Random Recipe

The Cook Book

Recipe for the week of January 30 - February 5

Macaroni: Disordered Wave Fragments

Recall that the Greenberg-Hastings Model is a simple CA rule for an excitable medium. In the present case, 5 colors advance around a cycle. Any site x with color 0 needs to see at least four sites of color 1 over its range 2 Box neighborhood (a 5 by 5 box centered at x) in order to advance to color 1. The remaining colors advance automatically, with color 4 cycling to 0. For this week's soup we start from a uniform random mix on a 500 by 400 array. The threshold is sufficiently high that wave fragments have a hard time bending inwards to make spiral cores. Consequently, cyclic bands self-organize over time into weakly aligned parades of wave fragments. Here we see a typical result after 200 updates.

The pattern is not unlike that of the monolayer of macaroni that adheres to a colander during the preparation of a $.059 box of macaroni and cheese. We have chosen a palette suggestive of pasta and the Agent Orange substance added later in the preparation. CAM6 experiments have revealed that if these dynamics are allowed to run for thousands of updates on a larger array, say 2048 by 2048, then the wave fragments continue to align into ever longer parades. But more than this, the pasta also merge end to end until they transform into a new variety: spaghetti. We expect that on an infinite array the process would continue to enormous length scales, giving rise to linguine, then angel hair, then ...

For more about the mathematics behind Greenberg-Hastings Models, 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 GH and CCA dynamics.

You can cook your own macaroni 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. The second beta should be ready in a couple more weeks. An experiment script macaroni.xpt and palette macaroni.pal should be added to the respective application subdirectories; these data files can be found at the same location as the beta.

Take me higher...
Introduction to the PSK PSK Search Recent Additions CA Archive CA Links Feedback Appreciated !