# 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 $k_{0}=1$.

## 2D neighborhood
### 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 $k_{0}x=k_{0}y=1$.

#### 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 $k_{0}x=k_{0}y=1$.

#### 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
### Hexagonal neighborhood
#### Symmetries

* reflection symmetry
* rotation symmetry 6-fold

### Small unaligned hexagonal neighborhood
#### Formal definition

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

#### Symmetries

* reflection symmetry
* rotation symmetry 3-fold

## References

* [mathworld] - [Neighborhood]