Von Neumann neighborhood

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Manhattan distance r = 1
Manhattan distance r = 2

In cellular automata, the von Neumann neighborhood comprises the four cells orthogonally surrounding a central cell on a two-dimensional square lattice. The neighborhood is named after John von Neumann, who used it for his pioneering cellular automata including the Universal Constructor. It is one of the two most commonly used neighborhood types, the other one being the 8-cell Moore neighborhood. It is similar to the notion of 4-connected pixels in computer graphics.

The concept can be extended to higher dimensions, for example forming a 6-cell octahedral neighborhood for a cubic cellular automaton in three dimensions.

The von Neumann neighbourhood of a point is the set of points at a Manhattan distance of 1.

von Neumann neighborhood of range r[edit]

An extension of the simple von Neumann neighborhood described above is to take the set of points at a Manhattan distance of r>1. This results in a diamond shaped region - the MathWorld link below has a nice diagram of what such neighborhoods look like. These are called von Neumann neighborhoods of range or extent r.

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