The Cook Book
Recipe for the week of October 16 - 22
Dynamic Formation of Poisson-Voronoi Tiles
- Here's an example of self-organization featured in my lecture at the
1995 Midwest Probability Colloquium this week. More than a half century
ago, Johnson and Mehl proposed various dynamic models for crystal formation
that combine Poisson-distributed nucleation centers, linear isotropic
growth of crystals, and "standoff" interaction at the boundary. Through
computer experimentation with WinCA, CAM8, and other simulations, we
have found large families of CA rules with the same phenomenology, except
that the norm for growth is given by the Threshold Growth Shape Theorem
(cf. last July
24).
- This week's soup depicts one such example. Start by coloring the lattice
randomly with 64 colors. At each step, the color k at site x
changes to a new color k' if it sees at least 4 sites of color k'
within its range 2 Diamond neighborhood for a unique k' ; otherwise there
is no change at x. The resulting evolution is sometimes called a
multitype threshold voter model. Our graphic shows the state of a
512 by 400 such system (with free boundary) after 100 updates. The model
has almost fixated, but not quite. Can you find the last remnants of
activity? Another puzzle: notice the small red dot in the middle of the blue
cluster at the far lower right. What happened there?
- To see this same trajectory earlier in the process of nucleation,
look at Time
40. Evidently the final design is some sort of random tiling. In
fact one can prove that the tile boundaries, suitably interpreted and rescaled,
tend toward a mathematical structure known as a Poisson-Voronoi tessellation
as the number of colors grows. In this instance the proper scaling turns out to
be (398)^.5 (# colors)^(-1.5). Details will appear in a forthcoming
research paper with Janko Gravner.
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