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Recipe for the week of January 9 - 15

Self-Correcting Perfect Spirals

The Cyclic Cellular Automaton with range 1 box ("Moore") neighborhood, threshold 3, and 4 colors is a particular interesting excitable system. From a uniform random configuration it quickly self-organizes into a chaotic soup with large length scale similar in structure to the "Fossil Record" evolution of last November 28 - December 4. But later on, typically after several hundred updates, widely separated stable spiral cores emerge and slowly take over the lattice. This week's soup shows two such structures that nucleated in a 500 by 400 array at about time 500. The graphic shows their gradual expansion after another 2000 updates. The palette is inspired by a 4-color Keith Haring print on my office wall at home.

The characteristic spirals of this rule are particularly interesting for three reasons. First, they are NOT stable periodic objects (spo's) in the sense Pritikin's monster (November 21 - November 27, '94) or the spiral cores of last week's CCA soup. Rather, external patterns can destabilize the boundary layer of the spiral. But secondly, the emergent spiral pattern away from the boundary is perfect, i.e. without any 'glitches.' The spirals of more aggressive excitable dynamics often work their way around particularly stubborn obstacles leaving residual anomalies. But based on extensive experimentation it appears that the spirals of this rule overrun any disordered environment without the slightest trace of the displaced configuration. Thirdly, it would appear that spiral cores arise either after a few hundred updates or essentially not at all. Our evidence for the last observation is the most impressionistic, but it seems that the time to nucleation from uniform randomness has an extremely thin tail. Perhaps the formation of a core is nearly impossible once the surrounding soup acquires a sufficiently long correlation length?

For more about the mathematics behind Cyclic Cellular Automata, 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. You can watch the growth of perfect spirals 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 perfect.xpt, 4-color palette perfect.pal and bitmap perfect.bmp should be added to the appropriate program subdirectories; these data files can be found at the same location as the beta. Perhaps you can help shed light on some of the fascinating issues raised by this simple rule.

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