von Neumann Neighborhood


A diamond-shaped neighborhood that can be used to define a set of cells surrounding a given cell (x_0,y_0) that may affect the evolution of a two-dimensional cellular automaton on a square grid. The von Neumann neighborhood of range r is defined by


von Neumann neighborhoods for ranges r=0, 1, 2, and 3 are illustrated above. The number of cells in the von Neumann neighborhood of range r is the centered square number 2r(r+1)+1, the first few of which are 1, 5, 13, 25, 41, 61, ... (OEIS A001844).

See also

Cellular Automaton, Moore Neighborhood, Neighborhood

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Gray, L. "A Mathematician Looks at Wolfram's New Kind of Science." Not. Amer. Math. Soc. 50, 200-211, 2003.Sloane, N. J. A. Sequence A001844/M3826 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

von Neumann Neighborhood

Cite this as:

Weisstein, Eric W. "von Neumann Neighborhood." From MathWorld--A Wolfram Web Resource.

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