The pictures below show how a density perturbation leads to a sound wave in a cellular automaton fluid.

Note (f) for Continuous Cellular Automata…Implementation [of continuous cellular automata] The state of a continuous cellular automaton at a particular step can be represented by a list of numbers, each lying between 0 and 1.

Sequential cellular automata Ordinary cellular automata are set up so that every cell is updated in parallel at each step, based on the colors of neighboring cells on the previous step. … For each complete update of a rule 90 sequential cellular automaton, the pictures below show results with (a) left-to-right scan, (b) random ordering of all cells, the same for each pass through the whole system, (c) random ordering of all cells, different for different passes, (d) completely random ordering, in which a particular cell can be updated twice before other cells have even been updated once. … There were several studies of sequential or asynchronous cellular automata done following my work on ordinary cellular automata in the early 1980s.

The effect of changing the number of initial black cells in the rule 30 cellular automaton shown above.

Note (e) for Emulating Other Systems with Cellular Automata…RAM [emulated with cellular automata] The rules for the cellular automaton shown here are {{2, 4 | 8, 2 | 11, _, _}  2, {11 | 10, 4 | 8, 2 | 11, _, _}  11, {2, 4 | 8, _, _, _}  10, {11 | 10, 4 | 8, _, _, _}  2, {2, 0, _, _, _}  2, {11, 0, _, _, _}  11, {3 | 7 | 6, _, 10, _, _}  1, {x : (3 | 7 | 6), _, _, _, _}  x, {_, _, 6, 4, 10}  5, {_, _, 6, 8, 10}  9, {_, 3, _, 10, _}  4, {_, 7, _, 10, _}  8, {_, _, 1, _, x : (5 | 9)}  x, {1, _, _, _, _}  1, {_, _, 1, _, _}  1, {_, _, _, _, 1}  1, {_, _, x : (4 | 8 | 0), _, _}  x} The initial conditions are divided into two parts: instructions on the left and memory on the right.

[Universality in] 2D cellular automata Universality was essentially built in explicitly to the underlying rules for the 2D cellular automaton constructed by John von Neumann in 1952 as a model for self-reproduction. … The pictures below show how 1D cellular automata can be implemented in the 4-color WireWorld cellular automaton of Brian Silverman from 1987, whose rules find the new value of a cell from its old value a and the number u of its 8 neighbors that are 1's according to a /. {0  0, 1  2, 2  3, 3  If[0 < u < 3, 1, 3]}

Cover image The image on the cover of this book is derived from the first 440 or so steps (with perhaps 10 at each end cut off by trimming) of the pattern generated by evolution according to the rule 110 cellular automaton discussed on page 32 , with an initial condition consisting of repeats of followed by repeats of .

Any rational function is the generating function for some additive cellular automaton.

Note (a) for Cellular Automata…Symmetric 5-neighbor [2D cellular automaton] rules Among the 32 possible 5-cell neighborhoods shown for example on page 941 there are 12 classes related by symmetries, given by s = {{1}, {2, 3, 9, 17}, {4, 10, 19, 25}, {5}, {6, 7, 13, 21}, {8, 14, 23, 29}, {11, 18}, {12, 20, 26, 27}, {15, 22}, {16, 24, 30, 31}, {28}, {32}} Completely symmetric 5-neighbor rules can be numbered from 0 to 4095, with each digit specifying the new color of the cell for each of these symmetry classes of neighborhoods.

Identifying particles [in networks] In something like a class 4 cellular automaton it is quite straightforward to start enumerating possible persistent structures—as we saw in Chapter 6 .