The Cook Book

Recipe for the week of April 29 - May 5

Period 4 Droplets in LtL Batik

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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