Two-Dimensional Cellular Automata from One-Dimensional Rules

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Some nontrivial two-dimensional (2D) cellular automata (CA) are reproduced with one-dimensional (1D) CA rules. Thus a large subclass of 2D CA can be conveniently labeled through the standard Wolfram indexing of 1D CA. One step of time evolution of a 2D CA is obtained in two stages. First, apply a 1D rule to all the ROWS of a 2D initial condition. Then to the result apply another 1D rule, but now to all the COLUMNS. Here this procedure is repeated 500 times, generating the evolution of a random 2D initial condition for 500 steps. This can be mapped onto the subclass of

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two-color 2D CA with a nine cell square neighborhood. It is useful to devise smaller subclasses of the general case to manage classification of such a large number of CA. Examples of such subclasses are totalistic CA, outer-totalistic CA, and the new subclass demonstrated here. Browse and animate bookmarks for interesting patterns. See the Details section for more a extensive explanation of the controls and images.