The Cook Book
Recipe for the week of November 13 - 19
More on NonMonotone CA Growth
- The shape theory for monotone CA growth rules is rather well understood
at this point. See the paper referenced in our recipe from the week of
July 31 for
most of the details. To summarize the highlights, for threshold growth on box
neighborhoods, either an initial seed stops growing at some point because it was too
small, or the occupied set A(t) spreads out linearly in time t and
A(t)/t attains a limiting shape L. Moreover,
- (i) L is independent of the initial seed from which it evolved;
- (ii) L is always convex;
- (iii) L is always a polygon.
Actually, this synopsis uses a new and very impressive argument by T. Bohman
to show that A(t) must fill the lattice if it keeps growing. We
intend to say more about Bohman's result in a future recipe.
- For the sake of comparison, and following up on last week's discussion,
let us consider some more examples of range 1 Box non-monotone CA growth. In
this week's rule, an unoccupied site becomes occupied if it has either 2 or
5 occupied neighbors, and remains occupied thereafter. Starting from a 21-cell seed
which consists of a 5 by 5 box with the 4 corner sites removed, one can
check that the convex octagon pictured in the thumbnail above emerges.
Click on the octagon to see this week's soup, obtained by running the same
rule from an initial 'radius 14' lattice ball. The limiting shape is not
clear, nor is it more apparent when the experiment is run almost twice as long:
radius 14, 1024 by 800 (209 kb)
But there is at least a suggestion that the limiting shape is not convex.
Additional seeds confirm that properties (i) and (ii) need not hold
for non-monotone growth: the limiting shapes are readily seen to be non-
convex polygons. Check out these rather elegant crystals, each with sides of
its own distinct slopes:
radius 9 (71 kb)
radius 12 (47 kb)
radius 13 (67 kb)
- The question of whether L can have smooth boundary, raised last
week, is of course more delicate. We have begun CAM8 experiments on 8K by
8K arrays to investigate. More later...