Moore neighbourhood

The Moore neighbourhood (in green) of a single cell

The Moore neighbourhood is the set of all cells that are orthogonally or diagonally-adjacent to the region of interest (the region of interest itself may or may not be considered part of the Moore neighbourhood, depending on context). For example, the Moore neighbourhood of a single cell consists of the eight cells immediately surrounding it. This neighbourhood is named after Edward F. Moore, one of the pioneers of cellular automata theory. The Moore neighborhood is the neighbourhood of interest in Conway's Game of Life and all Life-like cellular automata, though there are cellular automata that use other neighbourhoods such as the 4-cell von Neumann neighborhood.

The Moore neighbourhood naturally extends to cellular automata in higher dimensions, for example forming a 26-cell cubic neighborhood for a cellular automaton in three dimensions. The number of cells in the Moore neighbourhood of a single cell in an n-dimensional cellular automaton is 3ⁿ-1 (Sloane's  A024023).

The Moore neighbourhood of a cell can be thought of as the points at a Chebyshev distance of 1 from that cell.

Generalizations

Main article: Isotropic non-totalistic Life-like cellular automaton

Like in the hexagonal neighborhood, isotropic cellular automata using the Moore neighbourhood can be defined using Hensel notation, which was devised by Alan Hensel and represents the relative permutations of the cells using letters.

	0	1	2	3	4	5	6	7	8
— (no letter)	File:Neighborhood 0.png								File:Neighborhood 8.png
c (corner)			File:Neighborhood 2c.png	File:Neighborhood 3c.png	File:Neighborhood 4c.png	File:Neighborhood 5c.png	File:Neighborhood 6c.png	File:Neighborhood 7c.png
e (edge)		File:Neighborhood 1e.png	File:Neighborhood 2e.png	File:Neighborhood 3e.png	File:Neighborhood 4e.png	File:Neighborhood 5e.png	File:Neighborhood 6e.png	File:Neighborhood 7e.png
k (knight)			File:Neighborhood 2k.png	File:Neighborhood 3k.png	File:Neighborhood 4k.png	File:Neighborhood 5k.png	File:Neighborhood 6k.png
a (adjacent)			File:Neighborhood 2a.png	File:Neighborhood 3a.png	File:Neighborhood 4a.png	File:Neighborhood 5a.png	File:Neighborhood 6a.png
i			File:Neighborhood 2i.png	File:Neighborhood 3i.png	File:Neighborhood 4i.png	File:Neighborhood 5i.png	File:Neighborhood 6i.png
n			File:Neighborhood 2n.png	File:Neighborhood 3n.png	File:Neighborhood 4n.png	File:Neighborhood 5n.png	File:Neighborhood 6n.png
y				File:Neighborhood 3y.png	File:Neighborhood 4y.png	File:Neighborhood 5y.png
q				File:Neighborhood 3q.png	File:Neighborhood 4q.png	File:Neighborhood 5q.png
j				File:Neighborhood 3j.png	File:Neighborhood 4j.png	File:Neighborhood 5j.png
r				File:Neighborhood 3r.png	File:Neighborhood 4r.png	File:Neighborhood 5r.png
t					File:Neighborhood 4t.png
w					File:Neighborhood 4w.png
z					File:Neighborhood 4z.png

For instance, B2-a/S12 (the Just Friends rule) indicates that a dead cell will be born with 2 neighbors, except when they are adjacent, and that a live cell will survive with 1 or 2 neighbors in any configuration.

Higher ranges

The Moore neighbourhood can also be defined with a higher range; that is, so that it captures cells that are further than one cell away from the region of interest. The standard Moore neighbourhood has range 1. The Moore neighbourhood of range 2 is the set of all cells that are orthogonally or diagonally-adjacent to the Moore neighbourhood itself. The Moore neighbourhood of range n can be defined recursively as the set of all cells that are orthogonally or diagonally-adjacent to the Moore neighbourhood of range n-1. The number of cells in the Moore neighbourhood of range n is given by (2n+1)²-1 (Sloane's  A033996).

Symmetries

Main article: Symmetry

The Moore and von Neumann neighbourhoods rely on a different grid than the hexagonal neighbourhood and thus features a different set of inherent symmetries when dealing with isotropic rules:

• C2_1
• C2_2
• C2_4
• C4_1
• C4_4
• D2_+1
• D2_+2
• D2_x
• D4_+1
• D4_+2
• D4_+4
• D4_x1
• D4_x4
• D8_1
• D8_2 (only occasionally preserved)
• D8_4

Image gallery

The Moore neighbourhood (in green) of an eater 1

The Moore neighbourhood of range 2 of a single cell

The Moore neighbourhood of range 3 of a single cell

See also

• Neighbourhood
  • Hexagonal neighbourhood
  • Margolus neighbourhood
  • von Neumann neighbourhood
  • Triangular Moore neighbourhood
  • Triangular von Neumann neighbourhood
  • Euclidean neighbourhood
  • Circular neighbourhood
• Zone of influence

External links

• Moore neighborhood at Wikipedia

• Moore neighbourhood at the Life Lexicon

Moore neighborhood at Wolfram Mathworld