Invention versus discovery in mathematics One generally considers things invented if they are created in a somewhat arbitrary way, and discovered if they are identified by what seems like a more inexorable process.

Ordering of [mathematical] constructs One can deduce some kind of ordering among standard mathematical constructs by seeing how difficult they are to implement in various systems—such as cellular automata, Turing machines and Diophantine equations.

(One can also construct an infinite tree from a general network by following all its possible paths, as on page 277 , but in most cases there will be no simple way to apply symbolic system rules to such a tree.)

Lyapunov exponents If one thinks of cells to the right of a point in a 1D cellular automaton as being like digits in a real number, then linear growth in the region of differences associated with a change further to the right is analogous to the exponentially sensitive dependence on initial conditions shown on page 155 .

Hamming distances [in networks] In the so-called loop switching method of routing messages in communications systems one lays out a network on an m -dimensional Boolean hypercube so that the distance on the hypercube (equal to Hamming distance) agrees with distance in the network.

But in the late 1930s Gerhard Gentzen showed that if proofs are instead encoded as ordinal numbers (see note above ) then any proof can validly be reduced to a preceding one just by operations in logic. … And from the unprovability of consistency one can conclude that this must be impossible using the ordinary operation of induction in Peano arithmetic. (Set theory, however, allows transfinite induction—essentially induction on arbitrary sets—letting one reach such ordinals and thus prove the consistency of arithmetic.)

Searching for logic [axioms] For axiom systems of the form {…  a} one finds: {((b ∘ b) ∘ a) ∘ (a ∘ b)  a} allows the k = 3 operator 15552 for which the Nand theorem (p ∘ p) ∘ q  (p ∘ q) ∘ q is not true. … If one adds a ∘ b  b ∘ a to any of the other 23 axioms above then in all cases the resulting axiom system can be shown to reproduce logic. … It has been known since the 1940s that any axiom system for logic must have at least one axiom that involves more than 2 variables.

Higher-dimensional generalizations [of iterated maps] One can consider so-called Anosov maps such as {x, y}  Mod[m .

For rugs, it is typically desirable to have each cell correspond to more than one tuft, since otherwise with most rules the rug looks too busy.

With a background consisting of repetitions of the block , insertion of a single initial white cell yields a largely random pattern that expands by one cell per step.