Exploring Fractals

A suite of Java Applets illustrating fractal geometry and and chaotic dynamics

Written by Ginger Booth

Designed by Michael Frame

Edited by Richard Bedient

These Java programs run on both Mac and Windows systems, best on FIREFOX.

Saving and Printing Images

Most browsers, including the most current, print Java applets badly.

The easiest way to handle this problem is to capture the image with your machine's screen capture mechanism, paste the picture into a graphics program, and print from there.

Windows: Use < ALT >< PrintScreen > and Accessories | Paint
Mac OS 9: < Command >< Shift > 3
Mac OS X: Use the Grab utility

If you are printing, rather than just saving, you will get better results if you use the program's color controls (where available) to set the background to white and the foreground to dark colors.

You may also find it useful to include in your capture an open parameter window showing the settings used to create the image.

Enter the transformations and the starting picture. The software applies all the transformations to the picture, producing a new picture. Iterating this process, we observe the pictures converge to the fractal determied by the transformations. Deterministic IFS Java 1.3+

Enter the transformations. The software picks a starting point and applies one of the transformations to that point. Then it applies another transformation to the resulting point, and continues, selecting the transformations in random order. Random IFS Java 1.3+

Enter three starting points and their images. The software computes the affine transformation taking the starting points to the image points. Affine Transformation Calculator Flashplayer 7

This is a modification of the random IFS software. Here the order of the transformations is determined by a time series of data. Patterns in the resulting picture detect patterns in the data. Driven IFS Java 1.4+

Instead of allowing all combinations of transformations, in this software we can specify forbidden pairs or forbidden triples of transformations. We explore relations between forbidden combinations and features of the resulting pictures. IFS with Memory Java 1.4+

Instead of allowing all combinations of transformations, in this software we can specify forbidden pairs or forbidden triples of transformations. With this we can produce less symmetric, hence more realistic, trees, ferns, and such. random IFS with Memory Java 1.3+

Select an image and cover it with boxes of various sizes. Does the number of boxes scale as some power of the box size? Box Counting Dimension Java 1.2+

Draw Julia sets for points in and around the Mandelbrot set. Zoom in on regions in these pictures. Explore the corresponding sets for functions other than z² + c. Mandelbrot and Julia Sets Java 1.3+

Explore graphical iteration, histograms, time series, return maps, bifurcation diagrams, and Kelly plots for data generated by iterating any from a collection of functions. 1-Dimensional Dynamics Java 1.3+

Enter a time series and a selection of bins. The program generates a sequence of squares with colors assigned by the bins. This is particularly useful for detecting periodic and nearly periodic patterns in the data. Kelly Plots Java 1.3+

Couple together a collection of logistic maps, so the next iterate of each depends on the iterates of the surrounding maps. How do the patterns depend on which neighbors interact and how strongly they interact? Synchronized Oscillators Java 1.3+

Select the dimension of the space, the neighborhood of each cell, and the rule specifying which neighborhoods give rise to live cells. The software produces successive generations of patterns developing from a specified initial pattern. Cellular Automata Java 1.1+

Draw lines, curves, circles, and polygons. Observe how these shapes invert across a given circle. Circle Inversion Illustrator Java 1.3+

Input centers and radii of inverting circles. The software generates the limit set of inversion in these circles, using the random IFS algorithm with inversion in these circles replacing the affine transformations. Circle Inversion Limit Sets Java 1.2+

Input the starting string and the replacement rule. The program illustrates the results of successive applications of the replacement rule to the string. This approach was developed by Lindenmayer to simulate branching patterns of plants. L Systems Java 1.2+

With this software we explore the patterns resulting from many variations of the familiar Pascal's triangle construction. Pascal's triangle Java 1.3

Enter the amplitudes and frequencies of sine waves and observe the pattern produced by superimposing these waves. Spectrum Illustrator Flashplayer 7

Compose and play fractal music. Musical phrases can be copied, shifted, dilated, compressed, and combined to make compositions having a self-similar structure. Fractal Music Composer Java 1.3+

Draw histogtrams of the data and try to fit a normal curve. Compute the Hurst exponent. Compute the f(alpha) curve. Fractal Statistics Java 1.3+

These pieces of software, running in Java 1.0, were written by Shawn Shopmyer.

Chaos Game Click the mouse in the window to select the vertices. Check starting point and click the mouse to run.

Deterministic IFS Enter the parameters in the table (DO NOT PRESS RETURN AFTER ENTERING A NUMBER). Scroll to the bottom of the screen and click submit to bring up the drawing window. With the mouse, draw the initial picture in the left window.

Random IFS Enter the parameters in the table (DO NOT PRESS RETURN AFTER ENTERING A NUMBER). Scroll to the bottom of the screen and click submit to bring up the IFS window.

Enter the transformations and the starting picture. The software applies all the transformations to the picture, producing a new picture. Iterating this process, we observe the pictures converge to the fractal determied by the transformations.	Deterministic IFS	Java 1.3+
Enter the transformations. The software picks a starting point and applies one of the transformations to that point. Then it applies another transformation to the resulting point, and continues, selecting the transformations in random order.	Random IFS	Java 1.3+
Enter three starting points and their images. The software computes the affine transformation taking the starting points to the image points.	Affine Transformation Calculator	Flashplayer 7
This is a modification of the random IFS software. Here the order of the transformations is determined by a time series of data. Patterns in the resulting picture detect patterns in the data.	Driven IFS	Java 1.4+
Instead of allowing all combinations of transformations, in this software we can specify forbidden pairs or forbidden triples of transformations. We explore relations between forbidden combinations and features of the resulting pictures.	IFS with Memory	Java 1.4+
Instead of allowing all combinations of transformations, in this software we can specify forbidden pairs or forbidden triples of transformations. With this we can produce less symmetric, hence more realistic, trees, ferns, and such.	random IFS with Memory	Java 1.3+
Select an image and cover it with boxes of various sizes. Does the number of boxes scale as some power of the box size?	Box Counting Dimension	Java 1.2+
Draw Julia sets for points in and around the Mandelbrot set. Zoom in on regions in these pictures. Explore the corresponding sets for functions other than z² + c.	Mandelbrot and Julia Sets	Java 1.3+
Explore graphical iteration, histograms, time series, return maps, bifurcation diagrams, and Kelly plots for data generated by iterating any from a collection of functions.	1-Dimensional Dynamics	Java 1.3+
Enter a time series and a selection of bins. The program generates a sequence of squares with colors assigned by the bins. This is particularly useful for detecting periodic and nearly periodic patterns in the data.	Kelly Plots	Java 1.3+
Couple together a collection of logistic maps, so the next iterate of each depends on the iterates of the surrounding maps. How do the patterns depend on which neighbors interact and how strongly they interact?	Synchronized Oscillators	Java 1.3+
Select the dimension of the space, the neighborhood of each cell, and the rule specifying which neighborhoods give rise to live cells. The software produces successive generations of patterns developing from a specified initial pattern.	Cellular Automata	Java 1.1+
Draw lines, curves, circles, and polygons. Observe how these shapes invert across a given circle.	Circle Inversion Illustrator	Java 1.3+
Input centers and radii of inverting circles. The software generates the limit set of inversion in these circles, using the random IFS algorithm with inversion in these circles replacing the affine transformations.	Circle Inversion Limit Sets	Java 1.2+
Input the starting string and the replacement rule. The program illustrates the results of successive applications of the replacement rule to the string. This approach was developed by Lindenmayer to simulate branching patterns of plants.	L Systems	Java 1.2+
With this software we explore the patterns resulting from many variations of the familiar Pascal's triangle construction.	Pascal's triangle	Java 1.3
Enter the amplitudes and frequencies of sine waves and observe the pattern produced by superimposing these waves.	Spectrum Illustrator	Flashplayer 7
Compose and play fractal music. Musical phrases can be copied, shifted, dilated, compressed, and combined to make compositions having a self-similar structure.	Fractal Music Composer	Java 1.3+
Draw histogtrams of the data and try to fit a normal curve. Compute the Hurst exponent. Compute the f(alpha) curve.	Fractal Statistics	Java 1.3+

Chaos Game	Click the mouse in the window to select the vertices. Check starting point and click the mouse to run.
Deterministic IFS	Enter the parameters in the table (DO NOT PRESS RETURN AFTER ENTERING A NUMBER). Scroll to the bottom of the screen and click submit to bring up the drawing window. With the mouse, draw the initial picture in the left window.
Random IFS	Enter the parameters in the table (DO NOT PRESS RETURN AFTER ENTERING A NUMBER). Scroll to the bottom of the screen and click submit to bring up the IFS window.