INTERACTING CELLULAR AUTOMATA & KÅHRE'S CONJECTURE

The interaction or collision between two systems governed by different rules is common in nature. For illustration, we use a simple experiment of two interacting cellular automata rules. Let one rule be NKS rule 16. Starting from one black element, it creates a fragile regular pyramid of nested triangles (NKS page 55). Let it interact with a vertical line of black elements representing a wall or a mirror. The pyramid undergoes a dramatic change of behavior (ranging through all Wolfram classes 1 to 4) depending on the position of the wall in respect to the centerline of the pyramide.

Left side blanked out, right side regular strips.

Left side completely regular, right side random pattern partly with nested triangles, rightmost border regular.

A mixture of regular and random patterns.

Random nested triangles, some vertically striped.

An important class of interactions between systems in the real world can be described as solutions of an isoperimetric problem. The prototype of an isometric problem is to maximize volume given surface size. Here, the interaction is between the volume and the limiting surface, exemplified by a soap bubble or a quantum wave function (as described in my NKS poster). In contrast e.g. a snowflake is modelled by free expansion and a circular form appears only as an effect of rule symmetry. I have not, however, seen any CA experiment even remotely reminding of a solution to an isoperimetric problem.

My conjecture is that cellular automata can represent snowflakes but not soap bubbles.