# Cellular Automata/Neighborhood

## 1D neighborhood[edit | edit source]

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

### Radial neighborhood[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

### von Neumann neighborhood[edit | edit source]

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 neighborhood[edit | edit source]

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[edit | edit source]

reversible

see also [1]

### Unaligned rectangular neighborhood[edit | edit source]

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[edit | edit source]

### Hexagonal neighborhood[edit | edit source]

- Symmetries

- reflection symmetry
- rotation symmetry 6-fold

### Small unaligned hexagonal neighborhood[edit | edit source]

- 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[edit | edit source]

- [mathworld] - [Neighborhood]