The particular models I used were based on mobile automata—in which the presence of a single active cell forces only one event ever to occur in the universe at once.

base 2, the presence of carry digits in the multiplication process makes the system not quite an ordinary cellular automaton. It turns out, however, that multiplication by 3 in base 6, or by 2 or 5 in base 10, never leads to carry digits, with the result that in such cases the system can be thought of as following a purely local cellular automaton rule of the kind illustrated in the second set of pictures. … Note that only certain specific choices of base and multiplier lead to ordinary cellular automata; with other choices there are carries that propagate arbitrarily far.

The third rule yields behavior that appears to be computationally irreducible, so that its outcome can effectively be found only by explicitly tracing each step.

And presumably the only way is to have another system that already violates the Principle of Computational Equivalence.

Relativism and equivalence Although the notion has been discussed since antiquity, it has become particularly common in the academic humanities in the past few decades to believe that there can be no valid absolute conclusions about the world—only statements made relative to particular cultural contexts.

Often these can be thought of as one-way versions of axioms for operator systems (see page 1172 ), but applied only once per step (as /. does), rather than in all possible ways (as in a multiway system)—so that the evolution is just given by NestList[#/.rule &, init, t] .

Typically they imagine that while the pictures may look complicated, they would actually seem simple if only they were subjected to the appropriate kind of analysis.

Additive [continuous cellular automaton] rules In the case a = 0 the systems on page 159 are purely additive. … 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 .

Regular polytopes In 3D, of the five regular polyhedra, only the tetrahedron, cube and dodecahedron have three edges meeting at each vertex, corresponding to a trivalent network. … In 4D the six regular polytopes have 4, 4, 6, 8, 4 and 12 edges meeting at each vertex, and in higher dimensions the simplex ( d + 1 vertices) and hypercube ( 2 d vertices) have d edges meeting at each vertex, while the co-cube ( 2d vertices) has 2(d - 1) .

Continuous Cellular Automata Despite all their differences, the various kinds of programs discussed in the previous chapter have one thing in common: they are all based on elements that can take on only a discrete set of possible forms, typically just colors black and white. … Case (c) does not show such sensitivity to initial conditions, but instead always evolves to 0, independent of its initial conditions.