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 N_(x_0, y_0)^v = {(x, y): left bracketing bar x - x_0 right bracketing bar + left bracketing bar y - y_0 right bracketing bar <=r}. 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).

cellular automaton | Moore neighborhood | neighborhood