# A. Appendices

## A.2 Software Resources

I amassed a large collection of software available for Mac OS computers from 1985~2005. Most of these are shareware or freeware. Some were used to create the graphics included in this book. Others were used for inspiration or entertainment. Files archived on this server were compressed using StuffIt 5.5.

## OS X

dynamical‑systems, fractal‑dimension, ifs, l‑systems, mandelbrot‑julia, music, newton, terrain, miscellaneous, screen‑savers

### Dynamical Systems

**1D Maps**

James D. Meiss, 2001

James D. Meiss

Seven different folded mappings (logistic, cosine, cubic, exponential, tangent, tent, and circle map). Five different ways to analyze them (cobweb, bifurcation, lyapunov exponent, invariant density, and exit time plot). Interesting in a limited sort of way. Also available in a version for the Classic OS.

**ABC map**

James D. Meiss, 2003

James D. Meiss

A 3D map editor. An example of what I would call an "unripened" application. As the author himself warns, "This is NOT a full blown nice Mac application… and using this will possibly crash the program if you type in the wrong numbers. USE AT YOUR OWN RISK!" I concur.

**Chaos Viewer**

Lightchaos, 2002

members4.cool.ne.jp/~lightchaos/ [last url]

Staz Software [last referrer]

Disappeared before I could download it. Can someone out there send me a copy?

**Chaotic Flows**

John Lindner, Bryan Prusha, Josh Bozeday, 2001

CoWCPP

Explore the Chua, Duffing, Lorenz, and Rössler attractors. Zoom in and out, rotate, and play with the parameters. Also available in a version for the Classic OS. One of many scientific applications for the Macintosh from the very interesting College of Wooster Computational Physics Project. Other applications at this site related to chaos and fractals include:

**C-Ball**: Interactively explore the chaotic dynamics of a ball bouncing between two circles.**Chaotic Light Scattering**: Infinite hall of mirrors effect involving 4 reflecting globes stacked like cannon balls.**DLA**: Simulate diffusion limited aggregation.**Duffing**: Interactively explore the phase space flow of the famous Duffing nonlinear oscillator.**Jerk Chaos**: Phase space attractors of jerk (j = da/dt) differential equations.

**Gerry's Attraction**(formerly Gerry's Lorenz Attractor)

Gerry Beggs, 2004

Gerry Beggs

Start with a group of points on the xy plane, run them through the right equations, and watch them congeal on the Lorenz Attractor or the less interesting Rössler Attractor. Also comes with an option that allows you to fly around the attractors while they evolve. An inspired way to view them. Much better than fixed diagrams. Carbonized to run in both the Classic and OS X environments.

**Populus**

Don Alstad, 2003

Don Alstad

A comprehensive tutorial on population and evolutionary dynamics with something like 20 different models to play with. The help files are more interesting than the program itself, however, which hesitates in the most annoying manner. Still has a lot of potential as a great teaching tool for undergraduate students in this field. Also available in a version for Windows, Linux, Unix, DOS and Java. (The OS X version is actually just a "packaged" Java application.)

**StageGrow**

Ian G. Gillespie, 2001

Ian Gillespie

A program for exploring population dynamics in a single species. Very technical. Designed for people who know what it's all about (which excludes me). Staged, geometric, exponential, and logistic models are supported.

**StdMap**

James D. Meiss, 2002

James D. Meiss

Standard map orbits in this book were drawn using the Classic version of this program. Has other features for exploring nine different area preserving maps. A nice piece of software with good documentation. Fast and fun to play with, too.

### Lindenmayer Systems

**Fractal Grower**

Joel Castellanos, 2007

Joel Castellanos

Java software for growing L-systems fractals. Being Java means it works on multiple platforms, including the browser you're using to view this web page. Make your own system or use one of the many presets (paper folding, bush, carpet, dragon, fern, big-h, twig, weed, koch snowflake, sierpinski triangle, etc.).

**FractalTrees X**(also known as FTX)

Simon Woodside, 2002

Simon Woodside

Draws simple stick figure, fractal trees. Comes with an easy to operate set of sliders for changing parameter values. Orders of magnitude better than the original Classic application. The author is also working on a screen saver version called FTXSaver.

**iFractales**

Didier Straus, 2005

Software-DS

Draws 9 different fractal figures including the Koch snowflake, Sierpinski triangle, and Peano and Hilbert curves.

**VonKoch**

Julien Salort, 2003

Julien Salort

A simple, little program for drawing the Koch snowflake and its variants.

### Mandelbrot & Julia Sets

**AltiVec Fractal Carbon**

Dauger Research, 2001-2004

Dauger Research

A program for testing the computing power of a distributed computing network. Generates images of the standard and quartic Mandelbrot sets, then tells you the speed of your parallel processor network in megaflops. If you only have one computer, then it tells you the speed of just that computer. Allows you to zoom in and that's about all. Comes with very few options as it is primarily a benchmarking tool. Also available as a Classic application with no options — zero. It runs and it's done.

**Carbon Fractal**

Harold Cooper, 2001

physoft [last url]

carbon-fractal.sit

It appears as if it just draws Mandelbrot and Julia sets, but the website claims it has additional features. It doesn't really matter as it is too slow to be much fun. Simple programs like this one should run fast and be intuitively obvious to operate. This one is neither.

**Cheap Mandel**

Conor McCarthy, 2002

Conor McCarthy

Examine the Mandelbrot set. A beginner's attempt at programming for the Mac. The original Classic OS version was not much better.

**Complex Explorer**

Thomas Hallock, 2002

Thomas Hallock

A very simple program for exploring the Mandelbrot set. Comes with a unique color palette editor that I really like. Allows the user to save coordinates, but not images. A version for the Classic OS apparently exists, but I've never tested it.

**Dynamics**

Richard Koch, Pierre d'Herbemont, 2001

Richard Koch

A very basic fractal set explorer. Zoom and pan around a colored Mandelbrot set in the main window. View the associated black and white Julia set in a side window. A bit slow, given all the speed claims for OS X native applications.

**EasyFractal**

Berkhan Software, 2004

Berkhan Software

Zoom into the Mandelbrot set, check out the corresponding Julia set, play with the colors, try a different function. Comes with a collection of preset parameter files, some of which are interesting and rather unusual. Available in German and English versions for both the Classic OS and OS X.

**EDFractal**

Edward Baskerville, ED Development, 2001

ED Development [last url]

edfractal.sit

The first chaos/fractal application written for OS X that I found. Explore the Mandelbrot set. Create Julia sets and explore them, too. A very straightforward example of programming.

**FFFF - Fast Floating Fractal Fun**

Daniele Paccaloni, 2005

Source Forge Developer Profile

The crude beginings of an application that draws the Mandelbrot set, allows you to zoom in (but not out), rotate the color scheme and a bit more. Runs in terminal mode on OS X. Very speedy. Also available in a Windows version.

**Fractal Domains**(formerly FracPPC)

Dennis C. De Mars, 1994-2004

Fractal Domains

Explore the Mandelbrot set. Switch to the accompanying Julia set. Zooming in and out is rather tiresome, but otherwise this is a fine application. The switch from the Classic version to OS X has brought about a much improved look and feel.

**Fractale de Julia, Fractale de Mandelbrojt**(a.k.a. Fracatle Julia, Fracatle Mandel)

Didier Straus, 2002

Software-DS

Student programming projects that don't work. Draws a filled Julia or Mandelbrot set, then quits (or never completes the startup process). These application should not have been made public. They are effectively locked in an alpha phase of arrested development.

**Gaston**

Leo Fink, 2003

gaston.sit

A nice little program that renders three dimensional cross sections of quaternion Julia sets. A quaternion is a type of complex number with four parts (r, i, j, and k) rather than two (real and imaginary). Adjust all four parts of the quaternion constant with sliders. Try different escape radii until you find an interesting looking surface and export it as a TIFF. Also has sliders for adjusting the precision and resolution. Given the complexity of quaternion mathematics, you will want to keep the resolution low during casual exploration. High quality images are rendered very slowly. My only complaint is that the sliders are somewhat crude and hard to adjust finely. There should be some method for enetring the constant values directly.

**Mandel Blot**

Nixanz, 2002

nixanz.com

Amateur night at the Mac. Draw and explore the Mandelbrot set. Drag to zoom in, poof, the image window disappears, poof, the image window reappears with your new image. Very simple color options: your choice of red, green, blue, cyan, magenta, or yellow with varying brightness and contrast. Note how the author has chosen to hide behind a screen name.

**Mandel Bot**

Robert Covington, 2003

Artly There

According to the author, "A simple Mandelbrot viewer based on a demo by Robert Purves." Quite right.

**Mandelbrot**

Christoph Lauer, 2002

Christoph Lauer

A painfully slow Java program for exploring the Mandelbrot set and rendering corresponding Julia sets. Java is not the right environment for applications that require intensive calculation. Also available for Linux and Windows.

**Mandelbrot**

David Ross, 2001

David Ross

An easily forgettable program. Draw and explore Mandelbrot sets from order 2 through 59. This may sound impressive but really isn't. Very high order Mandelbrot sets all look like circles. Switch over to the Julia set associated with a particular location of the Mandelbrot set. Very simple and plain color schemes.

**Mandelbrot on Cocoa**

SADA, 2004

SADA

Explore the Mandelbrot set. Zoom in. Zoom out. Adjust the color scheme. Save your results as png files. A nice simple little program. Also available in a Windows version.

**Mandelbrot Viewer**(a.k.a. Custom View)

Michael C Thornburgh, 2001

Mr. Mike

A good basic Mandelbrot explorer. Simple and fast with very few options. Zoom in, but not out. Adjust the escape radius, number of iterations, number of threads (not quite sure what that means), and location of the origin (which renders distorted Mandelbrot-Julia hybrids), but not the color scheme.

**mandelPath**

Aaron Golden, 2003

Aaron Golden

Doesn't draw the Mandelbrot set in the traditional manner (counting the number of iterations to escape). Instead it traces the orbit of points in the complex plane. As the author says, "It's not particularly useful but I think the pictures are pretty." This program has to be run through a terminal window. Very odd.

**PhoenIX**

Adrian Platts, 2001

Adrian Platts

Deliberately styled interface with options, options, options. Auto zooms, generates relief maps, makes movies, plays background MIDI and MP3 files, and acts as a screen saver. It also draws several kinds of Mandelbrot and Julia sets. Comes with yet more options that I haven't described. Also available in a version for the Classic OS.

**TransWorld**(a.k.a. WindowFreak)

Junzo Sato, 2001

Junzo Sato

A waste of time. Very slowly renders a very low resolution image of the full Mandelbrot set or one specific Julia set in a transparent window. No zooming. No variations of any sort other than the

*transparency*level (thus the name

*Trans*World). A "look what I can do" application that is of interest to no one.

### Newton's Method

**NewtonFractal**

Stefan Messmer, 2001

Stefan Messmer

Draw fractals using Newton's root approximation method in the complex plane. Faster and more reliable than the original Classic application, but still looks and acts like a beta edition in need of a solid debugging. Also available in Windows and Java versions.

### Terrain & Landscape Modeling

**Cosmos3d**

Jeff Biggus & Heinz Nabielek, 2004

osx.hyperjeff.net

A program whose sole purpose is to render Hoyle's galaxy clumping model as mentioned on page 218 of Mandelbrot's

*The Fractal Geometry of Nature*. A good fractal for those interested in the history of cosmology, but I think Hoyle's model was disproved by galactic surveys.

**MojoWorld Generator Demo**

Ken Musgrave, Pandromeda Inc, 2002

Pandromeda

A surrealistic (super-realistic?) landscape generator. A sophisticated, commercial application that is not for the casual user. The gallery of images on the company's website hints at the possibilities for someone willing to put in the time and effort: mountain ranges, beaches, rivers, clouds, sand dunes, moons, entire planets. Also available in a version for Windows.

### Miscellaneous

**Cellular**

Christian Grunenberg, 2001

Christian Grunenberg

A program essentially designed to investigate cellular automata that also happens to draw Mandelbrot sets and model galactic collisions. Not bad, but nothing special.

**Escape**(a.k.a. EscapePPC)

Graham Anderson, 2004

EscapePPC

A comprehensive application for exploring over 20 different types of fractal images and strange attractors. Unregistered versions can only render Mandelbrot sets, however. Many, many features to explore for those with a serious interest. Carbonized to run in both the Classic and OS X environments. (Not to be confused with EscapeFractals: a Newton's Method application for the Classic OS.)

**Gerry's Mandelbrot Set**

Gerry Beggs, 1998

Gerry Beggs

More than just the Mandelbrot set, it also creates Julia sets, IFS images, a fractal tree, the Lorenz attractor, bifurcation diagrams, Newton's method, and Martin's mapping (a.k.a. hopalong). Gives you the ability to check out a lot of different fractals using just one application. Very fast. Also available in a version for the Classic OS.

### Screen Savers

**Apfelmaennle**,

**Huepfer**

Heiko Kretschmer, 2002

wurst-wasser.net

Apfelmaennle draws the whole Mandelbrot set, selects a region, and then magnifies it. Selects a subregion and magnifies that. Keeps going and going. I've never run it long enough to see it run out of steps, but eventually it must. According to the author, Huepfer is "One more screen saver displaying fractal-like graphics known as Hopper or Hüpfer." He does not explain what this is, however, nor have I ever heard of it before. Sometimes it looks like a diffusion pattern, sometimes like a checkerboard. Apfelmaennle is the better of the two programs.

**Amseq**

André and Paul van der Kouwe and Thomas Benner, 2003

Thomas Benner

Amseq stands for "Animated Mandelbrot Sequence Generator". There is no documentation to explain what it does, but it appears to trace the orbit of points in the complex plane through successive iterations of the quadratic mapping. Recall that the Mandelbrot set is the set of all points that don't escape to infinity after an infinite number of iterations of the quadratic mapping. This screen saver draws the path of the points used to draw the Mandelbrot set, not the set itself.

**Coral**,

**Flame**,

**Hopalong**,

**ifs**

Tommaso Pecorella, 2001, 2002

Uselesssoft

Open source Unix programs written in the late '80s and early '90s transformed into screen savers for OS X.

*Coral*fills the screen with DLA (diffusion limited aggregation) fractals that look something like corals when shaded in.

*Flame*draws "weird cosmic fractals" (whatever those are).

*Hopalong*draws the scribbly fractal of the same name plus Martin's Method, Renaldo Recuerdo (called RR), and EJK fractals (whatever those are).

*ifs*draws an iterated function system that evolves smoothly over time. The latest addition to this group, ifs is by far the best. Using Hopalong to save your screen seems risky as it renders images that stay static on the screen for long periods.

**FTXSaver**

Simon Woodside, 2004

Simon Woodside

FTX means FractalTrees X — the name of an application from which the screen saver was derived. Draws simple stick figure, fractal trees that evolve on the screen. Still in the alpha phase but has potential. Looks like it will make interesting images, but is just too computationally rough in its current form.

**Fracture**

Stick Software, 2001

Stick Software

A screen saver for OS X that creates a wide variety of fractal images: Mandelbrot set, Julia sets, Newton's and Halley's root approximation method, etc. Pleasant color schemes and rendering techniques. Interesting yet innocuous, which is what a good screen saver should be.

**Hénon**

Marco Coïsson, 2003

Marco Coïsson

This screen saver gives a graphical representation of the so-called Hénon correlations. Simple and uneventful. Available in English and Italian versions.

**IFS3D**

David Leppik, 2001

David Leppik

Flies over and around an evolving IFS fractal, but I don't like the way it looks. The animation isn't smooth and the images are unnaturally dim.

**Sierpinski 3D**

Epicware, 2001

Epicware

screen saver of a flying, tumbling Sierpinski pyramid. Part of an Open GL bundle of OS X screen savers.

**Xeverywhere**

Stéphane Sudre, 2000-2002

White Box

Start with a cube, replaced it with smaller cubes, replace the smaller cubes with even smaller cubes, and so on, then work backward. Comes with a warning not to set the number of iterations too high. Take it seriously or the Xeverywhere will seize control of your CPU.

## Classic OS

dynamical‑systems, fractal‑dimension, ifs, l‑systems, mandelbrot‑julia, music, newton, terrain, miscellaneous

### Dynamical Systems

**1D Maps**

James D. Meiss, 2001

James D. Meiss

Seven different folded mappings (logistic, cosine, cubic, exponential, tangent, tent, and circle map). Five different ways to analyze them (cobweb, bifurcation, lyapunov exponent, invariant density, and exit time plot). Interesting in a limited sort of way. Also available in a version for OS X.

**1-D Chaos Explorer**

Matthew Hall, 1992

1-d-chaos-explorer.sit

Bifurcation diagrams, web diagrams, time series, etc. Program your own functions for exploration. This is a good program for those who want to understand the basic behavior of iterated systems. Many of the web and bifurcation diagrams in this book were rendered with this program.

**Bifurcation**

Ronald T. Kneusel, 1995

bifurcation.sit

Faster than 1‑D Chaos Explorer, but does half as much (bifurcation, and time-series only).

**Bouncing Ball**

T. Abbott, N. B. Tufillaro, J. P. Reilly, 1993

bouncing-ball.sit

Do you find the quadratic map too abstract and the logistic function an oversimplification? Steady state, periodic, and chaotic phase space orbits can all 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. One of the authors has a page describing the program, but it is no longer available for download. This program and Quadratic Map (see below) go with the book

*An Experimental Approach to Nonlinear Dynamics and Chaos*, the entire contents of which are online in html or pdf format.

**ChaosPlot**

Jason Regier, 1994

chaosplot.sit

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.

**Chaotic Flows**

John Lindner, Bryan Prusha, Josh Bozeday, 1997

CoWCPP

Explore the Chua, Duffing, Lorenz, and Rössler attractors. Zoom in and out, rotate, and play with the parameters. Also available in a version for OS X. One of many scientific applications for the Macintosh from the very interesting College of Wooster Computational Physics Project. Other applications at this site related to chaos and fractals include:

**Celestial Chaos**: Illustrates chaos in the restricted 3-body problems by simulating an asteroid moving in the gravitational field of a binary star system.**Chaotic Sphere**: Interactively explore the spatiotemporal nonlinear dynamics of a sphere of coupled bistable oscillators.**Grid**: Interactively explore the spatiotemporal nonlinear dynamics of a grid of coupled bistable oscillators.**Pendulum**: Explore the dynamics of a nonlinear pendulum.

**Cliff's World**

John B. Matthews, Gem City Software, 1992

cliffs-world.sit

Iterates Dr. Cliff Pickover's dynamical system and plots the resulting coordinate pairs.

**Cycle Explorer**

James C. Burgess, 1995

Jim Burgess

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 interface with a wise guy attitude. Pulling down the "Options" menu gets "No Options" as a reply.

**Gerry's Attraction**(formerly Gerry's Lorenz Attractor)

Gerry Beggs, 2002

Gerry Beggs

Start with a group of points on the xy plane, run them through the right equations, and watch them congeal on the Lorenz Attractor or the less interesting Rössler Attractor. Also comes with an option that allows you to fly around the attractors while they evolve. An inspired way to view them. Much better than fixed diagrams. Carbonized to run in both the Classic and OS X environments.

**Intelligent Chaos**,

**Intelligent Mandelbrot**,

**HyperKaos**

Fabian Lidman, 1998

Fabian Lidman

Applications for viewing the behavior of actual numbers. Like a programmable calculator, but faster. Intelligent Chaos iterates the logistic function in the real numbers. Intelligent Mandelbrot iterates the quadratic map in the complex numbers. HyperKaos is a discontinued HyperCard version of Intelligent Chaos.

**Orbit**

Stephen Eubank, University of Texas, Austin, 1986

orbit.sit

Investigate a variety of well-known one dimensional oscillators and two dimensional discrete maps. Requires a lot of pampering to get it to work on a PPC machine. (Note the copyright date.)

**Quadratic Map**

T. Abbott, N. B. Tufillaro, J. P. Reilly, 1993

quadratic-map.sit

Draws time-series, bifurcation diagrams (in color), and phase space diagrams (which are 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. One of the authors has a page describing the program, but it is no longer available for download. This program and Bouncing Ball (see above) go with the book

*An Experimental Approach to Nonlinear Dynamics and Chaos*, the entire contents of which are online in html or pdf format.

**StdMap**

James D. Meiss, University of Colorado, 1994-2000

James D. Meiss

Standard map orbits were drawn using this program. Has other features for exploring nine different area preserving maps. A nice piece of software with good documentation. Fun to play with, too. Also available in a version for OS X.

### Fractal Dimension

**Coastline**

Boston University Polymer Center, 1995

Exploring Patterns in Nature

A great little tutorial that walks you through the conceptual steps behind fractal dimension. Construct a simulated coastline by successively crumpling a line segment. Repeatedly measure its length by covering it with ever smaller boxes or lines. Plot size versus number on a log-log graph. The slope of the line of best fit is an empirical measure of your coastline's fractal dimension. Perfect for teachers looking for a way to demonstrate this procedure to their students. One of nine related simulations from Boston University's Polymer Center. Also available in a version for Windows.

**Fractal Dimension**

Boston University Polymer Center, 1995

Exploring Patterns in Nature

An application for determining fractal dimension from pict or MacPaint files (also comes with a few sample images). Accepts color and grayscale images, but does its analysis on a black and white bitmapped rendition of the original. Uses the box counting and circle methods. Does not chose box sizes automatically; which means a lot of pointing, clicking, and watching on the part of the user. Automatically fits both data sets to its own straight line so that the two techniques may be compared. One of nine related simulations from Boston University's Polymer Center. Also available in a version for Windows.

**Fractal Dimension Calculator**

Paul Bourke, 1993

Paul Bourke

A box-counting application. Give it a black and white Pict image and it will count the number of boxes needed to cover the image. Automatically runs through different sized boxes but does not analyze its own data, which means you'll need some sort of statistical application to come up with a final number. Fractal dimensions calculated in this book are based on data from this program. The accompanying instruction manual is worth reading for its own sake. Includes a 3D version that I have never tested. (3D stuff is just too hard to learn) Also available in a version for OS X running X-Server and Linux.

### Iterated Function Systems

**Fract**

Bob Wiseman, Wiseman Software, 1990

fract.sit

A program for drawing IFS fractals. Comes with several parameter files.

**Fractal Attraction**,

**IFS Fractal Movie I**

Kevin D. Lee & Yosef Cohen, Sandpiper Software, 1991

fractal-attraction.sit

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 — a stand alone program that does nothing other than play a short movie showing an evolving IFS called "The Claw". This movie is still available in several software archives even though Fractal Attraction disappeared a long time ago. Apparently it was sold commercially with an 80 page instruction book. Check out the reference at amazon.com.

**IFS**

Paul Bourke, 1989

Paul Bourke

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 from a haze of dots) and polygon (which is a bit more intuitive). The author also has a version for generating random IFS patterns.

**IFS**

James D. Meiss, University of Colorado, 2000

James D. Meiss

As difficult to learn as all the rest, but much faster and more stable. The interface can be a bit unfriendly. Comes with a slide show feature for the truly lazy.

**IFS**

Stephen Scandalis, 1990

ifs-scandalis.sit

Another obscure little program. It draws what its name suggests. Nothing special.

**Koyn Fractal Studio Demo**

Terry Koyn, 1993

koyn-fractal-studio-demo.sit

The interface and instructions are far from intuitive. Comes with a few sample IFS files, but I was unable to make my own. Probably because this is just a demo. The retail version costs an unbelievable $119.95 according to the Read Me file or $149.95 according to an article in

*Byte*magazine. (Order before June 30, 1995 and get it for the "discount" price of $89.95.) The author bought koyn.com in December of 1997 but as of June 2002 a related website still does not exist.

**Sierpinski-Triangeln**

Martin Wiss, 1997

sierpinski-triangeln.sit

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 that even Americans can figure it out.

### Lindenmayer Systems

**3D-LParser**

David H. Goeb, 2000

David H. Goeb [last url]

3d-lparser.sit

Creates 3D files from L-System instruction files. Requires too much effort to learn (a problem typical of 3D programs). Based on Lparser by Laurens Lapre.

**Fractal Trees**

Simon Woodside, 1997

Simon Woodside [last url]

fractal-trees.sit

Draws simple stick figure, fractal trees. The original, Classic version is an exceptionally unfriendly program with no user interface. None! Its parameters can only be modified by editing the source code. The version for OS X is orders of magnitude better, however.

**KochToy**

Jim Burgess, 2001

Jim Burgess

A conceptual, non mathematical program. Rearrange the generator with your mouse and see the effect on the fractal simultaneously. No dialog boxes. No numerical parameters to enter. No programming language to learn. L-Systems reduced to their essence.

**L-Systems, 3D L-System**

Paul Bourke, 1991

Paul Bourke

Draws recursively defined fractals like the Peano monster, Koch coastline, Sierpinski gasket, and many others that look like plants or corals. The L-System Manual is worth reading for its own sake. Also available in a 3D version.

**Line Fractal**

Gabriel Nivasch, 1999

Gabriel Nivasch

Renders over a dozen different IFS fractals including the usual — Koch coastline, Sierpinski triangle, etc. Has two basic features: run through the iterations step-by-step or zoom in. The second feature is surprisingly fast at times.

**LParser**

Nick Porcino, 1998

Nick Porcino

I was able to get it to render a few 3D images, but I don't really know how this program works or what it does. LParser is too specialized for a generalist like myself. Based on Lparser by Laurens Lapre.

**LSystems**

Bryan Horling, 1996

Bryan Horling

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.).

**pfg**(Plant and Fractal Generator)

Przemyslaw Prusinkiewicz, 1988-92

University of Calgary [ftp link]

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). The author has a home page, but the only way to acquire this program is through an obscure ftp site.

### Mandelbrot & Julia Sets

**Aros Fractals**

Aros Magic Research, 1996

Aros Magic Research

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. Also available in a Windows version.

**CFG**(ColorFractalGenerator)

John Schlack, 1990-95

cfg.sit

Mandelbrot sets, Julia sets, random walk, etc. A basic rendering program for drawing color fractals. Unregistered versions aren't PPC native and run slowly.

**Cheap Mandel**

Conor McCarthy, 2001

Conor McCarthy

Examine the Mandelbrot set. A beginner's attempt at programming for the Mac. The version for OS X is not much better.

**EasyFract**

Alessandro Guzzini, 1997

easyfract.sit

Easy, yes, but there are better programs out there. Zoom into the Mandelbrot set, play with the colors, but that's all. Available in English and Italian versions.

**EasyFractal**

Berkhan Software, 2003

Berkhan Software

Zoom into the Mandelbrot set, check out the corresponding Julia set, play with the colors, try a different function. Comes with a collection of preset parameter files, some of which are interesting and rather unusual. Available in German and English versions for both the Classic OS and OS X.

**f_zoom**

Andreas Warnke, 1996

f-zoom.sit

"Zoom in die Mandelbrotmenge." A cute little nothing of a program for zooming into the Mandelbrot set. A rare piece of überminimalism from Germany. According to the AltaVista Translator, "Zoom in die Mandelbrotmenge" means "Zoom shot into the almond bread quantity."

**Floating Fractals**

Adam Smith, 1994

floating-fractals.sit

Zoom in and explore 11 different fractals (half of them are variations on the Mandelbrot set). Fast and easy to use.

**Fractal**(a.k.a.

**Fractal Generation Program**)

Cathy & Tom Saxton, 1999

Tom & Cathy Saxton

Idle Loop Software

A nice simple program. Draw the Mandelbrot set. Zoom in a step at a time. Draw the Julia set associated with a point in the Mandelbrot set. Zoom into that a step at a time. Also available in a version for Windows. (Not to be confused with Fractal Generation by Laurent Thil listed below under Miscellaneous).

**Fractal Artist**

Alexei Lebedev, Phronesis Software, 1992

fractal-artist.sit

Another Julia-Mandelbrot explorer. An obscure program I found in a Japanese archive of the comp.binaries.mac newsgroup.

**Fractal Domains**(formerly FracPPC)

Dennis C. De Mars, 1994-2000

Fractal Domains

Explore the Mandelbrot set. Switch to the accompanying Julia set. Rather plain interface with lots of control panels floating around. The version for OS X is much nicer. FracPPC was the original, very compact freeware version.

**Fractal Observatory**

Marcio Luis Teixeira, Trilobyte Software, 1990

fractal-observatory.sit

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 Super MANDELZOOM.

**Fractastic!**

Jake Olevsky, 1998

Jake Olevsky

jakeo.com

A nice piece of fractal eye candy. Draws eighteen different fractals including Hénon attractors, Julia sets, Newton's method, and Mandelbrot sets up to the sixth power and does it fast. Includes dynamic zooming and numerous coloring methods with crazy-fast animation as an option.

**Jason's Fractal Generator**

Jason's Geeky Software, 2000

Jason Woofenden

Draws Mandelbrot and Julia sets. Quite fast despite permanently operating in full screen mode. Registered versions can render movies of Julia set cascades and Mandelbrot zooms. Only one color scheme.

**JavaQuat**

Garr Lystad, 2000

Garr Lystad

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. Oversized interface makes it hard to work with on smaller monitors.

**JLB's Dirty Mandelbrot**

Jean-Luc Brousseau, 1998

jlbs-dirty-mandelbrot.sit

Somebody's weekend project. A decent job for two days work, but nothing special.

**Julia O' Matic, newer.JOM**

Jim Burgess, 2001

Jim Burgess

A split window Julia-Mandelbrot explorer. Click on the Mandelbrot set on the right and see the accompanying Julia set on the left. Then zoom into either set. A minimal program with few options and documentation that's a little hard to find. A version that draws higher order sets, called newer.JOM, appeared one day at xahlee.org and is still there. (Some versions have the apostrophe on the wrong side of the O.)

**Julia!**

Julie Mitchell, 1993

julia.sit

Another Julia-Mandelbrot explorer. Not fast or sophisticated. In fact, it's downright slow. Interlaced graphics and minimal options. There are better programs available.

**Julia's Dream**

Reinoud Lamberts, 1991

julias-dream.sit

Many of the Julia sets in this book were drawn using this program. Generates real-time images of Julia sets as you roam around the complex plane with your cursor. The author's home page from 1996 still exists, but it doesn't mention the program. (Julia's Dream was also the name of a pizza: basil pesto, ricotta and mozzarella, topped with garlic, broccoli, and spinach. $9.99 for a 12" pie. According to a now defunct web page, anyway .)

**Julia's Nightmare**

Ben Davenport, 1995

julias-nightmare.sit

A full-color sequel to Julia's Dream. I have nothing else to say about this program.

**MandelBrain**

Danny Brewer, 1995

mandelbrain.sit

Explore the Mandelbrot set. Comes with more preset color palettes than any other program so far.

**MandelBrowser, Mandella, MicroMandella**

Jesse Jones, 1995

mandelbrowser.sit

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. MicroMandella is a stripped-down version that requires an FPU. Mandella was once sold with a book (now out of print) called

*Fractals for the Macintosh*published by Waite Group Press (now a part of Macmillan Computer Publishing).

**MandelMovie**

Shinichiro Hirama, 2001

Shinichiro Hirama [last url]

mandelmovie-hirama.sit

Roam around the Mandelbrot set until you find a region of interest, then create a QuickTime movie that zooms into that spot. Available in Japanese and English.

**MandelMovie**

Michael Larsen, Dynamic Software, 1995

Dynamic Software

The Mandelbrot-Julia hybrid and Julia cascade movies in this book were rendered in 1992 at the University of Wisconsin-Milwaukee, School of Education using this commercial program. Haven't used it since, so I can't comment on it. Dynamic Software claims to offer other chaos and fractal programs, but sending them any money seems risky. Some of my favorite pages on their website include: a calendar of upcoming events dated 1995, an on-line book composed entirely of broken links, and a grossly overpriced $500 program designed when Macs ran on System 7. Crazy backgrounds render some pages unreadable.

**MandelNet**

Christoph Pfisterer, 1996

mandelnet.sit

Draw Mandelbrot sets using the combined power of several computers on an AppleTalk network. Set up master and slave applications on your network. Link the master to the slaves and away you go. Very simple to set up and use, but completely unnecessary. Can only draw one simple, small Mandelbrot set that does not require massive parallel processing power. Still, it's a pretty neat idea.

**MandelZot**

David Platt, Think Technologies, 1998

David Platt

Most of the Mandelbrot sets and some of the Julia sets in this book were drawn using this program. I have been using 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 any of this. Comes with extensions for drawing Lyapunov space images that I can't get to work.

**MandleAcid**

Derek Greenberg, The Bone Factory, 1993

mandleacid.sit

"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. Version 2.0 is orders of magnitude faster than 1.0, comes with a great wacky interface, and will completely coat even the largest monitor with psychedelic glop. I really like this program, but it should be spelled "MandelAcid".

**Object Mandelbrot**

Bryan Prusha, 1996

object-mandelbrot.sit

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.

**Parallel Fractal Carbon**

Dean Dauger, 2001

Project Appleseed

Dauger Research

A program for testing the computing power of a distributed computing network. Generates a preset image of the Mandelbrot set, then tells you the speed of your parallel processor network in megaflops. If you only have one computer, then it tells you the speed of just that computer. A test program with no options. — zero. It runs and it's done. Also available in a version for OS X with more than zero option called AltiVec Fractal Carbon.

**Phoenix**

Adrian Platts. no date

Adrian Platts

Deliberately styled interface with options, options, options. Auto zooms, generates relief maps, makes movies, plays background MIDI and MP3 files, and acts as a screen saver. It also draws several kinds of Mandelbrot and Julia sets. Comes with yet more options that I haven't described. A version for OS X is reportedly in the works.

**PowerXplorer**

Allesandro Levi Montalcini, 1996

Alessandro Levi Montalcini

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. Also available in a Japanese version.

**RatioJulia**

Christoph Stroh, 1998

Christoph Stroh

Draws Julia sets of rational functions. A unique program that is difficult to use. Has a combination command-line, menu-driven interface. Not much fun.

**Super MANDELZOOM**

Robert P. Manufo, 1988

super-mandelzoom.sit

An antique. Meant to be viewed on a Mac SE or thereabouts. Uses halftones instead of grays. It's been a long time, but the author is still out there working on fractals and whatnot.

**TurboMandel**

Silverware Software, Ross Anderson, 1999

Ross Anderson

A nice, simple Mandelbrot set viewer. Zoom in and out, pan, change color schemes, and take snapshots of your favorite scenes. That's pretty much it. The author's website is horribly polluted with pop up ads. Think twice before you visit it.

**XaoS Fractal Zoomer**

Jan Hubicka, 1997

xaos.theory.org

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. Available for nearly every platform: Amiga, BeOS, DOS, Linux, Mac OS, OS/2, and even Windows (or so they claim).

### Music

**BifurcationOscillator**,

**Logistic Synth**

Jae Ho Chang, 1996.

*bifurcationoscillator.sit*

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.

**ChaoticPianola**

The Boltzmann Toy Factory (Lars Rosenberg), 1998

Boltzmann Toy Factory

Listen to the behavior of the iterated logistic function played on the QuickTime Roland digital piano. A different approach to the exploration of a dynamical system. Sounds like Philip Glass when periodic and Schönberg when chaotic. Since there is a sort of structure to the chaotic regime, ChaoticPianola is briefly listenable from time to time. Also plays a random function for comparison or to reseed a stuck or stale pattern.

### Newton's Method

**EscapeFractals**

Yves Meynard, 1996

Yves Meynard

This program draws fractals using an escape-time method developed by Clifford Pickover. Visually appealing and fast. Not to be confused with EscapePPC (filed above under Mandelbrot & Julia Sets).

**Halley Maps**

Yves Meynard, 1997

Yves Meynard

This program produces interesting looking fractals using Halley's root approximation method for equations up to the twelfth degree. Windows must all be square and less than 512 pixels in size. No animated colors or dynamic zooming and yet I still like it.

**NewtonFractal**

Stefan Messmer, 2001

Stefan Messmer

The latest Classic version of NewtonFractal works, but just barely. This is a very slow application. Also available for OS X (see above), Windows, and Java.

**NewtRaph^3, NewtRaph^4**

Conor McCarthy, 2001

Conor McCarthy

Two programs for rendering fractal images using Newton's root approximation method (cubic and quartic functions only). A beginner's attempt at programming for the Mac. Very rough.

**Object Newton**

Bryan Prusha, 1996

object-newton.sit

Object Newton allows you to explore Julia sets created using Newton's root approximation method. Fast interface like Object Mandelbrot.

### Terrain & Landscape Modeling

**Fractal Contours**

Jim Cathey, 1985. Eric Traut, 1995

fractal-contours.sit

ftp.mactech.com [ftp link]

A wire frame fractal terrain generator based on algorithms developed by Lucas Film. I have never been able to run the original program without having it crash immediately. Looking at the icon I thought someone was playing a joke on me. Then I came across a

*MacTech*article called "Extreme Advantage" describing how Fractal Contours could be completely overhauled to run on a PPC. The author has made all eight reincarnations with their source code available to the general public. I bet this thing was pretty advanced in 1985 when the Mac was still in its infancy, but it doesn't look like anything special now.

**Fractal Islands**

Scott Berfield, Parity Productions, 1985

fractal-islands.sit

Another relic from 1985. Guess what? It crashes after every run on a PPC Mac.

**Fractal!**

Ed Rotberg, Gonzo Systems, 1992-93

Ed Rotberg

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.

**Matt's Fract**

Matson Dawson, 1995

matts-fract.sit

Fly around over a fractal-generated terrain. Not detailed or realistic in any way.

### Miscellaneous

**Carpet**

Kevin Whitley, Think Technologies, 1987-88

carpet.sit

A simple program for stamping carpet or gasket fractals. Ancient, but it still runs. Comes with a severely damaged tutorial file, however.

**ChaosGame, SnowCrystal**

K. Hirai, 1997

K. Hirai

Two small programs that use some sort of recursive action to generate what appear to be fractal images. The user interface is a set of chords of varying thickness on a pair of circles: x and y. (SnowCrystal has three circles: i, j, and k.) Moving the chords around the circle transforms the image. Anything beyond that is a mystery to me as the user manual is written in Japanese.

**Chaos Under Control**

Brian S. Macherone, 1994

Fractal Geometry Summer Workshop

Found this obscure suite of applications in the class notes for an August 2000 workshop at Yale University, taught by, of all people, Benoit Mandelbrot (the father of fractals) and Michael Frame (coauthor of Chaos Under Control, a book on chaos, fractals, and cellular automata for nonscientists). Included with the notes was the software to accompany Frame's book. All five applications are creaky and slow, which obscures their utility as learning tools. They are also a bit hard to appreciate when removed from the context of this one particular class.

**Cellabration**: one- and two-dimensional cellular automata.**DesperatelySeekingChaos**: data analysis by driven IFS, near returns, Kelly plots, and music.**IterateAgainSam**: one-dimensional dynamics, graphical iteration, histograms, time series, and bifurcation diagrams.**TreenessEmerging**: iterated function systems, deterministic and random algorithms, chaos game.**WaitingForMandelbrot**: Mandelbrot and Julia sets.

**Cubic Oscillator Explorer**

Bruno Degazio, 1995

University of Stuttgart

Can't download it for review. "Communications exception (-244)." Would someone out there send me a copy?

**Escape**(a.k.a. EscapePPC)

Graham Anderson, 2004

EscapePPC

A comprehensive application for exploring over 20 different types of fractal images and strange attractors. Unregistered versions can only render Mandelbrot sets, however. Many, many features to explore for those with a serious interest. Carbonized to run in both the Classic and OS X environments. (Not to be confused with EscapeFractals: a Newton's Method application for the Classic OS.)

**Fractal**

Paul Bourke. No date

Paul Bourke

This program renders a diverse array of fractal objects: the Mandelbrot set, Julia sets, Newton's method, diffusion patterns, Hénon orbits, several strange attractors, plus a few more. Does a good job drawing strange attractors. (Allows you to play around with the parameter values and see what changes.) Does a poor job drawing the Mandelbrot set. (It's slow and ugly.) Has a strange quirk to it that I found annoying: accessing the menu bar wipes out the display's memory and starts the rendering process over from the beginning.

**Fractal Explorer**

Peter Stone, 1999

Peter Stone

Easily the largest collection of preset fractals: Mandelbrot sets, Julia sets, escape fractals, Newton's method, quaternions, and others. Enter your own complex functions and see what happens. Fairly fast, but the interface is too clumsy for this to be fun. Zooming in is especially tiresome. There were only 74 visitors to the program's home page from March 1999 to February 2000. Something doesn't seem right. Won't you please visit this lonely site and cheer it up?

**Fractal Lab Kit**

Ronald T. Kneusel, 1994

fractal-lab-kit.sit

A command-line-driven IFS program. Why bother programming for the Mac if you're not going to make use of the GUI interface? 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 each before they crash.

**Fractal Studio**(formerly Fractals)

Keiron Liddle, Aftex Software, 1999

Aftex Software

Draws 30 different fractals. Has a ton of different options to play with. I haven't had the energy to fully explore this monster.

**FractalDesigner**

Martin Hairer, 2000

HairerSoft

Renders preset as well as user-defined functions. CodeWarrior users can even program it to draw Markus-Lyapunov fractals, but others have to live with the presets. I appreciated the flexibility, but was disappointed by the lack of speed and mediocre user interface.

**Fractal Generation**

Laurent Thil, 2001

Laurent Thil

Draws Mandelbrot, Julia, and "Magnetisme" fractals in your choice of colors and rendering schemes in a teeny tiny little window. Also renders disorienting QuickTime movies that make you feel like an ant crawling over a fractal textbook. Comes with preset parameter files to get you started — an absolute necessity given that the program is basically a "black box". The author is probably the only person who really knows how to use it. (Not to be confused with Fractal Generation Program by Cathy and Tom Saxton filed above under Mandelbrot & Julia Sets.)

**Fraxious**

Alan Smith, 1995

fraxious.sit

Mandelbrot and Julia sets, Newton's method, Hénon and other strange attractors, Brownian motion, and Lyapunov fractals. Crashes whenever I try to draw a strange attractor. The Read Me admits it's a buggy program. Not PPC optimized, so it's slow.

**Gerry's Mandelbrot Set**

Gerry Beggs, 1998

Gerry Beggs

More than just the Mandelbrot set, it also creates Julia sets, IFS images, a fractal tree, the Lorenz attractor, bifurcation diagrams, Newton's method, and Martin's mapping (Hopalong). Gives you the ability to check out a lot of different fractals using just one application. Also available in a version for OS X.

**IconMaker**

Jim Burgess, 2001

Jim Burgess

Plots points in the orbit of four unique iterative mappings up to the twelfth power. Color indicates the density of points at a particular location. Sliders permit you to change all five of the parameters that control the mapping. You can also bookmark interesting combinations and return to them later. This is a necessary feature given that it is quite easy to get lost among all the numbers.

**Iterative Functions**

Harold Brochmann, 1992

iterative-functions.sit

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. Great in terms of content, however. Someone should rewrite this suite and wrap it in an up-to-date interface.

**Lyapunov**

Andrea Pellizzon, 1994

lyapunov.sit

Lyapunov diagrams in this book were drawn using this program. It is the only program I know dedicated to drawing this kind of fractal. The author has a home page, but it never mentions this program.

**MacFractint**

Tim Gilman, 1999

Fractint for the Macintosh

Room for one more. Won't you join us? A well loved Windows-Intel classic, Fractint may finally make it to the Mac OS. The current beta version has a lot of preset fractals to play with, but the program as a whole is obviously still under construction.

**Persian Rug**

Paul Cantrell, 1996

persian-rug.sit

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, but unfortunately I was never able to run it for very long without having my machine freeze.

**Stone's Free Mac Programs**

Ishihama Yoshiaki, 1996-2001

Stone's Free Mac Programs

A prolific programmer, Stone has written around a hundred freeware applications for the Mac OS including about a dozen chaos and fractal programs. All are quite primitive and unpolished.

**ChaosGame**: an implementation of the classic IFS routine.**DLA**: a rather slow diffusion limited aggregation simulation.**Magnet**: a chaotic assembly of free-floating magnets.**MandelBrotMovie**: self-explanatory.**Mountain**: a terrain generator.**MRCM**: "multiple reduction copy machine" — some kind of pattern generator.**Terra**: another terrain generator.**Trajectory**: draws strange attractors.

## Relics

### After Dark Modules

I have not used the screen saver After Dark since 1998 and will no longer review After Dark modules for this book.

**Clouds**

Robert Geisler, 1993-95

clouds.sit

This module draws fractal clouds that drift across your desktop. Nicely done.

**Fractal Forest**

Scott Armitage, Berkeley Systems Inc, 1992

fractal-forest.sit

Draws simple, cartoon-like fractal trees on your desktop.

**Fractals Bundle**

Alessandro Levi Montalcini, 1992

fractals-bundle.sit

A suite of programs designed to run on 68k Macs with a math coprocessor. The After Dark module (called "Fractal" in the last version, but known as "Mandelbrot" in earlier versions) draws regions of the Mandelbrot set, which it then saves as Pict files. Comes with a program for viewing the picts and a mover based on the old Font/DA mover. As the author seems to admit, such a system is of little use now. There is no mention of this bundle at his website.

**Newton Map, Newton-EXP**

Huangxin Wang, 1994

newtonmap.sit

Far too slow. I assume they draw Julia sets using Newton's method, but I couldn't stand the wait. Requires an FPU.

**Ocean Child**

Paul Cantrell, 1994

ocean-child.sit

Perpetual, random Julia cascade with options for different surreal effects. A nice piece of eye candy.

**Planetmaker**

Adam Miller, 1992

planetmaker.sit

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. Uncomfortably slow.

**Sierpinski's Gasket**

WinterBright Software (David Thompson), 1992-93

sierpinskis-gasket.sit

Crash Warning. Gives "Error 10" when launched, even with an FPU emulator. Thus, it does nothing.

**TerrainMaker**

Jakub Buchowski, Adam Miller, 1991

terrainmaker.sit

Fractal terrain modeler. Old-fashioned looking color scheme and resolution.

### HyperCard Stacks

With the ascent of the World Wide Web there really doesn't seem to be much of a need for HyperCard anymore. It was a great application for its time, but I doubt I will add anything new to this category.

**BirdSong Engine**

David Benz, 1993

birdsong-engine.sit

Clever HyperCard stack that converts L-System-type fractals into bird songs.

**Chaos Theory**

Mike Atanasio, Wild Card Software. No date

chaos-theory.sit

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 Chaos**

A.J. Roberts, University of Adelaide, 1991

hypercard-chaos.sit

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. I loved many of them, but a few were complete mysteries.

**Mandelbrot Fractal Companion**

Evan T. Yeager, 1993

mandelbrotfractalcompanion.sit

A brief HyperCard tutorial of complex numbers, fractals, the Mandelbrot set, and how to construct a computer program. In living color (something rare in HyperCard).

**MandelMusic**

Roger R. Espinosa, Donna Iadipaolo, Jim Brunberg. No date

mandelmusic.sit

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.

### Floating Point Unit Required

A floating point unit (FPU) is a coprocessor that handles intensive mathematical calculations to relieve some of the burden on a computer's central processor unit (CPU). Applications designed to access an FPU's subroutines will run faster than those that don't; that is, if your computer has one. FPUs are currently extinct and have been so since about 1995. Computers are no longer designed or manufactured with these units. If you try to run an FPU program on a machine without one it will likely crash. If you're lucky you might just get an error message. There is a workaround for those who insist on trying things for themselves. I've managed to execute almost all of these applications on a 603 PPC Mac using a control panel called SoftwareFPU, but with only mixed results. Some programs run awkwardly under the emulator. Some run just fine. I have not tried this on a G3 or newer machine nor will I ever. There are so many non-FPU programs out there with similar capabilities that these few dinosaurs won't be missed. I present them here in an effort to maintain a complete archive.

**ab Fractal**

Eden Software, L. Pieniazek, 1994

ab-fractal.sit

I don't have time for this. It draws the Mandelbrot set, Julia sets, and maybe some other sets, but is intolerably slow. I couldn't believe that anyone would ask a shareware fee for this. Requires an FPU.

**Animated Fractal Generator**

Jerry Goldstein, 1990

animated-fractal-generator.sit

One of the slowest programs around. I estimate that a 300 MHz PPC machine would take an hour to complete one 640 × 480 image. I can't confirm this as I didn't feel like waiting. According to the manual, this program can render Mandelbrot, Julia, and "other specialized varieties of fractals." Requires an FPU.

**ARTAbrot**

George Warner, 1994

artabrot.sit

Does

*not*require an FPU but does require a DSP3210, which means it will only run on a Quadra 840AV or Centris 660AV — the two machines that used the AT&T 3210 Digital Signal Processor to run the ARTA (whatever that is). I cannot review this program since I do not have either of those machines.

**FracHill, FracBlob**

Paul Bourke, 1991

Paul Bourke

Wire frame fractal terrain generator. Runs slow and doesn't look very good. I'm not impressed. FracHill models on a plane while FracBlob models on a sphere. Requires an FPU.

**Fractal Wizard**

Thomas Okken, 1992

fractal-wizard.sit

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.

**Kaos**

Reinoud Lamberts, 1990

kaos.sit

Kaos produces IFS images that are quite unique, but it is 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. The author's home page from 1996 still exists, but it doesn't mention the program (or much of anything else).

**MacMandelMovieMaker**

pad-parrot productions, J. Geagan, 1993

macmandelmoviemaker.sit

Creates a sequence of PICT files spiraling into a point on the Mandelbrot set. In only 28 hours I was able to generate a ten second movie of a 101 frames. It was a truly stunning accomplishment [read sentence dripping with sarcasm]. Requires an FPU. Geagan is also responsible for the Mac shareware classic 99 Bottles — an application that sings the famous time-killing, travel song for you. Set it at a thousand bottles and pretend your in the audience at an Andy Kaufman performance. Set it to a million bottles and have it sing along with MacMandelMovieMaker while it generates a half hour video for you. On second thought, a million bottles may not be enough. 99 Bottles does not require an FPU.

**MandelScope, MandelMaker**

Anthony S. Ku, 1993

mandelscope.sit

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 the time, but looks ridiculous now. MandelMaker uses a divide and conquer algorithm that should speed things up, but still runs slowly. Doesn't use the symmetry of the Mandelbrot set across the real axis. Requires an FPU.

**MandelTV**

Ed Ludwig & Ken Abbott, Abbott Systems, 1990

mandeltv.sit

A relic from the days of System 6. MandelTV is a

*desk accessory*for exploring the Mandelbrot set. Abbott Systems also sells a commercial fractal software package called Kaleidoscope that I have not reviewed. It also requires an FPU!