Additive [continuous cellular automaton] rules

In the case a=0 the systems on page 159 are purely additive. A simpler example is the rule

Mod[RotateLeft[list] + RotateRight[list], 1]

With a single nonzero initial cell with value 1/k the pattern produced is just Pascal's triangle modulo k. If k is a rational number only a limited set of values appear, and the pattern has a nested form analogous to those shown on page 870. If k is irrational then equidistribution of Mod[Binomial[t, x], k] implies that all possible values eventually appear; the corresponding patterns seem fairly irregular, as shown below. (Compare pages 953 and 1092.)