The Cook Book
Recipe for the week of January 16 - 22
An Excitable CA Spiral Core
- This week we advertise a simple algorithm for growing 'synthetic'
spiral cores in excitable cellular automata, certain stable periodic
objects that are invariant under GH and CCA dynamics. In the present
case, we exhibit a 48-color core for excitable CA rules with threshold 42
on the range 8 Box neighborhood. The idea is to start from a suitable
band of 1's (excited) followed by a similar band of 2's, both vertical
say, near the top of the 250 by 200 array. Setting the rule's number of
colors equal to 47 originally, we find that a wave of excitation travels
clockwise, bending as it advances, and leaving refractory bands in its
wake. The value 47 is chosen so that excitation can make an entire
revolution without encountering interference from the initial 2's before
they finally relax to the resting state 0. After 50 updates it becomes
clear that exactly one additional color will fit to produce a stable,
closed spiral core, so we add a 48'th color to the cycle. After another
revolution, by time 100, the geometry of the spiral is unchanging. As a
final touch we recolor the central 'hole' black since the dynamics of the
wave ends are unaffected by this region. It is interesting to note that
microscopic imaging of actual oscillating chemical reactions produces
strikingly similar patterns.
- For more about the mathematics behind spiral cores, stable periodic
objects and their dynamics, see R. Durrett and D. Griffeath, "Asymptotic
Behavior of Excitable Cellular Automata." Experimental Mathematics
2 (1993), 183-208. You can also grow this week's spiral core 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 spiral.xpt, initial
configuration band.bmp and time 50 configuration
spiral.bmp should be added to the respective application
subdirectories; these data files can be found at the same location as the
beta. The design of the program makes it extremely easy to change
parameters, in this instance the number of colors, 'on the fly.'
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