Cellular Automata/Neighborhood

1D neighborhood
Since in 1D there are no shapes, the definition of the neighborhood is usually very simple.

Radial neighborhood


Usually the neighborhood in 1D is described by its radius $$r$$, 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
 * $$ N = \{-r,-(r-1),\dots,-1,0,1,\dots,r-1,r\}$$

or simply the neighborhood size $$k=2r+1$$ with the output cell at the center $$k_0=r$$.


 * Symmetries
 * reflection symmetry

Stephen Wolfram's notation
In Wolframs's texts and many others the number of available cell states $$|S|$$ and the radius $$r$$ are combined into a pair
 * $$ (|S|,r) \,$$


 * See also
 * Stephen Wolfram, Statistical Mechanics of Cellular Automata (1983)

Brickwall neighborhood


An unaligned neighborhood, usually the smallest possible $$k=2$$. The output cell is positioned at $$k_0=0.5$$ between the two cells of the neighborhood. It is usually processed by alternatively shifting the output cell between $$k_0=0$$ and $$k_0=1$$.



von Neumann neighborhood


It is the smallest symmetric 2D aligned neighborhood usually described by directions on the compass $$N=\{N,W,C,E,S\}$$ sometimes the central cell is omitted.


 * Formal definition

Formally the von Neumann neighborhood is the set of neighbors
 * $$ N = \lbrace \{0,-1\}, \{-1,0\}, \{0,0\}, \{+1,0\}, \{0,+1\} \rbrace $$

or a subset of the rectangular neighborhood size $$k_x=k_y=3$$ with the output cell at the center $$k_0x=k_0y=1$$.


 * Symmetries
 * reflection symmetry
 * rotation symmetry 4-fold


 * See also
 * [mathworld] - [von Neumann Neighborhood]

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 $$N=\{NW,N,NE,W,C,E,SW,S,SE\}$$ sometimes the central cell is omitted.


 * Formal definition

Formally the Moore neighborhood is the set of neighbors
 * $$ N = \lbrace

\{-1,-1\}, \{0,-1\}, \{1,-1\}, \{-1,0\}, \{0,0\},  \{+1,0\}, \{-1,+1\}, \{0,+1\},  \{1,+1\} \rbrace $$ or simply a square size $$k_x=k_y=3$$ with the output cell at the center $$k_0x=k_0y=1$$.


 * Symmetries
 * reflection symmetry
 * rotation symmetry 4-fold


 * See also
 * [mathworld] - [Moore Neighborhood]

Margolus neighborhood
reversible

see also

Unaligned rectangular neighborhood


An unaligned (brickwall) rectangular neighborhood, usually the smallest possible $$k_x=k_y=2$$. The output cell is positioned at $$k_{0x}=k_{0y}=0.5$$ between the four cells of the neighborhood. It is usually processed by alternatively shifting the output cell to $$k_{0x}=k_{0y}=0$$ and $$k_{0x}=k_{0y}=1$$.



Hexagonal neighborhood



 * Symmetries
 * reflection symmetry
 * rotation symmetry 6-fold



Small unaligned hexagonal neighborhood


Formally the small (3-cell) unaligned hexagonal neighborhood represented on a rectangular lattice is the set of neighbors
 * Formal definition
 * $$ N = \lbrace \{0,0\}, \{1,0\}, \{0,1\} \rbrace $$


 * Symmetries
 * reflection symmetry
 * rotation symmetry 3-fold

