Chapter 4: Systems Based on Numbers

Section 8: Continuous Cellular Automata

Implementation [of continuous cellular automata]

The state of a continuous cellular automaton at a particular step can be represented by a list of numbers, each lying between 0 and 1. This list can then be updated using

CCAEvolveStep[f_, list_List] := Map[f, (RotateLeft[list] + list + RotateRight[list])/3]

CCAEvolveList[f_, init_List, t_Integer] := NestList[CCAEvolveStep[f, #] &, init, t]

where for the rule on page 157 f is FractionalPart[3#/2] & while for the rule on page 158 it is FractionalPart[# + 1/4] &.

Note that in the definitions above, the elements of list can be either exact rational numbers, or approximate numbers obtained using N. For rough calculations, standard machine- precision numbers may sometimes suffice, but for detailed calculations exact rational numbers are essential. Indeed, the presence of exponentially increasing errors would make the bottom of the picture on page 157 qualitatively wrong if just 64-bit double-precision numbers had been used. On page 160 the effect is much larger, and almost all the pictures would be completely wrong—with the notable exception of the one that shows localized structures.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]