b*dt-a*dt-z(n)*dt)*x(n) - (a*dt+dt)*y(n))/3 + (2*(a*dt-dt)*x(n)*y(n) + (b*dt+a*dt-z(n)*dt)*(x(n)^2-y(n)^2))/(3*norm) z(n+1) = z(n) +(3*x(n)*dt*x(n)*y(n)-y(n)*dt*y(n)^2)/2 - c*dt*z(n) Parameters are dt, a, b, and c. Select here for details. ═══ 4.37. lorenz3d4 ═══ Lorenz four lobe attractor - orbit in three dimensions. The original formulas were developed by Rick Miranda and Emily Stone. z(0) = y(0) = z(0) = 1; x(n+1) = x(n) +(-a*dt*x(n)^3 + (2*a*dt+b*dt-z(n)*dt)*x(n)^2*y(n) + (a*dt-2*dt)*x(n)*y(n)^2 + (z(n)*dt-b*dt)*y(n)^3) / (2 * (x(n)^2+y(n)^2)) y(n+1) = y(n) +((b*dt-z(n)*dt)*x(n)^3 + (a*dt-2*dt)*x(n)^2*y(n) + (-2*a*dt-b*dt+z(n)*dt)*x(n)*y(n)^2 - a*dt*y(n)^3) / (2 * (x(n)^2+y(n)^2)) z(n+1) = z(n) +(2*x(n)*dt*x(n)^2*y(n) - 2*x(n)*dt*y(n)^3 - c*dt*z(n)) Parameters are dt, a, b, and c. Select here for details. ═══ 4.38. magnetj1 formula ═══ z(0) = pixel; / z(n)^2 + (c-1) \\ z(n+1) = | ---------------- | ^ 2 \ 2*z(n) + (c-2) / Parameters: the real and imaginary parts of c Select here for details. ═══ 4.39. magnet1m formula ═══ z(0) = 0; c = pixel; / z(n)^2 + (c-1) \\ z(n+1) = | ---------------- | ^ 2 \ 2*z(n) + (c-2) / Parameters: the real & imaginary parts of perturbation of z(0) Select here for details. ═══ 4.40. magnet2j formula ═══ z(0) = pixel; / z(n)^3 + 3*(C-1)*z(n) + (C-1)*(C-2) \\ z(n+1) = | -------------------------------------------- | ^ 2 \ 3*(z(n)^2) + 3*(C-2)*z(n) + (C-1)*(C-2) - 1 / Parameters: the real and imaginary parts of c Select here for details. ═══ 4.41. magnet2m formula ═══ z(0) = 0; c = pixel; / z(n)^3 + 3*(C-1)*z(n) + (C-1)*(C-2) \\ z(n+1) = | -------------------------------------------- | ^ 2 \ 3*(z(n)^2) + 3*(C-2)*z(n) + (C-1)*(C-2) - 1 / Parameters: the real and imaginary parts of perturbation of z(0) Select here for details. ═══ 4.42. mandel formula ═══ Classic Mandelbrot set fractal. z(0) = c = pixel; z(n+1) = z(n)^2 + c. Two parameters: real & imaginary perturbations of z(0) Select here for details. ═══ 4.43. mandel4 formula ═══ Special case of mandelzpower kept for speed. z(0) = c = pixel; z(n+1) = z(n)^4 + c. Parameters: real & imaginary perturbations of z(0) Select here for details. ═══ 4.44. mandelfn formula ═══ z(0) = c = pixel; z(n+1) = c*fn(z(n)). Parameters: real & imaginary perturbations of z(0), and fn Select here for details. ═══ 4.45. Martin formula ═══ Attractor fractal by Barry Martin - orbit in two dimensions. z(0) = y(0) = 0; x(n+1) = y(n) - sin(x(n)) y(n+1) = a - x(n) Parameter is a (try a value near pi) Select here for details. ═══ 4.46. mandellambda formula ═══ z(0) = .5; lambda = pixel; z(n+1) = lambda*z(n)*(1 - z(n)^2). Parameters: real & imaginary perturbations of z(0) Select here for details. ═══ 4.47. manfn+exp formula ═══ 'Mandelbrot-Equivalent' for the julfn+exp fractal. z(0) = c = pixel; z(n+1) = fn(z(n)) + e^z(n) + C. Parameters: real & imaginary perturbations of z(0), and fn Select here for details. ═══ 4.48. manfn+zsqrd formula ═══ 'Mandelbrot-Equivalent' for the Julfn+zsqrd fractal. z(0) = c = pixel; z(n+1) = fn(z(n)) + z(n)^2 + c. Parameters: real & imaginary perturbations of z(0), and fn Select here for details. ═══ 4.49. manowar formula ═══ c = z1(0) = z(0) = pixel; z(n+1) = z(n)^2 + z1(n) + c; z1(n+1) = z(n); Parameters: real & imaginary perturbations of z(0) Select here for details. ═══ 4.50. manowar julia formula ═══ z1(0) = z(0) = pixel; z(n+1) = z(n)^2 + z1(n) + c; z1(n+1) = z(n); Parameters: real & imaginary perturbations of z(0) Select here for details. ═══ 4.51. manzpower formula ═══ 'Mandelbrot-Equivalent' for julzpower. z(0) = c = pixel; z(n+1) = z(n)^exp + c; try exp = e = 2.71828... Parameters: real & imaginary perturbations of z(0), real & imaginary parts of exponent exp. Select here for details. ═══ 4.52. manzzpwr formula ═══ 'Mandelbrot-Equivalent' for the julzzpwr fractal. z(0) = c = pixel z(n+1) = z(n)^z(n) + z(n)^exp + C. Parameters: real & imaginary perturbations of z(0), and exponent Select here for details. ═══ 4.53. marksjulia formula ═══ A variant of the julia-lambda fractal. z(0) = pixel; z(n+1) = (c^exp)*z(n) + c. Parameters: real & imaginary parts of c, and exponent Select here for details. ═══ 4.54. marksmandel formula ═══ A variant of the mandel-lambda fractal. z(0) = c = pixel; z(n+1) = (c^exp)*z(n) + c. Parameters: real & imaginary perturbations of z(0), and exponent Select here for details. ═══ 4.55. marksmandelpwr formula ═══ The marksmandelpwr formula type generalized (it previously had fn=sqr hard coded). z(0) = pixel, c = z(0) ^ (z(0) - 1): z(n+1) = c * fn(z(n)) + pixel, Parameters: real and imaginary pertubations of z(0), and fn Select here for details. ═══ 4.56. newtbasin formula ═══ Based on the Newton formula for finding the roots of z^p - 1. Pixels are colored according to which root captures the orbit. z(0) = pixel; z(n+1) = ((p-1)*z(n)^p + 1)/(p*z(n)^(p - 1)). Two parameters: the polynomial degree p, and a flag to turn on color stripes to show alternate iterations. Select here for details. ═══ 4.57. newton formula ═══ Based on the Newton formula for finding the roots of z^p - 1. Pixels are colored according to the iteration when the orbit is captured by a root. z(0) = pixel; z(n+1) = ((p-1)*z(n)^p + 1)/(p*z(n)^(p - 1)). One parameter: the polynomial degree p. Select here for details. ═══ 4.58. pickover formula ═══ Orbit in three dimensions defined by: x(n+1) = sin(a*y(n)) - z(n)*cos(b*x(n)) y(n+1) = z(n)*sin(c*x(n)) - cos(d*y(n)) z(n+1) = sin(x(n)) Parameters: a, b, c, and d. Select here for details. ═══ 4.59. plasma formula ═══ Random, cloud-like formations. Requires 4 or more colors. A recursive algorithm repeatedly subdivides the screen and colors pixels according to an average of surrounding pixels and a random color, less random as the grid size decreases. One parameter: 'graininess' (.5 to 50, default = 2). Select here for details. ═══ 4.60. popcorn formula ═══ The orbits in two dimensions defined by: x(0) = xpixel, y(0) = ypixel; x(n+1) = x(n) - h*sin(y(n) + tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n) + tan(3*x(n)) are plotted for each screen pixel and superimposed. One parameter: step size h. Select here for details. ═══ 4.61. popcornjul formula ═══ Conventional Julia using the popcorn formula: x(0) = xpixel, y(0) = ypixel; x(n+1) = x(n) - h*sin(y(n) + tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n) + tan(3*x(n)) One parameter: step size h. Select here for details. ═══ 4.62. rossler3D formula ═══ Orbit in three dimensions defined by: x(0) = y(0) = z(0) = 1; x(n+1) = x(n) - y(n)*dt - z(n)*dt y(n+1) = y(n) + x(n)*dt + a*y(n)*dt z(n+1) = z(n) + b*dt + x(n)*z(n)*dt - c*z(n)*dt Parameters are dt, a, b, and c. Select here for details. ═══ 4.63. sierpinski formula ═══ Sierpinski gasket - Julia set producing a 'Swiss cheese triangle' z(n+1) = (2*x,2*y-1) if y > .5; else (2*x-1,2*y) if x > .5; else (2*x,2*y) No parameters. Select here for details. ═══ 4.64. spider formula ═══ c(0) = z(0) = pixel; z(n+1) = z(n)^2 + c(n); c(n+1) = c(n)/2 + z(n+1) Parameters: real & imaginary perturbation of z(0) Select here for details. ═══ 4.65. sqr(1/fn) formula ═══ z(0) = pixel; z(n+1) = (1/fn(z(n))^2 One parameter: the function fn. Select here for details. ═══ 4.66. sqr(fn) formula ═══ z(0) = pixel; z(n+1) = fn(z(n))^2 One parameter: the function fn. Select here for details. ═══ 4.67. test formula ═══ 'test' point letting us (and you!) easily add fractal types via the c module testpt.c. Default set up is a mandelbrot fractal. Four parameters: user hooks (not used by default testpt.c). Select here for details. ═══ 4.68. tetrate formula ═══ z(0) = c = pixel; z(n+1) = c^z(n) Parameters: real & imaginary perturbation of z(0) Select here for details. ═══ 4.69. tim's error formula ═══ A serendipitous coding error in marksmandelpwr brings to life an ancient pterodactyl! (Try setting fn to sqr.) z(0) = pixel, c = z(0) ^ (z(0) - 1): tmp = fn(z(n)) real(tmp) = real(tmp) * real(c) - imag(tmp) * imag(c); imag(tmp) = real(tmp) * imag(c) - imag(tmp) * real(c); z(n+1) = tmp + pixel; Parameters: real & imaginary pertubations of z(0) and function fn Select here for details. ═══ 4.70. unity formula. ═══ z(0) = pixel; x = real(z(n)), y = imag(z(n)) One = x^2 + y^2; y = (2 - One) * x; x = (2 - One) * y; z(n+1) = x + i*y No parameters. Select here for details. ═══ 5. Fractal Types ═══ Overview Fractint starts by default with the Mandelbrot set. You can change that by using the command-line argument "TYPE=" followed by one of the fractal type names, or by using the command and selecting the type - if parameters are needed, you will be prompted for them. In the text that follows, due to the limitations of the ASCII character set, "a*b" means "a times b", and "a^b" means "a to the power b". Select a fractal type for details: The Mandelbrot Set Julia Sets Newton domains of attraction Newton Complex Newton Lambda Sets Mandellambda Sets Plasma Clouds Lambdafn Mandelfn Barnsley Mandelbrot/Julia Sets Barnsley IFS Fractals Sierpinski Gasket Quartic Mandelbrot/Julia Distance Estimator Pickover Mandelbrot/Julia Types Pickover Popcorn Peterson Variations Unity Scott Taylor / Lee Skinner Variations Kam Torus Bifurcation Orbit Fractals Lorenz Attractors Rossler Attractors Henon Attractors Pickover Attractors Gingerbreadman Test Formula Julibrots Diffusion Limited Aggregation Magnetic Fractals L-Systems Lyapunov Circle Martin Attractors ═══ 5.1. The Mandelbrot Set ═══ (type=mandel) This set is the classic: the only one implemented in many plotting programs, and the source of most of the printed fractal images published in recent years. Like most of the other types in Fractint, it is simply a graph: the x (horizontal) and y (vertical) coordinate axes represent ranges of two independent quantities, with various colors used to symbolize levels of a third quantity which depends on the first two. So far, so good: basic analytic geometry. Now things get a bit hairier. The x axis is ordinary, vanilla real numbers. The y axis is an imaginary number, i.e. a real number times i, where i is the square root of -1. Every point on the plane -- in this case, your PC's display screen -- represents a complex number of the form: x-coordinate + i * y-coordinate If your math training stopped before you got to imaginary and complex numbers, this is not the place to catch up. Suffice it to say that they are just as "real" as the numbers you count fingers with (they're used every day by electrical engineers) and they can undergo the same kinds of algebraic operations. OK, now pick any complex number -- any point on the complex plane -- and call it C, a constant. Pick another, this time one which can vary, and call it Z. Starting with Z=0 (i.e., at the origin, where the real and imaginary axes cross), calculate the value of the expression Z^2 + C Take the result, make it the new value of the variable Z, and calculate again. Take that result, make it Z, and do it again, and so on: in mathematical terms, iterate the function Z(n+1) = Z(n)^2 + C. For certain values of C, the result "levels off" after a while. For all others, it grows without limit. The Mandelbrot set you see at the start -- the solid- colored lake (blue by default), the blue circles sprouting from it, and indeed every point of that color -- is the set of all points C for which the value of Z is less than 2 after 150 iterations (150 is the default setting, changeable via the options screen or "maxiter=" parameter). All the surrounding "contours" of other colors represent points for which Z exceeds 2 after 149 iterations (the contour closest to the M-set itself), 148 iterations, (the next one out), and so on. We actually don't test for Z exceeding 2 - we test Z squared against 4 instead because it is easier. This value (FOUR usually) is known as the "bailout" value for the calculation, because we stop iterating for the point when it is reached. The bailout value can be changed on the options screen but the default is usually best. Some features of interest: 1. Use the options screen to increase the maximum number of iterations. Notice that the boundary of the M-set becomes more and more convoluted (the technical terms are "wiggly," "squiggly," and "utterly bizarre") as the Z- values for points that were still within the set after 150 iterations turn out to exceed 2 after 200, 500, or 1200. In fact, it can be proven that the true boundary is infinitely long: detail without limit. 2. Although there appear to be isolated "islands" of blue, zoom in -- that is, plot for a smaller range of coordinates to show more detail -- and you'll see that there are fine "causeways" of blue connecting them to the main set. As you zoomed, smaller islands became visible; the same is true for them. In fact, there are no isolated points in the M-set: it is "connected" in a strict mathematical sense. 3. The upper and lower halves of the first image are symmetric (a fact that Fractint makes use of here and in some other fractal types to speed plotting). But notice that the same general features -- lobed discs, spirals, starbursts -- tend to repeat themselves (although never exactly) at smaller and smaller scales, so that it can be impossible to judge by eye the scale of a given image. 4. In a sense, the contour colors are window-dressing: mathematically, it is the properties of the M-set itself that are interesting, and no information about it would be lost if all points outside the set were assigned the same color. If you're a serious, no-nonsense type, you may want to cycle the colors just once to see the kind of silliness that other people enjoy, and then never do it again. Go ahead. Just once, now. We trust you. Select below for details of the formula. mandel formula ═══ 5.2. Julia Sets ═══ (type=julia) These sets were named for mathematician Gaston Julia, and can be generated by a simple change in the iteration process described for the Mandelbrot Set. 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 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 the discussion of Mandellambda Sets. 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. Select below for details of the formula. julia formula ═══ 5.3. 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 -- 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. Select below for details of the formula. newtbasin formula See also: Newton ═══ 5.4. 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 seventh root on the twenty-third 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. Select below for details of the formula. newton formula ═══ 5.5. 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! Select below for details of the formula. complexnewton and complexbasin formula ═══ 5.6. 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. Select below for details of the formula. lambda formula ═══ 5.7. 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 in {Image Calculation Parameters} for a way to experiment with different orbit intializations). Select below for details of the formula. mandellambda formula ═══ 5.8. Circle ═══ (type=circle) This fractal types is from A. K. Dewdney's "Computer Recreations" column in "Scientific American". It is attributed to John Connett of the University of Minnesota. (Don't tell anyone, but this fractal type is not really a fractal!) Fascinating Moire patterns can be formed by calculating x^2 + y^2 for each pixel in a piece of the complex plane. After multiplication by a magnification factor (the parameter), the number is truncated to an integer and mapped to a color via color = value modulo (number of colors). That is, the integer is divided by the number of colors, and the remainder is the color index value used. The resulting image is not a fractal because all detail is lost after zooming in too far. Try it with different resolution video modes - the results may surprise you! Select below for details of the formula. circle formula ═══ 5.9. Plasma Clouds ═══ (type=plasma) Plasma clouds ARE real live fractals, even though we didn't know it at first. They are generated by a recursive algorithm that randomly picks colors of the corner of a rectangle, and then continues recursively quartering previous rectangles. Random colors are averaged with those of the outer rectangles so that small neighborhoods do not show much change, for a smoothed-out, cloud-like effect. The more colors your video mode supports, the better. The result, believe it or not, is a fractal landscape viewed as a contour map, with colors indicating constant elevation. To see this, save and view with the <3> command (see {\"3D\" Images}) and your "cloud" will be converted to a mountain! You've GOT to try color 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 random number seed used for each plasma image is displayed on the information screen, and can be entered with the command line parameter "rseed=" to recreate a particular image. The algorithm is based on the Pascal program distributed by Bret Mulvey as PLASMA.ARC. We have ported it to C and integrated it with Fractint's graphics and animation facilities. This implementation does not use floating-point math. Saved plasma-cloud screens are EXCELLENT starting images for fractal "landscapes" created with the {\"3D\" commands}. Select below for details of the formula. plasma formula ═══ 5.10. 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! Select below for details of the formula. lambdafn formula ═══ 5.11. 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 "PMFRACT 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. See GIF Save File Format. Select below for details of the formula. mandelfn formula ═══ 5.12. Barnsley Mandelbrot/Julia Sets ═══ (type=barnsleym1/.../j3) Michael Barnsley has written a fascinating college-level text, "Fractals Everywhere," on fractal geometry and its graphic applications. (See Bibliography.) 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. Be sure to try the rbit function while plotting these types. Select below for details on each formula. barnsleyj1 formula barnsleyj2 formula barnsleyj3 formula barnsleym1 formula barnsleym2 formula barnsleym3 formula ═══ 5.13. Barnsley IFS Fractals ═══ (type=ifs) One of the most remarkable spin-offs of fractal geometry is the ability to "encode" realistic images in very small sets of numbers -- parameters for a set of functions that map a region of two-dimensional space onto itself. In principle (and increasingly in practice), a scene of any level of complexity and detail can be stored as a handful of numbers, achieving amazing "compression" ratios... how about a super-VGA image of a forest, more than 300,000 pixels at eight bits apiece, from a 1-KB "seed" file? Again, Michael Barnsley and his co-workers at the Georgia Institute of Technology are to be thanked for pushing the development of these iterated function systems (IFS). When you select this fractal type, Fractint scans the current IFS file (default is FRACTINT.IFS, a set of definitions supplied with Fractint) for IFS definitions, then prompts you for the IFS name you wish to run. Fern and 3dfern are good ones to start with. You can press at the selection screen if you want to select a different .IFS file you've written. Note that some Barnsley IFS values generate images quite a bit smaller than the initial (default) screen. Just bring up the zoom box, center it on the small image, and hit to get a full-screen image. To change the number of dots Fractint generates for an IFS image before stopping, you can change the "maximum iterations" parameter on the options screen. Fractint supports two types of IFS images: 2D and 3D. In order to fully appreciate 3D IFS images, since your monitor is presumably 2D, we have added rotation, translation, and perspective capabilities. These share values with the same variables used in Fractint's other 3D facilities; for their meaning see {"Rectangular Coordinate Transformation"}. You can enter these values from the command line using: rotation=xrot/yrot/zrot (try 30/30/30) shift=xshift/yshift (shifts BEFORE applying perspective!) perspective=viewerposition (try 200) Alternatively, entering from main screen will allow you to modify these values. The defaults are the same as for regular 3D, and are not always optimum for 3D IFS. With the 3dfern IFS type, try rotation=30/30/30. Note that applying shift when using perspective changes the picture -- your "point of view" is moved. A truly wild variation of 3D may be seen by entering "2" for the stereo mode (see {"Stereo 3D Viewing"}), putting on red/blue "funny glasses", and watching the fern develop with full depth perception right there before your eyes! This feature USED to be dedicated to Bruce Goren, as a bribe to get him to send us MORE knockout stereo slides of 3D ferns, now that we have made it so easy! Bruce, what have you done for us *LATELY* ?? (Just kidding, really!) Each line in an IFS definition (look at FRACTINT.IFS with your editor for examples) contains the parameters for one of the generating functions, e.g. in FERN: a b c d e f p ___________________________________ 0 0 0 .16 0 0 .01 .85 .04 -.04 .85 0 1.6 .85 .2 -.26 .23 .22 0 1.6 .07 -.15 .28 .26 .24 0 .44 .07 The values on each line define a matrix, vector, and probability: matrix vector prob |a b| |e| p |c d| |f| P. The "p" values are the probabilities assigned to each function (how often it is used), which add up to one. Fractint supports up to 32 functions, although usually three or four are enough. 3D IFS definitions are a bit different. The name is followed by (3D) in the definition file, and each line of the definition contains 13 numbers: a b c d e f g h i j k l p, defining: matrix vector prob |a b c| |j| p |d e f| |k| |g h i| |l| You can experiment with changes to IFS definitions interactively by using Fractint's command. After selecting an IFS definition, hit to bring up the IFS editor. This editor displays the current IFS values, lets you modify them, and lets you save your modified values as a text file which you can then merge into an XXX.IFS file for future use with Fractint. The program FDESIGN can be used to design IFS fractals. You can save the points in your IFS fractal in the file ORBITS.RAW which is overwritten each time a fractal is generated. The program Acrospin can read this file and will let you view the fractal from in any angle using the cursor keys. ═══ 5.14. 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. Select below for details of the formula. sierpinski formula ═══ 5.15. 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. Select below for details on each formula. mandel4 formula julia4 formula ═══ 5.16. Distance Estimator ═══ (distest=nnn/nnn) This used to be type=demm and type=demj. These types have not died, but are only hiding! They are equivalent to the mandel and julia types with the "distest=" option selected with a predetermined value. The Distance Estimator Method can be used to produce higher quality images of M and J sets, especially suitable for printing in black and white. If you have some *.fra files made with the old types demm/demj, you may want to convert them to the new form. See the Mandelfn section for directions to carry out the conversion. ═══ 5.17. 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+zsqrd biomorph=0 corners=-8/8/-6/6 function=sin to see a big biomorph digesting little biomorphs! Select below for details on each formula. manfn+zsqrd formula julfn+zsqrd formula manzpowr formula julzpowr formula manzzpwr formula julzzpwr formula manfn+exp formula julfn+exp formula ═══ 5.18. Pickover Popcorn ═══ (type=popcorn/popcornjul) Here is another Pickover idea. This one computes and plots the orbits of the dynamic system defined by: x(n+1) = x(n) - h*sin(y(n)+tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n)+tan(3*x(n)) with the initializers x(0) and y(0) equal to ALL the complex values within the "corners" values, and h=.01. ALL these orbits are superimposed, resulting in "popcorn" effect. You may want to use a maxiter value less than normal - Pickover recommends a value of 50. As a bonus, type=popcornjul shows the Julia set generated by these same equations with the usual escape-time coloring. Turn on orbit viewing with the "O" command, and as you watch the orbit pattern you may get some insight as to where the popcorn comes from. Although you can zoom and rotate popcorn, the results may not be what you'd expect, due to the superimposing of orbits and arbitrary use of color. Just for fun we added type popcornjul, which is the plain old Julia set calculated from the same formula. Select below for details on each formula. popcorn formula popcornjul formula ═══ 5.19. Peterson Variations ═══ (type=marksmandel, marksjulia, cmplxmarksmand, cmplxmarksjul, marksmandelpwr, tim's_error) These fractal types are contributions of Mark Peterson. MarksMandel and MarksJulia are two families of fractal types that are linked in the same manner as the classic Mandelbrot/Julia sets: each MarksMandel set can be considered as a mapping into the MarksJulia sets, and is linked with the spacebar toggle. The basic equation for these sets is: Z(n+1) = ((lambda^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. In the process of coding MarksMandelPwr formula type, Tim Wegner created the type "tim's_error" after making an interesting coding mistake. Select below for details on each formula. marksmandel formula marksjulia formula cmplxmarksmand formula cmplsmarksjul formula marksmandpwr formula tim's error formula ═══ 5.20. 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 and let us know what you come up with! Select below for details of the formula. unity formula ═══ 5.21. Scott Taylor / Lee Skinner Variations ═══ (type=fn(z*z), fn*fn, fn*z+z, fn+fn, sqr(1/fn), sqr(fn), spider, tetrate, manowar) Two of Fractint's faithful users went bonkers when we introduced the "formula" type, and came up with all kinds of variations on escape-time fractals using trig functions. We decided to put them in as regular types, but there were just too many! So we defined the types with variable functions and let you, the, overwhelmed user, specify what the functions should be! Thus Scott Taylor's "z = sin(z) + z^2" formula type is now the "fn+fn" regular type, and EITHER function can be one of sin, cos, tan, cotan, sinh, cosh, tanh, cotanh, exp, log, sqr, recip, ident, or cosxx. Plus we give you 4 parameters to set, the complex coefficients of the two functions! Thus the innocent-looking "fn+fn" type is really 66 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. Select below for details on each formula. fn(z*z) formula fn*fn formula fn*z+z formula fn+fn formula fn+fn(pix) formula sqr(1/fn) formula sqr(fn) formula spider formula tetrate formula manowar formula manowar julia formula ═══ 5.22. Kam Torus ═══ (type=kamtorus, kamtorus3d) This type is created by superimposing orbits generated by a set of equations, with a variable incremented each time. x(0) = y(0) = orbit/3; x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a) y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a) After each orbit, 'orbit' is incremented by a step size. The parameters are angle "a", step size for incrementing 'orbit', stop value for 'orbit', and points per orbit. Try this with a stop value of 5 with sound=x for some weird fractal music (ok, ok, fractal noise)! You will also see the KAM Torus head into some chaotic territory that Scott Taylor wanted to hide from you by setting the defaults the way he did, but now we have revealed all! The 3D variant is created by treating 'orbit' as the z coordinate. With both variants, you can adjust the "maxiter" value ( options screen or parameter maxiter=) to change the number of orbits plotted. Select below for details on the formula. kamtorus formula ═══ 5.23. Bifurcation ═══ (type=bifxxx) The wonder of fractal geometry is that such complex forms can arise from such simple generating processes. A parallel surprise has emerged in the study of dynamical systems: that simple, deterministic equations can yield chaotic behavior, in which the system never settles down to a steady state or even a periodic loop. Often such systems behave normally up to a certain level of some controlling parameter, then go through a transition in which there are two possible solutions, then four, and finally a chaotic array of possibilities. This emerged many years ago in biological models of population growth. Consider a (highly over-simplified) model in which the rate of growth is partly a function of the size of the current population: 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. Two parameters apply to bifurcations: Filter Cycles and Seed Population. Filter Cycles (default 1000) is the number of iterations to be done before plotting maxiter population values. This gives the iteration time to settle into the characteristic patterns that constitute the bifurcation diagram, and results in a clean-looking plot. However, using lower values produces interesting results too. Set Filter Cycles to 1 for an unfiltered map. Seed Population (default 0.66) is the initial population value from which all others are calculated. For filtered maps the final image is independent of Seed Population value in the valid range (0.0 < Seed Population < 1.0). Seed Population becomes effective in unfiltered maps - try setting Filter Cycles to 1 (unfiltered) and Seed Population to 0.001 ("PARAMS=1/.001" on the command line). This results in a map overlaid with nice curves. Each Seed Population value results in a different set of curves. 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 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. bifstewart P = r*p*p - 1 Stewart Map. It took a while for bifurcations to appear here, despite them being over a century old, and intimately related to chaotic systems. However, they are now truly alive and well in Fractint! Select below for details on each formula. bifurcation formula bif+sinpi formula bif=sinpi formula biflambda formula bifstewart formula ═══ 5.24. Orbit Fractals ═══ Orbit Fractals are generated by plotting an orbit path in two or three dimensional space. See Lorenz Attractors, Rossler Attractors, Henon Attractors, Pickover Attractors, Gingerbreadman, Martin Attractors. The orbit trajectory for these types can be saved in the file ORBITS.RAW by invoking Fractint with the "orbitsave=yes" command-line option. This file will be overwritten each time you generate a new fractal, so rename it if you want to save it. A nifty program called Acrospin can read these files and rapidly rotate them in 3-D - see Acrospin. ═══ 5.25. Lorenz Attractors ═══ (type=lorenz/lorenz3d) The "Lorenz Attractor" is a "simple" set of three deterministic equations developed by Edward Lorenz while studying the non- repeatability of weather patterns. The weather forecaster's basic problem is that even very tiny changes in initial patterns ("the beating of a butterfly's wings" - the official term is "sensitive dependence on initial conditions") eventually reduces the best weather forecast to rubble. The lorenz attractor is the plot of the orbit of a dynamic system consisting of three first order non-linear differential equations. The solution to the differential equation is vector-valued function of one variable. If you think of the variable as time, the solution traces an orbit. The orbit is made up of two spirals at an angle to each other in three dimensions. We change the orbit color as time goes on to add a little dazzle to the image. The equations are: dx/dt = -a*x + a*y dy/dt = b*x - y -z*x dz/dt = -c*z + x*y We solve these differential equations approximately using a method known as the first order taylor series. Calculus teachers everywhere will kill us for saying this, but you treat the notation for the derivative dx/dt as though it really is a fraction, with "dx" the small change in x that happens when the time changes "dt". So multiply through the above equations by dt, and you will have the change in the orbit for a small time step. We add these changes to the old vector to get the new vector after one step. This gives us: xnew = x + (-a*x*dt) + (a*y*dt) ynew = y + (b*x*dt) - (y*dt) - (z*x*dt) znew = z + (-c*z*dt) + (x*y*dt) (default values: dt = .02, a = 5, b = 15, c = 1) We connect the successive points with a line, project the resulting 3D orbit onto the screen, and voila! The Lorenz Attractor! We have added two versions of the Lorenz Attractor. "Type=lorenz" is the Lorenz attractor as seen in everyday 2D. "Type=lorenz3d" is the same set of equations with the added twist that the results are run through our perspective 3D routines, so that you get to view it from different angles (you can modify your perspective "on the fly" by using the command.) If you set the "stereo" option to "2", and have red/blue funny glasses on, you will see the attractor orbit with depth perception. 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 second, 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. Select below for details on the formula. lorenz and lorenz3d formula lorenz3d1 formula lorenz3d3 formula lorenz3d4 formula ═══ 5.26. Rossler Attractors ═══ (type=rossler3D) This fractal is named after the German Otto Rossler, a non-practicing medical doctor who approached chaos with a bemusedly philosophical attitude. He would see strange attractors as philosophical objects. His fractal namesake looks like a band of ribbon with a fold in it. All we can say is we used the same calculus-teacher-defeating trick of multiplying the equations by "dt" to solve the differential equation and generate the orbit. This time we will skip straight to the orbit generator - if you followed what we did above with type Lorenz 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 Select below for details on the formula. rossler3D formula ═══ 5.27. Henon Attractors ═══ (type=henon) Michel Henon was an astronomer at Nice observatory in southern France. He came to the subject of fractals via investigations of the orbits of astronomical objects. The strange attractor most often linked with Henon's name comes not from a differential equation, but from the world of discrete mathematics - difference equations. The Henon map is an example of a very simple dynamic system that exhibits strange behavior. The orbit traces out a characteristic banana shape, but on close inspection, the shape is made up of thicker and thinner parts. Upon magnification, the thicker bands resolve to still other thick and thin components. And so it goes forever! The equations that generate this strange pattern perform the mathematical equivalent of repeated stretching and folding, over and over again. xnew = 1 + y - a*x*x ynew = b*x The default parameters are a=1.4 and b=.3. Select below for details on the formula. henon formula ═══ 5.28. 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 Select below for details on the formula. pickover formula ═══ 5.29. Gingerbreadman ═══ (type=gingerbreadman) This simple fractal is a charming example stolen from "Science of Fractal Images", p. 149. xnew = 1 - y + |x| ynew = x The initial x and y values are set by parameters, defaults x=-.1, y = 0. Select below for details on the formula. gingerbreadman formula ═══ 5.30. Martin Attractors ═══ (type=hopalong/martin) These fractal types are from A. K. Dewdney's "Computer Recreations" column in "Scientific American". They are attributed to Barry Martin of Aston University in Birmingham, Alabama. Hopalong is an "orbit" type fractal like lorenz. The image is obtained by iterating this formula after setting z(0) = y(0) = 0: x(n+1) = y(n) - sign(x(n))*sqrt(abs(b*x(n)-c)) y(n+1) = a - x(n) Parameters are a, b, and c. The function "sign()" returns 1 if the argument is positive, -1 if argument is negative. This fractal continues to develop in surprising ways after many iterations. Another Martin fractal is simpler. The iterated formula is: x(n+1) = y(n) - sin(x(n)) y(n+1) = a - x(n) The paramneter is "a". Try values near the number pi. Select below for details on the formula. hopalong formula Martin formula ═══ 5.31. Test ═══ (type=test) This is a stub that we (and you!) use for trying out new fractal types. "Type=test" fractals make use of Fractint's structure and features for whatever code is in the routine 'testpt()' (located in the small source file TESTPT.C) to determine the color of a particular pixel. If you have a favorite fractal type that you believe would fit nicely into Fractint, just rewrite the C function in TESTPT.C (or use the prototype function there, which is a simple M-set implementation) with an algorithm that computes a color based on a point in the complex plane. After you get it working, send your code to one of the authors and we might just add it to the next release of Fractint, with full credit to you. Our criteria are: 1) an interesting image and 2) a formula significantly different from types already supported. (Bribery may also work. THIS author is completely honest, but I don't trust those other guys.) Be sure to include an explanation of your algorithm and the parameters supported, preferably formatted as you see here to simplify folding it into the documentation. Select below for details on the formula. test formula ═══ 5.32. Formula ═══ (type=formula) This is a "roll-your-own" fractal interpreter - you don't even need a compiler! To run a "type=formula" fractal, you first need a text file containing formulas (there's a sample file - FRACTINT.FRM - included with this distribution). When you select the "formula" fractal type, Fractint scans the current formula file (default is FRACTINT.FRM) for formulas, then prompts you for the formula name you wish to run. After prompting for any parameters, the formula is parsed for syntax errors and then the fractal is generated. If you want to use a different formula file, press when you are prompted to select a formula name. There are two command-line options that work with type=formula ("formulafile=" and "formulaname="), useful when you are using this fractal type in batch mode. The following documentation is supplied by Mark Peterson, who wrote the formula interpreter: Formula fractals allow you to create your own fractal formulas. The general format is: Mandelbrot(XAXIS) { z = Pixel: z = sqr(z) + pixel, |z| <= 4 } | | | | | Name Symmetry Initial Iteration Bailout Condition Criteria Initial conditions are set, then the iterations performed until the bailout criteria is true or 'z' turns into a periodic loop. All variables are created automatically by their usage and treated as complex. If you declare 'v = 2' then the variable 'v' is treated as a complex with an imaginary value of zero. Predefined Variables (x, y) -------------------------------------------- z used for periodicity checking p1 parameters 1 and 2 p2 parameters 3 and 4 pixel screen coordinates Precedence -------------------------------------------- 1 sin(), cos(), sinh(), cosh(), cosxx(), tan(), cotan(), tanh(), cotanh(), sqr, log(), exp(), abs(), conj(), real(), imag(), flip(), fn1(), fn2(), fn3(), fn4() 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 functions fn1(...) to fn4(...) are variable functions - when used, the user is prompted at run time (on the screen) to specify one of sin, cos, sinh, cosh, exp, log, sqr, etc. for each required variable function. The formulas are performed using either integer or floating point mathematics depending on the floating point toggle. 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 toggle. ═══ 5.33. Julibrots ═══ (type=julibrot) The following documentation is supplied by Mark Peterson, who "invented" the Julibrot algorithm. There is a very close relationship between the Mandelbrot set and Julia sets of the same equation. To draw a Julia set you take the basic equation and vary the initial value according to the two dimensions of screen leaving the constant untouched. This method diagrams two dimensions of the equation, 'x' and 'iy', which I refer to as the Julia x and y. z(0) = screen coordinate (x + iy) z(1) = (z(0) * z(0)) + c, where c = (a + ib) z(2) = (z(1) * z(0)) + c z(3) = . . . . The Mandelbrot set is a composite of all the Julia sets. If you take the center pixel of each Julia set and plot it on the screen coordinate corresponding to the value of c, a + ib, then you have the Mandelbrot set. z(0) = 0 z(1) = (z(0) * z(0)) + c, where c = screen coordinate (a + ib) z(2) = (z(1) * z(1)) + c z(3) = . . . . I refer to the 'a' and 'ib' components of 'c' as the Mandelbrot 'x' and 'y'. All the 2 dimensional Julia sets correspond to a single point on the 2 dimensional Mandelbrot set, making a total of 4 dimensions associated with our equation. Visualizing 4 dimensional objects is not as difficult as it may sound at first if you consider we live in a 4 dimensional world. The room around you is three dimensions and as you read this text you are moving through the fourth dimension of time. You and everything around your are 4 dimensional objects - which is to say 3 dimensional objects moving through time. We can think of the 4 dimensions of our equation in the same manner, this is as a 3 dimensional object evolving over time - sort of a 3 dimensional fractal movie. The fun part of it is you get to pick the dimension representing time! To construct the 4 dimensional object into something you can view on the computer screen you start with the simple 2 dimensions of the Julia set. I'll treat the two Julia dimensions as the 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. 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. ═══ 5.34. 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. Select below for details on the formula. diffusion formula ═══ 5.35. Magnetic Fractals ═══ (type=magnet1m/.../magnet2j) These fractals use formulae derived from the study of hierarchical lattices, in the context of magnetic renormalisation transformations. This kinda stuff is useful in an area of theoretical physics that deals with magnetic phase-transitions (predicting at which temperatures a given substance will be magnetic, or non-magnetic). In an attempt to clarify the results obtained for Real temperatures (the kind that you and I can feel), the study moved into the realm of Complex Numbers, aiming to spot Real phase-transitions by finding the intersections of lines representing Complex phase-transitions with the Real Axis. The first people to try this were two physicists called Yang and Lee, who found the situation a bit more complex than first expected, as the phase boundaries for Complex temperatures are (surprise!) fractals. And that's all the technical (?) background you're getting here! For more details (are you SERIOUS ?!) read "The Beauty of Fractals". When you understand it all, you might like to 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 Logic}. See {Finite Attractors} for more information on this aspect of Fractint internals. (Thanks to Kevin Allen for Magnetic type documentation above). Select below for details on each formula. magnet1m formula magnet2m formula magnet1j formula magnet2j formula ═══ 5.36. L-Systems ═══ (type=lsystem) These fractals are constructed from line segments using rules specified in drawing commands. Starting with an initial string, the axiom, transformation rules are applied a specified number of times, to produce the final command string which is used to draw the image. Like the type=formula fractals, this type requires a separate data file. A sample file, FRACTINT.L, is included with this distribution. When you select type lsystem, the current lsystem file is read and you are asked for the lsystem name you wish to run. Press at this point if you wish to use a different lsystem file. After selecting an lsystem, you are asked for one parameter - the "order", or number of times to execute all the transformation rules. It is wise to start with small orders, because the size of the substituted command string grows exponentially and it is very easy to exceed your resolution. (Higher orders take longer to generate too.) The command line options "lname=" and "lfile=" can be used to over- ride the default file name and lsystem name. Each L-System entry in the file contains a specification of the angle, the axiom, and the transformation rules. Each item must appear on its own line and each line must be less than 160 characters long. The statement "angle n" sets the angle to 360/n degrees; n must be an integer greater than two and less than fifty. "Axiom string" defines the axiom. Transformation rules are specified as "a=string" and convert the single character 'a' into "string." If more than one rule is specified for a single character all of the strings will be added together. This allows specifying transformations longer than the 160 character limit. Transformation rules may operate on any characters except space, tab or '}'. Any information after a ; (semi-colon) on a line is treated as a comment. Here is a sample lsystem: Dragon { ; Name of lsystem, { indicates start Angle 8 ; Specify the angle increment to 45 degrees Axiom FX ; Starting character string F= ; First rule: Delete 'F' y=+FX--FY+ ; Change 'y' into "+fx--fy+" x=-FX++FY- ; Similar transformation on 'x' } ; final } indicates end The standard drawing commands are: F Draw forward G Move forward (without drawing) + Increase angle - Decrease angle | Try to turn 180 degrees. (If angle is odd, the turn will be the largest possible turn less than 180 degrees.) These commands increment angle by the user specified angle value. They should be used when possible because they are fast. If greater flexibility is needed, use the following commands which keep a completely separate angle pointer which is specified in degrees. D Draw forward M Move forward \nn Increase angle nn degrees /nn Decrease angle nn degrees 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 preceded by I for inverse, or Q for square root. (e.g. @IQ2 divides size by the square root of 2) [ Push. Stores current angle and position on a stack ] Pop. Return to location of last push Other characters are perfectly legal in command strings. They are ignored for drawing purposes, but can be used to achieve complex translations. ═══ 5.37. Lyapunov Fractals ═══ (type=lyapunov) The Bifurcation fractal illustrates what happens in a simple population model as the growth rate increases. The Lyapunov fractal expands that model into two dimensions by letting the growth rate vary in a periodic fashion between two values. Each pair of growth rates is run through a logistic population model and a value called the Lyapunov Exponent is calculated for each pair and is plotted. The Lyapunov Exponent is calculated by adding up log | r -2*r*x| over many cycles of the population model and dividing by the number of cycles. Negative Lyapunov exponents indicate a stable periodic behavior and are plotted in color. Positive Lyapunov exponents indicate chaos and are colored black. Order parameter. Each possible periodic sequence yields a two dimensional space to explore. The Order parameter selects a sequence. The default value 0 represents the sequence ab which alternates between the two values of the growth parameter. Here is how to calculate the space parameter for any desired sequence. Take your sequence of a's and b's and arrange it so that it starts with at least 2 a's and ends with a b. It may be necessary to rotate the sequence or swap a's and b's. Strike the first a and the last b off the list and replace each remaining a with a 1 and each remaining b with a zero. Interpret this as a binary number and convert it into decimal. An Example I like sonnets. A sonnet is a poem with fourteen lines that has the following rhyming sequence: abba abba abab cc. Ignoring the rhyming couplet at the end, let's calculate the Order parameter for this pattern. abbaabbaabab doesn't start with at least 2 a's \ aabbaabababb rotate it \ 1001101010 drop the first and last, replace with 0's and 1's \ 512+64+32+8+2 = 618 An Order parameter of 618 gives the Lyapunov equivalent of a sonnet. "How do I make thee, let me count the ways..." Population Seed When two parts of a Lyapunov overlap, which spike overlaps which is strongly dependant on the initial value of the population model. Any changes from using a different starting value between 0 and 1 may be subtle. Reference: A.K. Dewdney Mathematical Recreations Scientific American Sept. 1991 ═══ 6. Miscellaneous Topics ═══ The following are interesting topics from the Fractint for DOS documentation. o Biomorphs o Distance Estimator Method o Acrospin o Decomposition o GIF File Format o Bibliography o Palette Maps ═══ 6.1. Biomorphs ═══ Related to 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 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. ═══ 6.2. Distance Estimator Method ═══ This is Phil Wilson's implementation of an alternate method for the M and J sets, based on work by mathematician John Milnor and described in "The Science of Fractal Images", p. 198. While it can take full advantage of your color palette, one of the best uses is in preparing monochrome images for a printer. Using the 1600x1200x2 disk-video mode and an HP LaserJet, we have produced pictures of quality equivalent to the black and white illustrations of the M-set in "The Beauty of Fractals." The distance estimator method widens very thin "strands" which are part of the "inside" of the set. Instead of hiding invisibly between pixels, these strands are made one pixel wide. Though this option is available with any escape time fractal type, the formula used is specific to the mandel and julia types - for most other types it doesn't do a great job. To turn on the distance estimator method with any escape time fractal type, set the "Distance Estimator" value on the options screen (or use the "distest=" command line parameter). Setting the distance estimator option to a negative value -nnn enables edge-tracing mode. The edge of the set is display as color number nnn. This option works best when the "inside" and "outside" color values are also set to some other value(s). In a 2 color (monochrome) mode, setting to any positive value results in the inside of the set being expanded to include edge points, and the outside points being displayed in the other color. In color modes, setting to value 1 causes the edge points to be displayed using the inside color and the outside points to be displayed in their usual colors. Setting to a value greater than one causes the outside points to be displayed as contours, colored according to their distance from the inside of the set. Use a higher value for narrower color bands, a lower value for wider ones. 1000 is a good value to start with. The second distance estimator parameter ("width factor") sets the distance from the inside of the set which is to be considered as part of the inside. This value is expressed as a percentage of a pixel width, the default is 71. You should use 1 or 2 pass mode with the distance estimator method, to avoid missing some of the thin strands made visible by it. For the highest quality, "maxiter" should also be set to a high value, say 1000 or so. You'll probably also want "inside" set to zero, to get a black interior. Enabling the distance estimator method automatically toggles to floating point mode. When you reset distest back to zero, remember to also turn off floating point mode if you want it off. Unfortunately, images using the distance estimator method can take many hours to calculate even on a fast machine with a coprocessor, especially if a high "maxiter" value is used. One way of dealing with this is to leave it turned off while you find and frame an image. Then hit to save the current image information in a parameter file (see {Parameter Save/Restore Commands}). Use an editor to change the parameter file entry, adding "distest=1", "video=something" to select a high- resolution monochrome disk-video mode, "maxiter=1000", and "inside=0". Run the parameter file entry with the <@> command when you won't be needing your machine for a while (over the weekend?) ═══ 6.3. Acrospin ═══ ACROSPIN, by David Parker - An inexpensive commercial program that reads an object definition file and creates images that can be rapidly rotated in three dimensions. The Fractint "orbitsave=yes" option creates files that this program can read for orbit-type fractals and IFS fractals. Contact: David Parker 801-966-2580 P O Box 26871 800-227-6248 Salt Lake City, UT 84126-0871 ═══ 6.4. 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. 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. ═══ 6.5. GIF Save File Format ═══ 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. Until version 14, Fractint saved images as .FRA files, which were a non-standard extension of the then-current GIF87a specification. The reason was that GIF87a did not offer a place to store the extra information needed by Fractint to implement the feature -- i.e., the parameters that let you keep zooming, etc. as if the restored file had just been created in this session. The .FRA format worked with all of the popular GIF decoders that we tested, but these were not true GIF files. For one thing, information after the GIF terminator (which is where we put the extra info) has the potential to confuse the on-line GIF viewers used on Compuserve. For another, it is the opinion of some GIF developers that the addition of this extra information violates the GIF87a spec. That's why we used the default filetype .FRA instead. Since version 14, Fractint has used a genuine .GIF format, using the GIF89a spec - an upwardly compatible extension of GIF87a, released by Compuserve on August 1 1990. This new spec allows the placement of application data within "extension blocks". In version 14 we changed our default savename extension from .FRA to .GIF. There is one significant advantage to the new GIF89a format compared to the old GIF87a-based .FRA format for Fractint purposes: the new .GIF files may be uploaded to the Compuserve graphics forums (such as Fractint's home forum, 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. Fractint can still read .FRA files generated by earlier versions. If for some reason you wish to save files in the older GIF87a format, for example because your favorite GIF decoder has not yet been upgraded to GIF89a, use the command-line parameter "GIF87a=yes". Then any saved files will use the original GIF87a format without any application-specific information. An easy way to convert an older .FRA file into true .GIF format suitable for uploading is something like this at the DOS prompt: FRACTINT MYFILE.FRA SAVENAME=MYFILE.GIF BATCH=YES Fractint will load MYFILE.FRA, save it in true .GIF format as MYFILE.GIF, and return to DOS. GIF and "Graphics Interchange Format" are trademarks of Compuserve Incorporated, an H&R Block Company. ═══ 6.6. 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*!!! WEGNER, Timothy & PETERSON, Mark: "Fractal Creations", Waite Group Press, 1991. If we tell you how *wonderful* this book is you might think we were bragging, so let's just call it: THE definitive companion to Fractint! ═══ 6.7. Palette Maps ═══ If you have a VGA, MCGA, Super-VGA, 8514/A, XGA, TARGA, or TARGA+ video adapter, you can save and restore color palettes for use with any image. To load a palette onto an existing image, use the File/Read Color Map command. To save a palette, use the File/Write Color Map command. To change the default palette for an entire run, use the command line "map=" parameter. The default filetype for color-map files is ".MAP". These color-maps are ASCII text files set up as a series of RGB triplet values (one triplet per line, encoded as the red, green, and blue [RGB] components of the color). Note that .MAP file color values are in GIF format - values go from 0 (low) to 255 (high), so for a VGA adapter they get divided by 4 before being stuffed into the VGA's Video-DAC registers (so '6' and '7' end up referring to the same color value). PMFRACT is distributed with some sample .MAP files: ALTERN.MAP the famous "Peterson-Vigneau Pseudo-Grey Scale" BLUES.MAP for rainy days, by Daniel Egnor CHROMA.MAP general purpose, chromatic DEFAULT.MAP the VGA start-up values FIRESTRM.MAP general purpose, muted fire colors GAMMA1.MAP and GAMMA2.MAP Lee Crocker's response to ALTERN.MAP GLASSES1.MAP used with 3d glasses modes GLASSES2.MAP used with 3d glasses modes GOODEGA.MAP for EGA users GREEN.MAP shaded green GREY.MAP another grey variant GRID.MAP for stereo surface grid images HEADACHE.MAP major stripes, by D. Egnor (try cycling and hitting <2>) LANDSCAP.MAP Guruka Singh Khalsa's favorite map for plasma "landscapes" NEON.MAP a flashy map, by Daniel Egnor PAINTJET.MAP high resolution mode PaintJet colors ROYAL.MAP the royal purple, by Daniel Egnor TOPO.MAP Monte Davis's contribution to full color terrain VOLCANO.MAP an explosion of lava, by Daniel Egnor COV1.MAP and COV2.MAP the continuous spectrum 256-color palettes described in the article "HAX 9 - A Better 256-Color VGA Palette", PC Techniques Vol. 1, No. 2, June/July 1990, page 8. PLASMA.MAP the Fractint for DOS 256-color palette used for the PLASMA display. ═══ 7. Settings Selections ═══ Settings are the parameters and options that determine how the program will evaluate the selected fractal formula. Settings come in the following flavors: o Extents o Parameters o Options o Image Settings o Palette Switching ═══ 7.1. Extents ═══ The Extents of the fractal are the range of the complex plane over which the fractal will be calculated. The numbers that can be entered are limited, and will be automatically adjusted if entered out of limits. This can be because of restrictions in the fractal calculation algorithms, or just because looking at a wider range lacks detail and is therefore uninteresting. If you are interested in zeroing in at a particular complex number, that value can be entered as the "Center" X and Y values. The Left, Right, Top, and Bottom values will then be automatically adjusted to make that value the center but stay in acceptable limits. When entry is complete, select OK, or press Enter. To exit with no changes, select Cancel, or press Escape. To see the default (built-in) values, press Default. To proceed with the default values, press Default, then OK. Exiting with OK will cause the fractal to be recalculated over the new extents. Extents come in the following flavors: o X Range o Y Range ═══ 7.1.1. X Range ═══ The "X Range" values are the left and right (lower and upper) decimal numbers that define the rectangle's range in the X, or real, or left-to-right range of the complex plane. The numbers that can be entered are limited, and will be automatically adjusted if entered out of limits. This can be because of restrictions in the fractal calculation algorithms, or just because looking at a wider range lacks detail and is therefore uninteresting. If you are interested in zeroing in at a particular complex number, that value can be entered as the "Center" X and Y values. The Left, Right, Top, and Bottom values will then be automatically adjusted to make that value the center but stay in acceptable limits. When entry is complete, select OK, or press Enter. To exit with no changes, select Cancel, or press Escape. To see the default (built-in) values, press Default. To proceed with the default values, press Default, then OK. Exiting with OK will cause the fractal to be recalculated over the new extents. ═══ 7.1.2. Y Range ═══ The "Y Range" values are the top and bottom (upper and lower) decimal numbers that define the rectangle's range in the Y, or imaginary, or top-to-bottom range of the complex plane. The numbers that can be entered are limited, and will be automatically adjusted if entered out of limits. This can be because of restrictions in the fractal calculation algorithms, or just because looking at a wider range lacks detail and is therefore uninteresting. If you are interested in zeroing in at a particular complex number, that value can be entered as the "Center" X and Y values. The Left, Right, Top, and Bottom values will then be automatically adjusted to make that value the center but stay in acceptable limits. When entry is complete, select OK, or press Enter. To exit with no changes, select Cancel, or press Escape. To see the default (built-in) values, press Default. To proceed with the default values, press Default, then OK. Exiting with OK will cause the fractal to be recalculated over the new extents. ═══ 7.2. Parameters ═══ The Parameters of each fractal algorithm are described by a short title, which appears above the entry field for each of the (up to) four parameters. Parameters that don't apply to a given fractal type will have no title, and the entry field will be inaccessible. For information on what the parameters mean, just select "Help" and the fractal formula with the explaination of the parameters will be displayed. When entry is complete, select OK, or press Enter. To exit with no changes, select Cancel, or press Escape. To see the default (built-in) values, press Default. To proceed with the default values, press Default, then OK. Exiting with OK will cause the fractal to be recalculated with the new parameters. ═══ 8. Operation Instructoins ═══ Copyright Copyright (C) 1992 The Stone Soup Group. FRACTINT for OS/2 2.0 may be freely copied and distributed, but may not be sold. GIF and "Graphics Interchange Format" are trademarks of Compuserve Incorporated, an H&R Block Company. Select a topic from the following list for information on using this program. What's New Introduction History of this program Distribution policy Contacting the author Fractal Types Miscellaneous topics Program Operation ═══ 8.1. How to operate FRACTINT for OS/2 2.0 ═══ FRACTINT for OS/2 2.0 operates from the menu via either a mouse or a keyboard. However, zooming and panning using either View/Pan Center or the View/Zoom In/Zoom Out works only with a mouse. To change the are of the fractal viewed using the keyboard, change the extents and/or center using Settings/Extents. Select a topic below for additional information. Main Menu Zooming and Panning File menu Edit menu View menu Settings menu Help menu ═══ 8.2. Main Menu ═══ The main controls are the fifth and sixth menu bar entries. The fifth entry switches between "Halt!" and "Freeze!". When "Halt!" is displayed, it implies that a fractal is being calculated and drawn, and if you want to stop the calculation, select "Halt!" and it will stop shortly. When "Freeze!" is displayed, a calculation is not in progress and selecting "Freeze!" will cause a calculation to not automatically start, as it would after selecting values from either the View, Settings, or File pull-downs, but will wait for an explicit "Go!" menu selection. When you get tired of consuming computer resources looking at pretty pictures, the File/Exit selection will shut down the program. ═══ 8.3. Zooming or Panning ═══ To see a smaller part of the fractal, blown up to full screen size, use selections from the View/Zoom In or View/Zoom Out menu. View changing with the mouse is Object/Action oriented, as the IBM SAA guidelines expect. Zoom In or Zoom Out select magnification or reduction by either a fixed or selectable amount around the current center of the display. Click the left mouse button to display cross-hairs. View/Pan Center then pans to this point. To Zoom to a window, click and drag the left mouse button to outline a rectangle. Then either pick View/Zoom In/to Window to zoom in, or View/Zoom Out/to Window to zoom out. ═══ 8.4. File menu ═══ The File menu allows you to select a fractal, read a .GIF, .BMP, or .PCX file into the program, write the current fractal/window out as a .GIF, .BMP, or .PCX, print the current window contents, or load or save palette files. The following selections are on the File menu. New Open Save as Print Target Printer Read Color Map Write Color Map ═══ 8.4.1. New ═══ The File/New selection allows you to select a fractal type from a dialog box. The current fractal, even if not yet complete, will be completely replaced by the new fractal. To set extents, parameters, or options for this fractal, use selections from the Settings menu. ═══ 8.4.2. Open ═══ File/Open reads in a .GIF file (saved by one of the FRACTINT family programs) to view, a variety of Bitmap (BMP) formats, or the PC Paintbrush .PCX format. A .GIF file created by other than a FRACTINT family program, or any bitmap or PCX file, will be restored as a PLASMA fractal, which is not zoomable or otherwise editable. The program can read the following types of bitmaps: o OS/2 1.x bitmaps o Windows 3.0 device independent bitmaps (DIBs) o Windows 3.0 DIBs compressed as RLE4 or RLE8 o OS/2 2.0 bitmaps o OS/2 2.0 bitmaps compressed as RLE4 or RLE8 Indicating any of the bitmap formats will allow you to read any of the indicated formats: i.e. the program will figure out what the format is from the file contents. This is because I don't see any easy way for you as the user to know what format bitmap you have just by looking at the filename (at least I can't). After indicating the type of file to read, a standard file selection dialog box will be displayed. You may then search around for the file you are interested in. File/Open is available only if a calculation is not currently in progress. ═══ 8.4.3. Save as ═══ File/Save as writes a .GIF file, any of the supported bitmap formats noted, or a PC Paintbrush .PCX file. The file format for the Bitmap file will be determined by your selection from the list of BMPs indicated (some programs will only read one of those supported). o OS/2 1.x bitmaps o Windows 3.0 device independent bitmaps (DIBs) o Windows 3.0 DIBs compressed as RLE4 or RLE8 o OS/2 2.0 bitmaps o OS/2 2.0 bitmaps compressed as RLE4 or RLE8 After indicating the type of file to write, a standard file selection dialog box will be displayed. You may then specify a file name or search around for an existing file to select and replace. File/Save as is available only if a calculation is not currently in progress. Note: The compressed forms may be use if you want to transfer the image to a program that can process compressed bitmaps, and the image contains significant areas of one color. I leave "significant" undefined, as it depends on the file as a whole. The Run Length Encoded (RLE) compression algorithms are fairly simple, and my implementations of them is even more simple-minded. The GIF format is still much better for complicated images. ═══ 8.4.4. Print ═══ File/Print prints to the selected Presentation Manger printer. The Print Dialog will display letting you set the number of copies to print. Note: If you are printing to a Presentation Manager printer that only prints black and white (well, actually, only prints black or the paper color), then you must select one of the 2-color palette settings (either "Black/White" or "White/Black") before printing to get a usable result. Printing attempts to map the displayed colors to the capabilities of the printer as best it can. Unfortunately, the rules for printing to a 2-color printer are that the "Foreground" color (usually white) will come out as black, and all the rest will come out as white (background). The result is usually not as interesting as the multi-color display. This support is here so that a color printer will attempt to map displayed colors as best it can to the colors available in the printer. Unfortunately, that then makes printing to a black and white printer a special case. File/Print is only available if a calculation is not currently in progress. For more information, see Palette switching. ═══ 8.4.5. Read Color Map ═══ File/Read Color Map reads a FRACTINT family .MAP file and makes it available as the User Palette selection in the Settings/Set Palette dialog, discussed below. For a description of some of the available files, see Palette Maps. ═══ 8.4.6. Write Color Map ═══ File/Write Color Map writes the palette values currently selected by the Settings/Set Palette dialog to a FRACTINT family .MAP file. For a description of some of the available files, see Palette Maps. ═══ 8.5. Edit menu ═══ The Edit menu allows access to the OS/2 Presentation Manager Clipboard. This allows the screen contents to be transfered to another program that can extract a Bitmap from the PM Clipboard, or to receive a bitmap generated by another program. The other program could be a paint program, ICON editor, or whatever. The menu has the following options: Copy Bmp Paste Clear ClipBoard ═══ 8.5.1. Copy Bmp ═══ Edit/Copy Bmp will place a copy of the current screen image on the PM Clipboard as a Bitmap. Color information is transmitted as part of the bitmap, but the fractal description information (that would be saved as part of a FRACTINT .GIF file) is not. This option is available only when a fractal is currently not being calculated, and will be grayed-out if it cannot be selected. ═══ 8.5.2. Paste ═══ Edit/Paste, brings a bitmap off the PM Clipboard back onto the screen. As no fractal description information is available, the display becomes a PLASMA fractal. Edit/Paste is available only if a calculation is not currently in progress, and there is actually a bitmap on the Clipboard. ═══ 8.5.3. Clear ClipBoard ═══ Edit/Clear ClipBoard causes Fractint for PM to discard any current contents of the PM Clipboard, whatever it is and from whatever source is came from. This is a useful way to save some memory if you don't what the current clipboard contents. Unless told to discard the Clipboard, Presentation Manger will hold on to the bitmap until you shut your system down. ═══ 8.6. View menu ═══ The View menu allows you to move around in the current fractal image. This is possible on most fractal types. Note: Zooming or Panning is not possible for the PLASMA fractal type. Since this is also the type used internally for an image (Bitmap or .GIF file) that is not from a Fractint family source, zooming and panning is also not possible on these images. The following selections are on the View menu. various Zoom selections Pan center ═══ 8.6.1. Zoom selections ═══ The various zoom selections provide zooming in or out to see more or less detail of the fractal. Details are given under Zooming and Panning. Variable zooming is provided by a prompting window for a zoom factor. A decimal number is allowed as a zoom factor. ═══ 8.6.2. Pan Center ═══ This selection chantges the point that is the center of the screen. Details are given under Zooming and Panning. ═══ 8.7. Settings menu ═══ Settings are the parameters and options that determine how the program will evaluate the selected fractal formula. The following selections are on the Settings menu. Swap Set Extents Set Parameters Set Options Reset Image Settings Palette switching ═══ 8.7.1. Swap ═══ The Settings/Swap to Mandel or Settings/Swap to Julia allows you to switch between related Mandelbrot and Julia sets if the specific fractal allows that. ═══ 8.7.2. Set Extents ═══ The Settings/Set Extents displays and allows modification of the X and Y extents of the complex plane (the numbers that the fractals are defined and calculated on) that the display window represents. This is a numeric display of the changes made by the View menu. More Detail ═══ 8.7.3. Set Parameters ═══ The Settings/Set Parameters displays and allows modification to various numeric parameters that each fractal calculation contains. The meaning and effects of these parameters can be determined by selecting help when the dialog box is displayed. The help window will detail the parameters for the current fractal type. More Detail ═══ 8.7.4. Set Options ═══ The Settings/Set Options selects various calculation options, such as integer or floating point math, number of passes, and calculation depth (max iterations). ═══ 8.7.5. Reset ═══ The Settings/Reset Above will, when in the Freeze state, allow the cancelation of changes made by one of the above choices, restoring the "current" calculation options from what is currently displayed. ═══ 8.7.6. Image Settings ═══ The Settings/Set Image Settings changes the number and color depth of the pixels being calculated. The colors can be 2 (black and white), 16 color, or 256 color. The pixel dimensions largely affect the resolution of a future saved or printed image, as whatever is being calculated will be compressed or expanded as needed to fit in the display window. Note: The fractal calculation engine has finally recieved a reprieve from the limits of the 64K segments of DOS and WINDOWS. As such, the fractal image can now be 4096 by 4096 pixels. Warning: Don't over do this new-found freedom. Do a little reality check first. The fact is PMFRACT needs a pixel memory array of the size you ask, and OS/2 NEEDS ONE, TOO. A little calculation: A 2-color image needs 1 bit per pixel, a 16-color image needs a half byte (4 bits), and a 256-color image needs 1 full byte (8 bits). An image 4096 by 4096 by 256 colors needs 4096x4096/1 = 16,777,216 bytes or 16 megabytes -- TIMES 2; or 32 MEGABYTES of memory. OS/2 2.0 can handle it. Can your system? Remember, what doesn't fit in your real memory comes out of your hard disk's SWAPPER.DAT file. Do you have that much space free to begin with? Can the drive live long enough to handle all the swapping? A test (well, really a finger check) during development revealed that you can easily end up with a situation that could only be described as disk-video at its worst. (And no, OS/2 2.0 Did Not Crash.) ═══ 8.7.7. Palette switching ═══ The Settings/Set Palette selects a dialog giving various palette options. The Black and White, 16-color VGA, 256-color VGA, and Physical palette are fixed by the program or your hardware. The User Loaded Palette selection will be enabled when sucessfully loaded by an external palette, such as reading a Color Map via the File/Read Color Map menu selection, or by the palette contained in a loaded .GIF file or bitmap. The "Black/White" and "White/Black" selections are available to allow you to set up for printing on a 2-color printer and view the result before pringing. For more information on printing, see Print. ═══ 8.8. Help ═══ At any time additional Help can be displayed. Help for Help describes how to use the help system. General Help goes to a panel giving access to topics from an overview of fractals, details of the fractal types and formulas implemented in this program, and details on the operation of this program. Keys Help details the single keystroke "Hot Keys" available. Help Index will display all topics included in this help system. It is also available by selecting the "Index" button at the bottom of this help system window. About displays copyright, author, and release information for the program. ═══ 8.9. Print ═══ Here you see confirmation of the printer you are about to print to and the name of the fractal you are about to print. If desired, set a number of copies in the "Copies" box, then select "Print" to start the print process. The screen will blank as the print process occurs in the background. If desired, you may switch to another application while the print is occuring. If you change your mind, select "Cancel". If you don't want to print to the indicated printer, select "Cancel" then change the printer from the "File/Targe Printer" menu. For more information, see: o File/Print for considerations of the printing process. o File/Target Printer for details of selecting a printer. ═══ 8.10. Select a File to Open ═══ Based on the file type previously selected, you see here all files of that type in the current directory. Select one, or select another drive or directory from the appropriate drop-down list, or type in a full path and file name in the entry box at the top (if you need the typing practice). For more information, select from below: o File/Open. o GIF Save File Format. ═══ 8.11. Save to What File? ═══ Based on the file type previously selected, you see here all files of that type in the current directory. Select one, or select another drive or directory from the appropriate drop-down list, or type in a partial or full file name in the entry box at the top. Warning: You will not be warned if you are about to overwrite an existing file. For more information, select from below: o File/Save As. o GIF Save File Format. ═══ 8.12. Specify the Color Map File ═══ You see here all files of type .MAP in the current directory. Select one to read or overwrite, or select another drive or directory from the appropriate drop-down list, or type in a partial or full file name in the entry box at the top. For more information, select from below: o Read Color Map. o Write Color Map. o Palette Maps ═══ 8.13. Select an IFS File ═══ You see here all files of type .IFS in the current directory. Select one to read, or select another drive or directory from the appropriate drop-down list, or type in a partial or full file name in the entry box at the top. For more information, select from below: o Barnsley IFS Fractals ═══ 8.14. Select a Formula File ═══ You see here all files of type .FRM in the current directory. Select one to read, or select another drive or directory from the appropriate drop-down list, or type in a partial or full file name in the entry box at the top. For more information, select from below: o Formula ═══ 8.15. Select an L-System File ═══ You see here all files of type .L in the current directory. Select one to read, or select another drive or directory from the appropriate drop-down list, or type in a partial or full file name in the entry box at the top. For more information, select from below: o L-Systems ═══ 8.16. Target Printer Selection. ═══ Select one of the printers displayed in the list box. The one initially highlighted is the last one you selected, or the OS/2 default you defined if you have not set your printer up before. The selected printer will then be the one that a subsequent "File/Print" action will print to. Initially, the attributes for the chosen printer (such as orientation, form, draft or final quality, etc.) will be the default attributes set in the printer setup. To change these attributes, select the printer and then press the "Job Properties..." button. A dialog box will be displayed which will allow you to see and change the attributes that are applicable to that printer. When you are satisfied, press "OK" from there, then "OK" on the "Printing Options" dialog box. Note: Printer selections, and printer attributes, are not saved between executions of this program. If you desire to use a non-default printer, you must select it each time you run this program. If you want non-default attributes, you must set them each time you run this program. ═══ 9. Keys Help ═══ The following keys have special meaning. To see a detailed explaination of the action, just tab to highlighted words and press enter, or click with the mouse. F3 Alias for File/Exit to terminate the program. F2 Alias for File/New. Alt-F2 Alias for File/Open. F4 Alias for File/Print. Esc Cancels out of Pan or Zoom mode. The following keys are a subset of the keys used by FRACTINT for Dos. Tab Alias for Settings/Extents. Alias for File/New. Alias for Settings/Set Options. Another alias for Settings/Set Options. Alias for Settings/Set Parameters. Alias for File/Save As. Alias for File/Open.

Alias for File/Print.