John B. Matthews, M.D.

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Java Web Start: Launch Cliff by clicking here.
Download jar: Get the Cliff jar file by clicking here.
Download app: Get the Cliff application bundle for Mac in .tgz format by clicking here.

About Cliff: Cliff is a Macintosh application for exploring the dreams of the Latööcarfians, as described in Cliff Pickover's Chaos in Wonderland. The program iterates Dr. Pickover's dynamical system and plots the resulting co-ordinate pairs:

given (-3 < a,b < 3) (0.5 < c,d < 1.5)
xi = 0.1 ; yi = 0.1
for i in 1…n do
xi+1 = sin(yib) + c sin(xib)
yi+1 = sin(xia) + d sin(yia)
 xi = xi+1 ; yi = yi+1
 shadePixel(xi, yi)

Mutation α:
xi+1 = sin(yib) + sin2(xib) + sin3(xib)
yi+1 = sin(xia) + sin2(yia) + sin3(yic)

Mutation β:
xi+1 = sin(yib) + sin2(xib)
yi+1 = sin(xia) + sin2(yia)

Mutation γ:
xi+1 = |sin(yib)| + sin2(xib)
yi+1 = |sin(xia)| + sin2(yia)

Certain combinations of a, b, c and d produce quite striking pictures.

Cliff screen shot

The shading uses a palette of equally spaced hues. The resulting plots have an attractive diaphanous quality, but they take a while to "grow". My approach has been to search for interesting parameter sets at lower resolution and re-run. The favorites menu includes some of my own samples, as well as parameters from appendix B.2 of the book.

Requirements: Cliff runs on any Mac with OS X 10.4 or later. The enclosed jar may run on other platforms with Java 5, but none have been tested.

Application: Near criticality, such dynamical systems may find application to the weak password problem.

References: These simple iterated mappings are the subject matter for an entire computer art/math book Dr. Pickover published in 1994, Chaos in Wonderland. He has written extensively about this and related topics; for more information visit his award-winning web site.

The Author: Cliff is copyright © 1992, 2008, Gem City Software. It may be distributed freely but without warranty. Send me your interesting parameter sets, and I'll include them in a future release.

Disclaimer: Sorry, neither Gem City Software nor I can accept liability arising out of use of this program including (but not limited to) the time you waste playing with it. Happy hunting!

Copyright: ©1992, 2008, John B. Matthews.

Last updated 3-Apr-2011

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