The Cook Book
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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