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Recipe for the week of June 12 - 18

Spiral Creation in the Basic CCA

This soup (ca. 1989) shows a characteristic spiral emerging from interaction of droplets in the 18-color nearest neighbor Cyclic Cellular Automaton. These dynamics, which were featured in a Scientific American article by Kee Dewdney, are surpisingly amenable to mathematical understanding. For many of the basic structural properties, see R. Fisch, J. Gravner, and D. Griffeath (1990) Cyclic cellular automata in two dimensions, Spatial Stochastic Processes. A festschrift in honor of Ted Harris. Birkhauser, 1990, 171-185. In this realization a spiral was nowhere to be seen at time 700, then well-established by the time our soup was captured after 850 updates. The natural question: how does it arise?

Painstaking interactive inspection of the trajectory, especially tedious with our limited visualization tools of five years ago, resolved the mystery. By a defect we mean a self-avoiding loop of sites in the lattice along which the colors wind through one or more complete cycles of length 18, the flux in state values being 0, +1, or -1 across each nearest neighbor bond. One cannot expect a defect in the initial random configuration on any finite array of worldly dimensions. But the dynamics self-organize until one is formed, in this case at time 727. Here is the automaton at that time Time 727, although you will need to analyze it quite carefully to spot the desired cyclic loop. Click on Zoom 727 to see an enlarged image of the region where defects first appear, with the final bond that completes the cycle marked in black.

It turns out that defects are invariant under basic CCA dynamics, i.e., once formed they can never be destroyed. Moreover, as time goes on, tighter and tighter defects arise until one of minimal length emerges to play the role of spiral core. At that point new layers are added and the spiral grows to serve as pacemaker for a sizable chunk of the ultimate locally periodic state. Such topological regularities can shed considerable light on complex dynamics, even in instances where detailed quantitative analysis seems beyond our grasp.

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