The Cook Book

Recipe for the week of October 30 - November 5


The Primordial Soup Kitchen was one year old this week. To celebrate, I'm reprising from last spring one of the very first color computer graphics I ever produced, another image from my Particle Postcards set of ten years ago. Here is the recipe, as I described it as that time:


Start with one painted dot at the center of the screen, surrounded by a "sea" of unpainted cells. Identify each unpainted dot which has at least one painted neighbor, and flip a coin to decide whether it will be painted next time. Repeat this (synchronous) procedure over and over. Of course the painted region will spread randomly. D. Richardson proved a beautiful theorem in the mid 1970's which asserts that the painted region spreads at a constant speed and always tends to the same shape as it grows. If the chance of tossing heads is p (between 0 and 1), then different values of p give rise to different shapes. This postcard produces all the shapes simultaneously by using a spectrum of colors. The p=1 process has no randomness; its shape is the "outer diamond". A theorem of R. Durrett and T. Liggett, proved in 1981, says that the shape has flat pieces if p is close to 1; this is why the tips of the diamond have their own colors.

A mathematical technique known as subadditivity can be applied to various deterministic and random growth models in order to prove shape theorems, but gives very little information as to the geometry of the limit set . As has been noted here in the Kitchen, threshold growth dynamics make polygons, but it is widely thought that most nonlinear growth models should produce limiting shapes with smooth boundary. It is also believed that these shapes should rarely if ever be Euclidean circles due to a residual effect of the underlying lattice. To this day there is widespread interest among researchers in understanding the basic causes of anisotropy and boundary singularities.

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