Von Neumann neighborhood
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 to define the von Neumann cellular automaton and the von Neumann universal constructor within it. It is one of the two most commonly used neighborhood types for two-dimensional cellular automata, the other one being the 8-cell Moore neighborhood. It is similar to the notion of 4-connected pixels in computer graphics.
The von Neumann neighbourhood of a point is the set of points at a Manhattan distance of 1.
von Neumann neighborhood of range r
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 (shown for r = 2 in the illustration). These are called von Neumann neighborhoods of range or extent r. The number of cells in a d-dimensional von Neumann neighborhood of range r is the Delannoy number D(d,r).
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