- 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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