Recipe for the week of November 21

The Cook Book

Recipe for the week of November 21 - 27

A Monster for a Threshold 2 Greenberg-Hastings Rule

Recall from our recipe for the week of November 7 that the Greenberg- Hastings Model is a simple CA rule for an excitable medium. In the present case, 10 colors advance around a cycle. Any site with color 0 needs to see at least two sites of color 1 among its 8 nearest neighbors (the so-called Moore neighborhood) in order to advance to color 1. The remaining colors advance automatically, with color 9 cycling to 0. Starting from a uniform random soup on any array of worldly proportions this system apparently dies out. However it is not hard to design little rectangular wave fragments that march along in a horizontal or vertical direction -- we call these bugs. A natural question arises: is there a stable periodic objects (s.p.o.) for this rule, i.e., a finite configuration that cycles indefinitely and cannot be disturbed by external influences?

Remarkably, Dan Pritikin has constructed s.p.o.'s for threshold 2, Moore neighborhood GH models with any number of colors up to 10. This week's soup shows the evolution of Dan's monster under the dynamics. Evidently, horizontal and vertical waves emanate from the core, spreading out until they encounter similar waves transmitted from another monster somewhere far far away. The stable core shown here is the smallest known for the 10-color rule. Whether an 11-color s.p.o. exists for threshold 2 remains an open question.

For more about the mathematics behind stable periodic objects and their dynamics, see R. Durrett and D. Griffeath, "Asymptotic Behavior of Excitable Cellular Automata." Experimental Mathematics 2 (1993), 183-208. That article contains numerous color graphics, including one of the central hole in a very large, nearly critical GH spiral.

You can watch Dan's monster grow 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 dan10.xpt and initial configuration dan10.bmp should be added to the respective application subdirectories; these data files can be found at the same location as the beta.