The Cook Book

Recipe for the week of March 13 - 19

Richardson's Growth Model

This week I'm on spring break, so the Kitchen is featuring an old chestnut soup, one of the first particle system color graphics I produced nearly ten years ago. [Next week we hope to offer a brand new image of critical nucleation in a threshold growth cellular automaton.]

Start with one painted dot at the center of the array, surrounded by a "sea" of unpainted cells. Identify each unpainted dot that 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 week's soup displays 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.

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