Cellular Automata/Neighborhood

1D neighborhood Since in 1D there are no shapes, the definition of the neighborhood is usually very simple. Radial neighborhood thumb|150px|right|Radial neighborhood Usually the neighborhood in 1D is described by its radius , meaning the number of cell left and right from the central cell that are used for the neighborhood. The output cell is positioned at the center. Formal definition Formally the radial neighborhood is the set of neighbors or simply the neighborhood size with the output cell at the center . Symmetries reflection symmetry Stephen Wolfram's notation In Wolframs's texts and many others the number of available cell states and the radius are combined into a pair See also Stephen Wolfram, Statistical Mechanics of Cellular Automata (1983) Brickwall neighborhood thumb|150px|right|Brickwall neighborhood An unaligned neighborhood, usually the smallest possible . The output cell is positioned at between the two cells of the neighborhood. It is usually processed by alternatively shifting the output cell between and . 2D neighborhood von Neumann neighborhood thumb|150px|right|von Neumann neighborhood It is the smallest symmetric 2D aligned neighborhood usually described by directions on the compass sometimes the central cell is omitted. Formal definition Formally the von Neumann neighborhood is the set of neighbors or a subset of the rectangular neighborhood size with the output cell at the center . Symmetries reflection symmetry rotation symmetry 4-fold See also [mathworld] - [von Neumann Neighborhood] Moore neighborhood thumb|150px|right|Moore neighborhood Is a simple square (usually 3×3 cells) with the output cell in the center. Usually cells in the neighborhood are described by directions on the compass sometimes the central cell is omitted. Formal definition Formally the Moore neighborhood is the set of neighbors or simply a square size with the output cell at the center . Symmetries reflection symmetry rotation symmetry 4-fold See also [mathworld] - [Moore Neighborhood] Margolus neighborhood reversible see also Unaligned rectangular neighborhood thumb|150px|right|Unaligned rectangular neighborhood An unaligned (brickwall) rectangular neighborhood, usually the smallest possible . The output cell is positioned at between the four cells of the neighborhood. It is usually processed by alternatively shifting the output cell to and . Hexagonal neighborhood Hexagonal neighborhood thumb|150px|right|Hexagonal neighborhood Symmetries reflection symmetry rotation symmetry 6-fold Small unaligned hexagonal neighborhood thumb|150px|right|Unaligned hexagonal neighborhood Formal definition Formally the small (3-cell) unaligned hexagonal neighborhood represented on a rectangular lattice is the set of neighbors Symmetries reflection symmetry rotation symmetry 3-fold References [mathworld] - [Neighborhood]