Links open in an external window or tab. These simulators run on most popular computers and Web browsers.

Fractals

These simulators were built for Michael Frame at Yale University. His Fractal Geometry website provides complete course materials.

More of these simulators may deserve listing in my "Most Popular" box, but I don't have usage statistics from the Yale site. Fractal Music Composer is only hosted at my site, and it's one of my most popular apps.

Close Pairs looks for signs of pattern in a time series. This app was commissioned by Gene Hamilton of Washington College, instead of Michael Frame at Yale. Michael once had a Close Pairs simulator, but the software senesced, and Gene wanted it back. Java 1.4.

Released 2007-04.

Mandelbrot-Julia Sets explores the famous Mandelbrot Set and its associated Julia Sets with paired plots. Each point of a Mandelbrot set generates a Julia set. Includes Julia orbits, several color schemes, and alternate functions besides the usual z². Java 1.4.

I also have a simpler Flash version, written as a Flash performance test. Flash 9.

Revised 2007-10, orig 2004-05.

Fractal Music Composer is a powerful music editor. Students build their own musical phrases out of notes from their choice of multiple keys and musical scales. Compositions are built from notes and phrases playing on assorted instruments. Fractal editing operations create self-similar sequences across different time and sound scales. Students can save MIDI output of their creations, which can play their music on almost any computer. Java 1.4.

Click to listen to a sample composition made by applying fractal operations on the two Pachelbel's Canon phrases shown at left.

Revised 2007-04, orig 2004-09.

Box Dimension explores the idea of fractal dimension by counting boxes along geographical edges. Latest version allows user to supply his own pictures. Java 1.4.

Revised 2007-11, orig 2002-10.

Fourier waveforms are built by adding sine waves, controlled by frequency/amplitude sliders similar to a graphic equalizer. Flash 7.

Released 2004-7.

Oscillator Synchronization explores coupled oscillators. Each oscillator is driven by a logistic map function, which can yield steady state, periodic, or chaotic behavior depending on its parameter s. Many different neighborhood coupling, initial value, and s distribution schemes are provided to explore. Java 1.4.

Revised 2007-05, orig 2004-10.

FracDyn explores iterated function dynamics via 6 different analytic plots - graphical iteration (with animation), binned histogram, return map, Kelly plot, time series, and bifurcation diagram. Java 1.4.

Revised 2007-05, orig 2004-06.

Kelly Plot allows you to explore Kelly plots further than in FracDyn, by pasting in your own data, and applying a choice of processing and binning schemes. Java 1.4.

Revised 2007-05, orig 2004-09.

FracStats does R/S analysis and comparison to a normal distribution, on a time series, with or without difference processing on the input data. Also does multifractal analysis (f(alpha) curves) on time series or planar data, using either the method of moments or histogram method. Several data sets are supplied, but the main use is to paste in your own data to explore. Java 1.4.

Revised 2007-06, orig 2005-05.

In Circle Inversion, a user drawn picture is inverted over one or more circles. Java 1.4.

Revised 2007-05, orig 2003-06.

Limit Sets of Circle Inversions generates fractals from a set of input circles. Java 1.4.

Revised 2007-05, orig 2003-04.

Cellular Automata runs an assortment of 1- and 2-dimensional cellular automata, allowing the user to set the rules graphically. Java 1.4.

Revised 2007-05, orig 2003-03.

Aristid Lindenmayer, a botanist, invented L-Systems to describe plant development. L-Systems generates 2-d fractal images, including plant-like branching structures, and familiar fractals such as the Koch curve and Sierpinski gasket. The user can alter the rules, colors, and specify error and variation in the rule application. Java 1.4.

Revised 2007-05, orig 2003-04.

Affine calculates the affine transformation to map three source points to three target points. Flash 7.

Revised 2007-04, orig 2004-07.

Deterministic IFS runs iterated function systems (affine transforms) on a user-drawn picture. Java 1.4.

Revised 2007-04, orig 2003-07.

Random IFS shows how random traversals of iterated function systems converge to the deterministic shape of the system. Java 1.4.

Revised 2007-04, orig 2004-09.

IFS with Memory applies selected sequences of affine transforms, subdividing an ancestral square, to generate fractal pictures. Sequence selection is in 2d or 3d. Java 1.4.

Revised 2008-02, original 2004-07?