Think Complexity, 2nd Edition

Chapter 7. Physical Modeling

The cellular automatons we have seen so far are not physical models; that is, they are not intended to describe systems in the real world. But some CAs are intended as physical models.

In this chapter we consider a CA that models chemicals that diffuse (spread out) and react with each other, which is a process Alan Turing proposed to explain how some animal patterns develop.

And we’ll experiment with a CA that models percolation of liquid through porous material, like water through coffee grounds. This model is the first of several models that exhibit phase change behavior and fractal geometry, and I’ll explain what both of those mean.

Diffusion

In 1952 Alan Turing published a paper called “The chemical basis of morphogenesis”, which describes the behavior of systems involving two chemicals that diffuse in space and react with each other. He showed that these systems produce a wide range of patterns, depending on the diffusion and reaction rates, and conjectured that systems like this might be an important mechanism in biological growth processes, particularly the development of animal coloration patterns.

Turing’s model is based on differential equations, but it can be implemented using a cellular automaton.

Before we get to Turing’s model, we’ll start with something simpler: a diffusion system with just one chemical. We’ll use a 2-D CA where the state of each cell is a continuous quantity (usually between 0 and 1) that represents the concentration of ...