The Cook Book

Recipe for the week of December 5 - 11

Turbulent Equilibrium in a Cyclic Cellular Automaton

What happens to CCA dynamics when the threshold is too high for spiral formation, but not so high that wave fronts are convex confined? Judging from a great many experiments on large arrays, there appears to be a turbulent phase. (This name is only suggestive; we do not claim any connection to classical fluid dynamics.)

Roughly speaking, the system tries to make spiral centers, but these would-be centers are unstable. As a result, when a vortex unravels it leaves behind a smattering of debris that can act as a seed for the formation of new wave fronts. This debris is a permanent and crucial part of the dynamics. The turbulent phase is characterized by very large length scales, so equilibrium behavior is only observed on sizable arrays. Here we see the 8-color, range 2 Box, threshold 5 CCA on a 500 x 400 array after 2500 updates, started from a uniform random configuration. I have felt for some time that these turbulent CCA equilibria make would make nice fabric designs.

For more about the mathematics behind Cyclic Cellular Automata, and an extensive classification of their phenomenology, see R. Fisch, J. Gravner and D. Griffeath, "Threshold-Range Scaling of Excitable Cellular Automata." Statistics and Computing 1 (1991), 23-39. That article contains numerous color graphics, cutoff tabulations, and phase diagrams for both CCA and GH dynamics.

You can watch turbulent equiulibrium evolve for yourself by downloading WinCA, our Windows-based interactive modeling environment for cellular automata. To pick up the first beta version of WinCA look in the kitchen sink. An experiment script turbulen.xpt and bright 8-color palette turbulen.pal should be added to the appropriate program subdirectories; these data files can be found at the same location as the beta.

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