On step t the color of a cell at position x is given by b 〚 Mod[x + 4 t, 14] + 1 〛 .

But division by 2 just does the opposite of multiplication by 2, so in base 2 it simply shifts all digits one position to the right.

In addition, instead of modelling the displacement of atoms, one can try to model directly the presence or absence of atoms at particular positions.

But other criteria can equally well be used—say the head reaching a particular position (see page 759 ), or a certain pattern of colors being formed on the tape. And in a system like a cellular automaton a halting problem can be set up by asking whether a cell at a particular position ever turns a particular color, or whether, more globally, the complete state of the system ever reaches a fixed point and no longer changes.

Identifying the 171 patterns [that satisfy 2D constraints] The number of constraints to consider can be reduced by symmetries, by discarding sets of templates that are supersets of ones already known to be satisfiable, and by requiring that each template in the set be compatible with itself or with at least one other in each of the eight immediately adjacent positions.

Shapes of [biological] cells Many types of cells are arranged like typical 3D packings of deformable objects (see page 988 )—with considerable apparent randomness in individual shapes and positions, but definite overall statistical properties.

Generating causal networks If every element generated in the evolution of a generalized substitution system is assigned a unique number, then events can be represented for example by {4, 5}  {11, 12, 13} —and from a list of such events a causal network can be built up using With[{u = Map[First, list]}, MapIndexed[Function[ {e, i}, First[i]  Map[(If[# === {}, ∞ , # 〚 1, 1 〛 ] &)[ Position[u, #]]) &, Last[e]]], list]]

The presence of this nested structure is an inevitable consequence of the fact that the rule for replacing an element at a particular position does not depend in any way on other elements.

The picture below shows the fluctuations around m/2 of the cumulative number of 1's up to position m in the sequence obtained at step 1000.

However, if one uses the function to generate a score—say playing a note at the position of each peak—then no such simplicity can be recognized.