Appendices

Chaos, Fractals, Dimension
mathematics in the age of the computer
©1995-1998 Glenn Elert
All Rights Reserved--Fair Use Encouraged

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chaos, ifs, l-systems, mandelbrot-julia, music, newton, terrain, miscellaneous, after-dark


A.2 Annotated Bibliography of Software Resources

There are thousands of software resources on chaos, fractals, and dimension. I have listed a small sample of the programs available for Mac OS computers (recommended programs are highlighted in yellow). Some of these were used to create the graphics that were included in this paper. Others were used for inspiration or entertainment. All programs were tested on a Power Mac (300 MHz 603e) running MacOS 8. Appendix 3 also contains links to software archives if you're looking for additional resources.

Chaos (Dynamical Systems)

icon 1-D Chaos Explorer. Matthew Hall. 1992.
Files Associated with 1-D-ChaosExplorer
Bifurcation diagrams, web diagrams, time series, etc. Program your own functions for exploration. Used extensively in Chapter 1.2 and Chapter 1.3. This is a good program for those who want to understand the basic behavior of an iterated system.
icon Bifurcation. Ronald T. Kneusel. 1995.
Bifurcation: Period Doubling Route to Chaos
Faster than 1-D Chaos Explorer, but does half as much. Bifurcation, and time-series only.
icon Bouncing Ball. T. Abbott, N. B. Tufillaro, J. P. Reilly. 1987-1993.
Directory of /pub/time-series/mac
Do you find the quadratic map to be too abstract and the logistic function an oversimplification? All the aspects of the above mentioned functions (steady state, periodic, and chaotic phase space orbits) can be illustrated by a ball bouncing on a vertically oscillating table. It's a great physical application, but I find the program hard to work with and slow. There are too many windows and too many options. It will take some effort to master all of them, so beware.
application icon ChaosPlot. Jason Regier. 1994.
Files Associated with ChaosPlot
Orbit diagrams for continuous systems. Plots the behavior of a damped, driven, anharmonic oscillator. Generates a chaotic path reminiscent of the shadow of a fly on a wall.
application icon Cliff. John B. Matthews-Gem City Software. 1992.
Files associated with Cliff's World
Cliff is a Macintosh application that iterates Dr. Cliff Pickover's dynamical system and plots the resulting co-ordinate pairs.
icon Cycle Explorer. James C. Burgess. n.d.
Source Unknown.
Click on the bifurcation diagram and draw the corresponding web diagram. Move the parabola around on the web diagram and see the corresponding location on the bifurcation diagram. Very limited with a wise guy interface. Pulling down the options menu gets "No Options" as a reply.
HyperCard icon HyperCard Chaos. A.J. Roberts-University of Adelaide. 1991.
Source Unknown.
The most technical of the HyperCard stacks. A tutorial in dynamical systems with applications and interactive demos. Serious stuff, but doesn't always display well. The interactive demos are a bit erratic also. I loved many of them, but a few were complete mysteries.
HyperCard icon HyperKaos. Fabian Lidman. n.d.
Software by Fabian Lidman
Simple HyperCard stack for iterating one-dimensional functions. Output is a column of numbers. Faster than a programmable calculator, plus you can enter your own functions.
icon Intelligent Chaos. Fabian Lidman. 1998.
Software by Fabian Lidman
Application for iterating the one-dimensional logistic function. Output is a column of numbers or a time-series graph. Faster than a programmable calculator. I wish I had this when I was doing my research for Chapter 1.1. A version for iterating complex numbers is also available.
icon Orbit. Dr. Stephen Eubank (Buff Miner, Jim Wiley, Toshi Tajima)-Department of Physics, University of Texas, Austin). 1986.
Source Unknown.
Note the copyright date. An antique lost in cyberspace. Investigate a variety of well-known 1-dimensional oscillators and 2-dimensional discrete maps. Some of the strange attractors in Chapter 2.1 were drawn on a 68k Mac with this program. Requires a lot of pampering to get it to work on a PPC machine. The primary author has a home page, but it doesn't mention Orbit.
icon Quadratic Map. T. Abbott, N. B. Tufillaro, J. P. Reilly. 1987-1993.
Directory of /pub/time-series/mac
Draws time-series, bifurcation diagrams (in color), and phase space diagrams (not very useful as far as I'm concerned). Users can't enter their own functions. There are menu items for adding sound, but they don't do anything on my machine.
icon Std Map. James D. Meiss, University of Colorado. 1994.
Programs of James Meiss
Standard map orbits were drawn using this program. Has other features for exploring two-dimensional iterated maps. A nice piece of code.

Iterated Function Systems (IFS)

icon Fract. Bob Wiseman-Wiseman Software. 1989-90.
fract-1-0-cpt.hqx
A program for drawing IFS fractals. Comes with several parameter files.
icon Fractal Attraction. Kevin D. Lee & Yosef Cohen-Sandpiper Software. 1991.
Source Unknown.
A reader was kind enough to email me a copy. Draws IFS fractals from a template of polygons. This program was used to create IFS Fractal Movie I which is available on the Web (see below) even though Fractal Attraction is not.
icon Fractal Lab Kit. Ronald T. Kneusel. 1994
Fractal Lab Kit: Draw Fractals Using IFS
A command-line-driven IFS program for the Mac. Why bother programming for the Mac if you're not going to make use of the Mac interface? I hate this program. The full package also includes one program for Julia set exploration and another for the Mandelbrot set. I can only get them to draw one image and then they crash.
icon IFS. Paul Bourke. 1989.
Software
So far, this is the easiest IFS program to use. I still find that the images generated look nothing like what I predict, however. Uses two different methods: hopalong (which makes images materialize out of a haze of points) and polygon (which is a bit more intuitive).
icon IFS. Stephen Scandalis. 1990.
Files associated with IFS by Scandalis
Another obscure little program. It draws what its name suggests. Nothing special.
icon IFS Fractal Movie I. Kevin D. Lee-Sandpiper Software. 1991.
ifs-fractal-movie.hqx
Whatever happened to IFS Fractal Movies II and III? A stand alone program that does nothing other than play a short movie showing an evolving IFS called "The Claw". The midpoint of the movie is a Sierpinski triangle. Created with some long lost program called "Fractal Attraction" (see above). Ah the bad old days: before QuickTime, before MPEG, before the Web.
icon Kaos. Reinoud Lamberts. 1990.
kaos-004r.hqx
Kaos produces IFS images that are quite unique. Painfully slow! After running half an hour on a 300 MHz 603e PPC, I had a smudge that occupied one-third of the screen. An interesting looking smudge, but it wasn't worth the wait. Kaos requires an FPU. The author has a home page, but it doesn't mention the program.
icon Sierpinski-Triangeln. Martin Wiss. 1997.
sierpinski-triangeln-1-swe.hqx
A minimal little program that draws the Sierpinski Triangle. You can play around with the parameters, but that's it. Written in Norwegian, but simple enough for Americans to figure out.

Lindenmayer Systems

icon Fractal Trees. Simon Woodside. 1997.
Download, Abstract
A really bad program. Draws fractal trees using an IFS, but has no interface. None! Its parameters can only be modified in the source code. My advice, don't waste your time.
icon L-Systems. Paul Bourke. 1991.
Software
Draws recursively defined fractals like the type shown in Chapter 3.2 (Peano monster, Koch coastline, Sierpinski gasket, etc.). The L-System Manual is worth reading for its own sake. Also available in a 3D version that crashes every time I launch it.
icon LSystems. Bryan Horling. 1996.
Senior Research: Lindenmayer-Systems
Renders fractals from simple recursion instructions. Colorful and visually appealing. The author has also written a nice, compact paper on branching systems in nature (plants, corals, etc.).
application icon PFG (Plant and Fractal Generator). Przemyslaw Prusinkiewicz. 1988-92.
Directory of /pub/projects/pfg
Draws fractal images and plant-like branching structures using L-systems with "turtle interpretation". Fast, but difficult to work with. No options. No interface for producing your own parameter sets (must be coded by hand using a text editor).

Mandelbrot & Julia Sets

ab Fractal. Eden Software (L. Pieniazek). 1994.
ab-fractal-1.2.sit.hqx
I don't have time for this. It draws the Mandelbrot set, Julia sets, and maybe some other sets, but unbelievably slow. I was insulted that anyone would ask a shareware fee for anything so bad as ab Fractal. Requires an FPU.
icon CFG (ColorFractalGenerator). John Schlack. 1990-95.
Shareware: Color Fractal Generator
Mandelbrot sets, Julia sets, random walk, etc. A basic rendering program for drawing color fractals. Unregistered versions aren't PPC-native and run slowly.
icon EasyFractPPC. Alessandro Guzzini. 1997.
easyfract-ppc-1.1.sit.hqx
Easy, yes, but there are better programs out there. Zoom into the Mandelbrot set, play with the colors, but that's all.
icon Floating Fractals. Adam Smith. 1994.
Files Associated with Floating Fractals
Zoom in and explore 11 different fractals (half of them are variations on the Mandelbrot set). Fast and easy to use.
icon Fractal Artist. Alexei Lebedev-Phronesis Software. 1992.
<http://ring.nacsis.ac.jp/archives/misc/NetNews/comp.binaries.mac/volume93/Jul/930710.06>
Another basic Mandelbrot-Julia set explorer.
icon Fractal Domains (formerly FracPPC). Dennis C. De Mars. 1994-97
Fractal Domains Home Page
Explore the Mandelbrot set. Switch to the accompanying Julia set. Rather plain interface with lots of control panels floating around. You only get one window size. Not my favorite, but it appears to be quite popular.
icon Fractal Observatory. Marcio Luis Teixeira-Trilobyte Software. 1990.
MarcioT's Software Archive
Another program that doesn't use the Mac interface. Halftone grays meant to be viewed on a black and white monitor. Looks and behaves like SuperMandelZoom.
icon Fractals. Keiron Liddle. 1997.
Fractals Home Page
Draws 30 different fractals. Has a ton of different options to play with. I haven't had the energy to fully explore this monster.
icon Intelligent Mandelbrot. Fabian Lidman. 1998.
Software by Fabian Lidman
A unique application for iterating the complex quadratic map. Dig around in the dirt of the Mandelbrot set. Explore the actual numbers involved in generating the pictures produced by other programs. Produces two columns of numbers as output. Faster than a programmable calculator.
Java Applet icon JavaQuat. Garr Lystad. 1997.
Lystad Family Home Page
A Java applet for exploring Mandelbrot sets, Julia sets, and the sets in between. Uses quaternions: complex-complex numbers of four real parameters! Stretch your visualization muscles as you view two-dimensional slices through the four-dimensional mother of all sets. Not as smooth running as a well-written, stand-alone application would be. Oversized interface looks like it was designed on a 21" monitor.
icon Julia! Julie Mitchell. 1993.
Files associated with Julia!
Another Julia-Mandelbrot explorer. Not fast or sophisticated. In fact, it's downright slow. Interlaced graphics and minimal options. There are better programs available. The author has a home page, but it doesn't mention Julia!
icon Julia 'O Matic. Jim Burgess. n.d.
Julia 'O Matic: a fractal generator
Tiny windows. No documentation. No explanation. Sort of fast, but who cares? Another minimal Julia-Mandelbrot explorer.
icon Julia's Dream. Reinoud Lamberts. 1991.
Files associated with Julia's Dream
Many of the Julia sets in this paper were drawn using this program. Generates real-time images of Julia sets as you roam around the complex plane with your cursor. The author has a home page, but it doesn't mention the program. (Julia's Dream is also the name of a pizza: basil pesto, ricotta and mozzarella, topped with garlic, broccoli, and spinach. $9.99 for a 12" pie. 804-978-7898.)
icon Julia's Nightmare. Ben Davenport. 1995.
Files associated with Julia's Nightmare
A full-color sequel to Julia's Dream. The author has a home page, but it doesn't mention the program.
icon MandelAcid. Derek Greenberg-The Bone Factory, 1993.
Files associated with MandelAcid
"A video drug for the Macintosh." Eye candy, pure and simple. Way cool, but useless for those interested in mathematics. No theory behind what you're looking at. (Peculiar sideline. Read "Virtual VIKKI's Twilight Gallery". MandelAcid used to clothe a computer-generated animal in tie dye thus rendering it safe for viewing by children. You can't get much weirder than this.)
icon MandelBrain. Danny Brewer. 1994.
MandelBrain
Explore the Mandelbrot set. Comes with more preset color palettes than any other program.
HyperCard icon Mandelbrot Fractal Companion. Evan T. Yeager. 1993.
mandelfractalcomp.cpt.hqx
A brief HyperCard tutorial of complex numbers, fractals, the Mandelbrot set, and how to construct a computer program. In living color (something quite rare in HyperCard).
icon MandelBrowser, Mandella, MicroMandella. Jesse Jones. 1990-95.
Files associated with MandelBrowser, A Plug for MandelBrowser
Mandelbrot sets, Julia sets, Newton's method, and a few others. Over 50 different sets and numerous color schemes. Fast, PPC-native code. Mandella is an older 68k version that also draws strange attractors (the Ikeda attractor in Chapter 2.1, for example). MicroMandella is a stripped-down version of the 68k program.
icon MandelMovie. Michael Larsen-Dynamic Software. 1990-95.
Dynamic Software Products: Software
Mandelbrot-Julia hybrid and Julia cascade movies in section 2.3 were rendered in 1992 at the University of Wisconsin-Milwaukee, School of Education using this program. Haven't used it since, so I can't comment on it. Dynamic Software offers other chaos and fractal programs.
icon MandelScope, MandelMaker. Anthony S. Ku. 1993.
Mandelscope Maker
Explore the Mandelbrot set with MandelScope and then draw it using MandelMaker. Why not combine these features into one program? MandelScope has an unusual interface that might have seemed clever at one time, but it looks ridiculous now compared to the hot rod programs out there now. MandelMaker uses a divide and conquer algorithm that should speed things up, but it still runs slowly. Doesn't use the symmetry of the Mandelbrot set across the real axis to save time either. Both programs require an FPU.
desk accessory icon MandelTV. Ed Ludwig & Ken Abbott-Abbott Systems, Inc. 1990.
mandeltv1.0.cpt.hqx
A relic from the days of System 6. MandelTV is a desk accessory for exploring the Mandelbrot set. Requires an FPU.
icon MandelZot. David Platt-Think Technologies. 1988­1993.
Directory of /~dplatt/mac
Most of the Mandelbrot sets and some of the Julia sets in this paper were drawn using this program. I have been using this it since 1991 so I'm used to it. Apparently, a commercial version called FractalMagic is also available from Sintar Software, but I can't confirm this. Comes with extensions for drawing Markus-Lyapunov fractals that I can't get to work.
icon Object Mandelbrot. Bryan Prusha. 1996.
My Originals
A personal favorite. Zoom into the Mandelbrot set, draw the corresponding Julia set. Fantastically fast, with dynamic zooming. Hold down the mouse and zoom away. Keep zooming until you reach the limits of your computer's numeric resolution.
icon PowerXplorer. Allesandro Levi Montalcini. 1996.
ALM Software
Bare bones simple. No interface, no documentation, no options. Draw a box around the region you want to explore and let go. PowerExplorer zooms you in. Intended primarily for testing computing speed. Has a bug that replaces the menu bar with a blank region in full screen mode.
icon Super MandelZoom. Robert P. Manufo. 1988.
Super Mandelzoom
Another antique still floating around. Meant to be viewed on a Mac SE or thereabouts. Uses halftones instead of grays.

Music

icon BifurcationOscillator, Logistic Synth. Jae Ho Chang. 1996.
Chaotic Sound Synthesis Projects
Orbit diagram and chaotic music generator. Watch the evolution of an orbit and then transform it into a music-like series of notes saved in AIFF format. Logistic Synth is an equivalent program for generating sounds in real time.
icon BirdSong Engine. David Benz. 1993.
fractal-birdsongs-1.2.sit.hqx
Clever HyperCard stack that converts L-System-type fractals into bird songs.
HyperCard icon Chaos Theory. Mike Atanasio-Wild Card Software. n.d.
chaos-theory.sit.hqx
HyperCard stack. Listen to the time-series of the logistic equation. A one card stack. No date, but it appears that it was created in 1990.
HyperCard icon MandelMusic. Roger R. Espinosa, Donna Iadipaolo, Jim Brunberg. n.d.
Mandelbrot Music for Macintosh
HyperCard stack. They call it a chaotic music generator, but I can't confirm this. No mathematical description of what it actually does. Your choice of 12 different instruments.

Newton's Method

icon NewtonFractal. Stefan Messmer. 1991.
NewtonFractal
Painfully slow. Stay away from this program if you value your time. I didn't fully evaluate this problem as I got tired of waiting for it.
icon Object Newton. Bryan Prusha. 1996.
My Originals
Object Newton allows you to explore Julia sets created using Newton's root-approximation method. Fast interface like Object Mandelbrot.

Terrain & Landscape Modeling

icon FracHill, FracBlob. Paul Bourke. 1991.
Software
Wire frame fractal terrains. Slow, requires an FPU, and doesn't look very good. I'm not impressed. FracHill models on a plane while FracBlob models on to a sphere.
icon Fractal! Ed Rotberg-Gonzo Systems. 1992-93.
Fractal-1.4-ppc.sit.hqx
Draws interesting snowcapped mountains with blue lakes in the foreground. Play with lighting and color schemes. Use it to render a desktop background if you wish.
Fractal Contours. Jim Cathey. 1985.
fractalcontours1.0.sit.hqx
Crash Warning. Gives "Error 2" when launched, even with PowerFPU. Using ResEdit, I found out it's a fractal terrain modeler of some sort based on algorithms developed by LucasFilm, Ltd. Looking at the icon, I'd say someone is playing a joke on us. Tell me what you think it looks like.
icon Fractal Islands. Scott Berfield-Parity Productions. November 1985.
fractisle.hqx
The oldest piece of shareware on the Net. Guess what? It crashes after drawing one wireframe pseudo-fractal surface.
application icon Matt's Fractal. Matson Dawson. 1995.
Files associated with Matt\'s Fract [I assume the backslash is a typo.]
Fly around over a fractal-generated terrain. Not detailed or realistic in any way. Plenty fun, though.
icon Venus, New Venus. Oleg. 1997.
Venus
Someone spent a lot of time working on this, I'm sure, but I find it amateurish and stupid. The ReadMe describes it as "Virtual circling around through the clouds." Looks like a bunch of nothing to me. New Venus is an older version with controls.

Miscellaneous

icon Aros Fractals. Aros Magic Research. 1996.
Aros Fractals
Mandelbrot set, Newton's method, and a third fractal type I've never seen before that looks something like a moire pattern. Zoom in, but not out. Animated colors. Unusual interface with limited options.
icon Carpet. Kevin Whitley-Think Technologies. 1987-88.
mycarpet2-0-sit.hqx
A simple program for stamping carpet or gasket fractals. Ancient, but it still runs.
? Cubic Oscillator Explorer. Bruno Degazio. 1995.
Cubic Oscillator Explorer
Can't download it for review. "Communications exception (-244)." Would someone out there send me a copy?
icon EscapeFractals. Yves Meynard. 1996.
Download, Abstract
This program draws fractals using an escape-time method developed by Clifford Pickover. Visually appealing and fast.
icon Fractal Dimension Calculator. Paul Bourke. 1993.
Software
Hausdorff-Besicovitch dimension calculations using the box-counting method. Can be slow. Data points in Chapter 3.3 were calculated with this program. The author has also published an instruction manual that's worth reading for its own sake. Also available in a 3D version that I haven't tested.
icon Fractal Wizard. Thomas Okken. 1992
fractal-wizard-15.hqx
Mandelbrot sets, Julia sets, Newton's method, IFS, and a few others. Quite a range of different sets, but it's basically another slow antique. Has a nice pull down menus briefly explaining the mathematics. Requires an FPU.
application icon Fraxious. Alan Smith. 1995.
Files associated with Fraxious
Mandelbrot and Julia sets, Newton's method, Henon and other Strange attractors, Brownian motion, and Lyapunov fractals. Crashes whenever I try to draw a strange attractor. The ReadMe admits it's a buggy program. Not PPC optimized, so it's slow.
application icon Iterative Functions. Harold Brochmann. 1992.
Source Unknown.
An odd suite of six programs: diffusion limited aggregation, real orbits, complex orbits, the Mandelbrot set, logistic equation, and population dynamics. Lousy interface that makes no use of Mac features. A time capsule from the bad old days of programming. Great in terms of content, however. Someone should rewrite this suite and wrap it in an up-to-date interface.
icon Lyapunov. Andrea Pellizzon. 1994
Files associated with Lyapunov
Lyapunov diagrams in this paper were drawn using this program as was the Lyapunov zoom movie in section 4.3 (included in the package). This is the only program I know dedicated to drawing this kind of fractal.
icon Persian Rug. Paul Cantrell. 1996.
persian-rug-10.hqx
Persian Rug is a control panel that generates random fractal patterns for your desktop. Don't like your current desktop? Open the control panel and click the "change" button. A really great idea. Unfortunately, there's a bug in it that causes my machine to hang.
application icon XaoS Fractal Viewer. Tapio Vocadlo. 1996.
XaoS: Real-Time Fractal Zoomer
Mandelbrot sets of powers 2-6. Zoom in dynamically, then switch to the Julia set at the same level of magnification and the same point. Nice for showing the quasi-self-similarity between the two sets. Other fractals included: "octal", Newton's method, "Barnsley", and "Phoenix". Comes with a really neat option that remaps the complex plane. Very fast, optimized code that doesn't try to redraw the entire window. Intentionally ugly interface takes some getting used to, however.

After Dark Modules

icon Clouds. Robert Geisler. 1993-95.
clouds2.01.sit.hqx
This module draws fractal clouds that drift across your desktop. Nicely done.
After Dark icon IFS Dream. Justin Sands. 1994.
ifsdream1.0.cpt.hqx
Draws animated IFS fractals from evolving parameter sets.
After Dark icon Fractal Forest. Scott Armitage-Berkeley Systems Inc. 1992.
ad-modfractal-forest.sit.hqx Where's it gone? You tell me.
Draws simple, cartoon-like fractal trees on your desktop.
After Dark icon Fractals. Alessandro Levi Montalcini. 1992.
Source Unknown.
This module draws regions of the Mandelbrot set which are then saved as pict files. An interesting idea, but the old source code makes this module run slowly on PPC Macs. Part of Fractals Bundle 1.33: a suite of programs designed to run on 68k Macs with a math coprocessor. Comes with a program for viewing the picts and a mover based on the old Font/DA mover. Both of these applications are of little value in the age of OS 8.
After Dark icon Newton Map, Newton-EXP. Huangxin Wang. 1994.
newton-map-ad.hqx
Far too slow. Each requires an FPU. I assume they draw Julia sets using Newton's method, but I couldn't stand the wait.
After Dark icon Ocean Child. Paul Cantrell. 1994.
ocean-child-101-ad-ppc.hqx
Perpetual, random Julia cascade with options for different surreal effects. An excellent piece of eye candy.
After Dark icon Planetmaker. Adam Miller. 1992.
planet-maker-10.hqx
Draws green on blue patterns that look something like continents and oceans. It's a real stretch to call it a "fractal planet maker," however. Very slow.
After Dark icon Sierpinski Gasket. WinterBright Software (David Thompson). 1992-93.
sierpinskis-gasket-11.hqx
Crash Warning. Gives "Error 10" when launched, even with PowerFPU.
After Dark icon Terrain Maker. Jakub Buchowski & Adam Miller. 1991.
terrain-maker-20.hqx
Fractal terrain modeler. Old-fashioned looking color scheme and resolution.


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