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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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