After a large number of steps t , the number of distinct positions visited will be proportional to t , at least above 2 dimensions (in 2D, it is proportional to t/Log[t] and in 1D √ t ). … The picture below, for example, shows the so-called extreme value distribution of positions furthest from the origin reached after 10 steps and 100 steps by random walks on various lattices. In the pictures in the main text, all particles start out at a particular position, and progressively spread out from there.

The number of steps for which a cell at position n will survive can be computed as Module[{q = n + k - 1, s = 1}, While[Mod[q, k] ≠ 0, q = Ceiling[(k - 1)q/k]; s++]; s] If a cell is going to survive for s steps, then it turns out that this can be determined by looking at the last s digits in the base k representation of its position.

Specially constructed transcendental numbers Numbers known to be transcendental include ones whose digit sequences contain 1's only at positions n!

Connection [of 2D substitution systems] with digit sequences Just as in the 1D case discussed on page 891 , the color of a cell at position {i, j} in a 2D substitution system can be determined using a finite automaton from the digit sequences of the numbers i and j . … Note that the excluded pairs of digits are in exact correspondence with the positions of which squares are 0 in the underlying rules for the substitution systems.

But their positions depend on the details of the initial conditions given, and in many cases the final arrangement of structures can be thought of as a kind of filtered version of the initial conditions.

[No text on this page] Examples of patterns set up so that a short computation can be used to determine the color of each cell from the numbers representing its position.

The basic reason for this is that as shown on page 84 the evolution of the substitution system always yields a tree, and the successive digits in n determine which branch is taken at each level in order to reach the element at position n . … In terms of this rule, the color of the element at position n is given by Fold[Replace[{#1, #2}, rule]&, 1, IntegerDigits[n - 1, 2]] The rule used here can be thought of as a finite automaton with two states. … It then turns out that if one expresses the position n as a generalized digit sequence of this kind, then the color of the corresponding element in substitution system (c) is just the last digit in this sequence.

Visualization [of 2D Turing machines] The pictures below show the 2D position of the head at 500 successive steps for the rules on page 185 .

In the 1700s and 1800s the idea of position and time as just two coordinates was widespread in mathematical physics—and this then led to notions like "travelling in time" in H. … For typically space and time are both just represented by abstract symbolic variables, and the formal process of solving equations as a function of position in space and as a function of time is essentially identical.

Counter [Turing] machine Turing machine (f) operates like a base 2 counter: at steps where its head is at the leftmost position, the colors of the cells correspond to the reverse of the base 2 digit sequences of successive numbers.