In the past century, the emphasis has been on physical constants such as the fine structure constant α ≃ 1/137.0359896 , and usually the idea is that such constants arise directly from counting objects of some specified type using traditional discrete mathematics.

(An example is NestList[Mod[2 #, 1]&, N[ π /4, 40], 200] ; Map[Precision, list] gives the number of significant digits of each element in the list.) … As an example, consider the iterated map x  Mod[2x, 1] discussed in the main text. … (Base 10 is used so that multiplying for example 1/3 by 3 gives exactly 1 rather than the more confusing result 0.9999... obtained with base 2.)

Tag systems [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 , the following will construct a tag system which emulates it: CAToTS[rules_] := {2, {{s[x_], s[y_]}  {d[x, y], d[x, y]}, {d[w_, x_], d[y_, z_]}  {s[{w, x, y} /. rules], s[{x, y, z} /. rules]}, {s[x_], d[y_, z_]}  {s[0], s[0], s[{0, y, z} /. rules]}, {d[x_, y_], s[z_]}  {s[{x, y, 0} /. rules], s[0], s[0]}}} The initial condition for the tag system that corresponds to a single black cell in the cellular automaton is {s[0], s[0], s[1], s[0], s[0]} .

Note that in terms of the ∘ operator 1  a ∘ a , b  (a ∘ a) ∘ b , and a · b  a ∘ ((a ∘ a) ∘ b) .

Simple examples in Mathematica include: First[Prepend[p, q]] === q Join[Join[p, q], r] === Join[p, Join[q, r]] Partition[Union[p], 1] === Split[Union[p]] One can set up axiom systems say by combining definitions of programming languages with predicate logic (as done by John McCarthy for Lisp in 1963).

(c) has an outer envelope whose edges grow at rates {-1.5, 0.3} √ t .

The overall density of black cells is not 1/2 but is instead approximately 0.35, just as for random initial conditions.

(The simple result (n + 1) n k based on the number of ways to connect up n nodes is a significant overestimate because of equivalence between automata with different patterns of connections.)

The state with no black cells always forms a cycle of length 1.

The width of the region of roughness on the surface of each cluster varies with the radius of the cluster approximately like r 1/3 .