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.
|
