And so long as the boundaries of the regions do not get stuck—as happens in many one-dimensional cellular automata—the result is that whichever color was initially more common eventually takes over the whole system. A one-dimensional cellular automaton in which the density of black cells obtained after a large number of steps changes discretely when the initial density of black cells is continuously increased.

As an example, the first set of pictures below show how nested patterns with larger and larger features can be built up by starting with a single black cell, and then following simple additive cellular automaton rules. … Nested patterns built by the evolution of the rule 90 and rule 150 additive cellular automata starting from a single black cell.

Endpapers The goldenrod pages inside the front cover show the center 900 or so cells of the first 500 or so steps in the evolution of the rule 30 cellular automaton of page 29 from a single black cell.

Difference patterns [in cellular automata] The maximum rate at which a region of change can grow is determined by the range of the underlying cellular automaton rule. … In 2D class 3 cellular automata, the region of change usually ends up having a roughly circular shape—a result presumably related to the Central Limit Theorem (see page 976 ). For any additive or partially additive class 3 cellular automaton (such as rule 90 or rule 30) any change in initial conditions will always lead to expanding differences.

Diffusion-limited aggregation (DLA) DLA is a model for a variety of natural growth processes that was invented by Thomas Witten and Leonard Sander in 1981, and which at first seems quite different from a cellular automaton. … To construct a cellular automaton analog of DLA one can introduce gray as well as black and white cells, and then have the gray cells represent pieces of solid that have not yet become permanently attached to the main cluster. … No doubt there are also simpler cellular automaton rules that yield similar results.

In base 6, (3/2) n is a cellular automaton with rule {a_, b_, c_}  3 Mod[a + Quotient[b, 2], 2] + Quotient[3 Mod[b, 2] + Quotient[c, 2], 2] (Note that this rule is invertible.) Looking at u (3/2) n then corresponds to studying the cellular automaton with an initial condition given by the base 6 digits of u .

Game of Life [cellular automaton] Invented by John Conway around 1970 (see page 877 ), the Life 2D cellular automaton has been much studied in recreational computing, and as described on page 964 many localized structures in it have been identified.

Excluded blocks [in cellular automaton evolution] As the evolution of a cellular automaton proceeds, the set of sequences that can appear typically shrinks, with progressively more blocks being excluded.

Cellular automaton rules as formulas The value a[t, i] for a cell on step t at position i in any of the cellular automata in this chapter can be obtained from the definition a[t_, i_] := f[a[t - 1, i - 1], a[t - 1, i], a[t - 1, i + 1]] Different rules correspond to different choices of the function f . … And in this case cellular automaton rules become logic expressions: • Rule 254: Or[p, q, r] • Rule 250: Or[p, r] • Rule 90: Xor[p, r] • Rule 30: Xor[p, Or[q, r]] • Rule 110: Xor[Or[p, q], And[p, q, r]] (Note that Not[p] corresponds to 1 - p , And[p, q] to p q , Xor[p, q] to Mod[p + q, 2] and Or[p, q] to Mod[p q + p + q, 2] .) Given either the algebraic or logical form of a cellular automaton rule, it is possible at least in principle to generate symbolic formulas for the results of cellular automaton evolution.

A simple cellular automaton system set up to emulate the microscopic behavior of molecules in a fluid.