- A year ago we featured the recipe
Self-organized
Batik, generated by one set of parameters for the Larger than Life
(LtL) rule. For each choice of local neighborhood, LtL comprises a 4-dimensional
phase space of CA dynamics that may be thought of as digital, spatial versions of the
classical logistic models from population biology. An empty site becomes occupied if the
population in its neighborhood lies in the interval [b1,b2], whereas an
occupied site becomes vacant unless the local population lies in
[d1,d2]. Conway's Life is the range 1 Box case with birth and survival
intervals [3,3] and [3,4], respectively. My student Kellie Evans is now
finishing a thesis which explores the LtL phase portrait for general range
box nieghborhoods and the full spectrum of parameter values. Over the next
few weeks I plan to present a series of soups illustrating some of the
exotic scenarios included in this fascinating family of nonlinear
dynamics.
- The forthcoming LtL series will feature a color scheme we have
used previously in the Kitchen to enhance the visualization of growth
models. Namely, with a palette of 201 colors (say), vacant sites always have color 0,
but we paint an occupied site x with color 1+(t mod 200), where t
denotes the last time of a 'birth' at x. Such a representation not only
produces cooler graphics, but often yields additional insight about the
rule's self-organization. This week's soup is a case in point.
- With the same parameters as a year ago, and starting from a symmetric
random distribution of vacant (0) and occupied (1) cells, the process quickly
relaxed to all 0's except for three seeds that symmetrized to form the
'crystal' centers of the red, nucleating target patterns. By about time
150 almost all of the space had fixated, either early on in concentric
waves of alternating 0's and 1's emanating from the seeds, or later (the greens and
yellows) in a more complex mix trying to fill in the gaps between the
waves. Curiously, though, once the residual pockets of activity are
sufficiently small they do not fill in with a final, stable arrangement.
Instead they continue to fluctuate, some for more than a thousand updates,
before locking into periodic cycles. The dark blue regions of our soup
represent occupied sites that continue to blink indefinitely. Using
WinCA, it is easy to check that they all settle into period 4
cycles despite their irregular shapes. We are led to conjecture that the
Batik regime of LtL space is non-uniformly locally periodic, meaning
that every cell eventually cycles deterministically, but different cells
cycle with different periods. Moreover, it seems that the period of every site
is either 1 or 4 (or perhaps 2 for a smattering of interfacial cells).
Over the next month or so we'll be cooking up additional LtL tidbits such as
pretzels, waffles, and fat bugs with stomachs. So drop on by
whenever you're hungry...
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