Cellular Automata/Neighborhood

1D neighborhoodEdit

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

Radial neighborhoodEdit

 
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 notationEdit

In Wolframs's texts and many others the number of available cell states   and the radius   are combined into a pair

 
See also

Brickwall neighborhoodEdit

 
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 neighborhoodEdit

von Neumann neighborhoodEdit

 
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

Moore neighborhoodEdit

 
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

Margolus neighborhoodEdit

reversible

see also [1]

Unaligned rectangular neighborhoodEdit

 
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 neighborhoodEdit

Hexagonal neighborhoodEdit

 
Hexagonal neighborhood
Symmetries
  • reflection symmetry
  • rotation symmetry 6-fold


Small unaligned hexagonal neighborhoodEdit

 
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


ReferencesEdit