Chapter 4: Systems Based on Numbers

Section 8: Continuous Cellular Automata

Properties [of addition continuous cellular automata]

At step t the background is FractionalPart[a t]. For rational a this always repeats, cycling through Denominator[a] possible values (compare page 255). In most patterns generated from initial conditions containing say a single black cell most cells whose values are not forced to be the same end up being at least slightly different—even in cases like a = 0.375. Note that in cases like a = 0.475 there is some trace of a pattern at every step—but it only becomes obvious when it makes values wrap around from 1 to 0. The pictures below show successive colors of (a) the background (compare page 950) and (b) the center cell for each a = n/500 from 0 to 1 for the systems on page 159. (Compare page 243.)

If a is not a rational number the background never repeats, but the main features of patterns obtained seem similar.

Image Source Notebooks:

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