CellularAutomaton
 Usage
• CellularAutomaton[rnum, init, t] generates a list representing the evolution of cellular automaton rule rnum from initial condition init for t steps.
• CellularAutomaton[rnum, init, t, { ,  , ... }] keeps only the parts of the evolution list with the specified offsets.
Notes
• Possible settings for rnum are:
| n |  ,  , elementary rule | | {n, k} | general nearest-neighbor rule with  colors | | {n, k, r} | general rule with  colors and range  | {n, k, { ,  , ... ,  }} |  -dimensional rule with  neighborhood | {n, k, {{ }, { }, ... , { }}} | rule with neighbors at specified offsets | | {n, {k, 1}} |  -color nearest-neighbor totalistic rule | | {n, {k, 1}, r} |  -color range  totalistic rule | {n, {k, { ,  , ... }}, rspec} | rule in which neighbor  is assigned weight  | | {fun, {}, rspec} | applies the function fun to each list of neighbors, with a second argument | | of the step number |
• CellularAutomaton[{n, k}, ... ] is equivalent to CellularAutomaton[{n, {k, {k^2, k, 1}}}, ... ]. • Common forms for 2D cellular automata include:
| {n, {k, 1}, {1, 1}} | 9-neighbor totalistic rule | | {n, {k, {{0, 1, 0}, {1, 1, 1}, {0, 1, 0}}}, {1, 1}} | 5-neighbor totalistic rule | | {n, {k, {{0, k, 0}, {k, 1, k}, {0, k, 0}}}, {1, 1}} | 5-neighbor outer totalistic rule |
• Normally, all elements in init and the evolution list are integers between 0 and  . • But when a general function is used, the elements of init and the evolution list do not have to be integers. • The second argument passed to fun is the step number, starting at 0. • Initial conditions are constructed from init as follows:
{ ,  , ... } | explicit list of values  , assumed cyclic | {{ ,  , ... }, b} | values  superimposed on a b background | {{ ,  , ... }, { ,  , ... }} | values  superimposed on a background of | | repetitions of  ,  ,  |
{{{{ ,  , ... },  }, {{ , ... },  }, ... }, bspec} | values  at offsets  on a background | {{ ,  , ... }, { , ... }, ... } | explicit list of values in two dimensions | | {aspec, bspec} | values in  dimensions with  -dimensional padding |
• The first element of aspec is superimposed on the background at the first position in the positive direction in each coordinate relative to the origin. This means that bspec[[1, 1, ... ]] is aligned with aspec[[1, 1, ... ]]. • Time offsets  are specified as follows:
| All | all steps  through  | | u | steps 0 through  | | -1 | last step (step  ) | | {u} | step  | { ,  } | steps  through  | { ,  , du} | steps  ,  +  ,  |
• CellularAutomaton[rnum, init, t] generates an evolution list of length  . • The initial condition is taken to have offset 0. • Space offsets  are specified as follows:
| All | all cells that can be affected by the specified initial condition | | Automatic | all cells in the region that differs from the background | | 0 | cell aligned with beginning of aspec | | x | cells at offsets up to  on the right | | -x | cells at offsets up to  on the left | | {x} | cell at offset  to the right | | {-x} | cell at offset  to the left | { ,  } | cells at offsets  through  | { ,  , dx} | cells  ,  +  ,  |
• In one dimension, the first element of aspec is taken by default to have space offset 0. • In any number of dimensions, aspec[[1, 1, 1, ... ]] is taken by default to have space offset {0, 0, 0, ... }. • Each element of the evolution list produced by CellularAutomaton is always the same size. • With an initial condition specified by an aspec of width  , the region that can be affected after  steps by a cellular automaton with a rule of range  has width  . • If no bspec background is specified, space offsets of All and Automatic will include every cell in aspec. • A space offset of All includes all cells that can be affected by the initial condition. • A space offset of Automatic can be used to trim off background from the sides of a cellular automaton pattern. • In working out how wide a region to keep, Automatic only looks at results on steps specified by  . • New in Version 4.2.
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