Square tiling: Difference between revisions
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{{Glossary}} | {{Glossary}} | ||
The '''square grid''', '''square tiling''' or '''square | The '''square grid''', '''square tiling''', '''square tessellation''' or '''square lattice''' is one of the three possible proper Euclidean tilings of 2D space, in which four squares are placed at each vertex. | ||
The square grid is the most familiar of all 2D tilings, with cell positions being easily and uniquely defined via Cartesian coordinates, and is the grid on which [[cellular automata]] are simulated most frequently. | The square grid is the most familiar of all 2D tilings, with cell positions being easily and uniquely defined via Cartesian coordinates, and is the grid on which [[cellular automata]] are simulated most frequently. | ||
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! saltire<br>''(trivial)'' | ! saltire<br>''(trivial)'' | ||
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Due to its simplicity, the von Neumann neighbourhood does not lend itself to particularly many interesting 2-state [[outer-totalistic]] rules; it does, however, see much more use in more complicated anisotropic rules, and outer-totalistic rules based on it can often be sighted in more complex rules; for example, 2D replicators tend to follow the rule B13/S13V to an extent. | |||
The Moore neighbourhood is the neighbourhood most commonly used in naturalistic rules, with many of the most famous rules such as [[Life]] and [[HighLife]] being based on it. Its larger size permits a much wider array of rules, many with their own interesting behaviours, in contrast to the more oppressive von Neumann neighbourhood, while not being so large as to include an unthinkably large amount of possibilities which makes finding interesting rules difficult. | |||
== Symmetries == | == Symmetries == | ||
Revision as of 13:50, 11 January 2022
The square grid, square tiling, square tessellation or square lattice is one of the three possible proper Euclidean tilings of 2D space, in which four squares are placed at each vertex.
The square grid is the most familiar of all 2D tilings, with cell positions being easily and uniquely defined via Cartesian coordinates, and is the grid on which cellular automata are simulated most frequently.
Neighbourhoods on the square grid
Range-1 cases
Most of the cellular automata which are investigated are isotropic and use neighbourhoods of range 1. These constraints permit two neighbourhoods of interest; the 4-cell von Neumann neighbourhood, consisting of the four cells orthogonally adjacent to a central cell, and the 8-cell Moore neighbourhood, consisting of the four cells orthogonally adjacent to a central cell as well as the four cells diagonally adjacent to said center cell. (A third neighbourhood, referred to on this wiki as the "saltire" neighbourhood, which consists only of the four cells diagonally adjacent to the central cell, functionally duplicates the von Neumann neighbourhood and is disregarded in this section for the sake of brevity.)
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Moore | saltire (trivial) |
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Due to its simplicity, the von Neumann neighbourhood does not lend itself to particularly many interesting 2-state outer-totalistic rules; it does, however, see much more use in more complicated anisotropic rules, and outer-totalistic rules based on it can often be sighted in more complex rules; for example, 2D replicators tend to follow the rule B13/S13V to an extent.
The Moore neighbourhood is the neighbourhood most commonly used in naturalistic rules, with many of the most famous rules such as Life and HighLife being based on it. Its larger size permits a much wider array of rules, many with their own interesting behaviours, in contrast to the more oppressive von Neumann neighbourhood, while not being so large as to include an unthinkably large amount of possibilities which makes finding interesting rules difficult.