The Cook Book
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 nonmonotone 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 halfsize 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 nonmonotone CA rules to
generate polygonal shapes was noted by Packard and Wolfram in their pioneering
paper "Twodimensional cellular automata" J. Stat. Phys. 38
(1985), 901946. Perhaps the most interesting case is n = 5, an instance of
the socalled 'Vichniac twist' that induces pseudostochastic fluctuations
at the boundary. Is the limiting shape here a polygon, or is it smooth?
 Especially for the Vichniactype experiment, a larger array size is clearly
needed to reach any compelling empirical conclusions. We intend to carry out
extensive CAM8based 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 ...
