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Recipe for the week of November 6 - 12

Shape Experiments with NonMonotone CA Growth

Following up on last week's soup, let us look at some CA dynamics that seem to interpolate between Threshold Growth and random systems such as Richardson's Model. Namely, we consider non-monotone deterministic range 1 Box growth models in which a site becomes occupied if either exactly 3 or exactly n neighboring sites are occupied, and occupied sites remain so forever. The case n = 4 generates the same octagonal shape as Threshold Growth with threshold 3, so we examine the more interesting rules n = 5,6,7,8. In each of these cases we let the occupied region grow until it reaches the edge of a 1024 by 800 array in order to get an initial impression of the limiting shape. To enhance visualization we display each successive update in a new color from a rainbow palette. This week's soup is the case n = 4. The complete set, reduced to half-size and original size, are:

3or5 (101 kb) - - 3or5 big (364 kb)

3or6 (137 kb) - - 3or6 big (477 kb)

3or7 (114 kb) - - 3or7 big (413 kb)

3or8 (077 kb) - - 3or8 big (280 kb)

Summarizing the results in reverse order, note first that n = 8 reduces to the Life without Death rule that has previously been featured here in the Kitchen on several occasions. This week's soup and the n = 6 case seem to indicate limiting shapes that are polygonal, an octagon and a square, respectively. The ability of complex non-monotone CA rules to generate polygonal shapes was noted by Packard and Wolfram in their pioneering paper "Two-dimensional cellular automata" J. Stat. Phys. 38 (1985), 901-946. Perhaps the most interesting case is n = 5, an instance of the so-called 'Vichniac twist' that induces pseudo-stochastic fluctuations at the boundary. Is the limiting shape here a polygon, or is it smooth?

Especially for the Vichniac-type experiment, a larger array size is clearly needed to reach any compelling empirical conclusions. We intend to carry out extensive CAM8-based simulations on very large arrays (up to 16K by 16K) in an attempt to shed some light on such delicate issues. With experiments of that size we suspect that one will be able to make educated guesses as to anisotropy and the emergence of corners. Moreover, one hopes for a glimmer of the hydrodynamic mechanism that destabilizes corners in the presence of certain nonlinear effects. As usual, any interesting discoveries will be described in future recipes ...

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