The Cook Book
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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