Cliff

Download jar: Get the Cliff jar file by clicking here.

Download dmg: Get a macOS application 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)

x_i = 0.1 ; y_i = 0.1

for i in 1…n do

x_i+1 = sin(y_ib) + c sin(x_ib)

y_i+1 = sin(x_ia) + d sin(y_ia)

x_i = x_i+1 ; y_i = y_i+1

shadePixel(x_i, y_i)

end.

Mutation α:

x_i+1 = sin(y_ib) + sin²(x_ib) + sin³(x_ib)

y_i+1 = sin(x_ia) + sin²(y_ia) + sin³(y_ic)

Mutation β:

x_i+1 = sin(y_ib) + sin²(x_ib)

y_i+1 = sin(x_ia) + sin²(y_ia)

Mutation γ:

x_i+1 = |sin(y_ib)| + sin²(x_ib)

y_i+1 = |sin(x_ia)| + sin²(y_ia)

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

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.

• http://sprott.physics.wisc.edu/pickover/home.htm.

• mailto:cliff@watson.ibm.com.

• Dr. Cliff Pickover, P.O. Box 549, Millwood, New York 10546-0549 USA.

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