The Cook Book
Recipe for the week of November 21  27
A Monster for a Threshold 2 GreenbergHastings 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
socalled 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
10color rule. Whether an 11color 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), 183208. 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 Windowsbased 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.
