Von Neumann neighborhood

Manhattan distance r = 1
Manhattan distance r = 2

In cellular automata, the von Neumann neighborhood is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells.[1] 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.[2] It is one of the two most commonly used neighborhood types for two-dimensional cellular automata, the other one being the Moore neighborhood.

This neighbourhood can be used to define the notion of 4-connected pixels in computer graphics.[3]

The von Neumann neighbourhood of a cell is the cell itself and the cells at a Manhattan distance of 1.

The concept can be extended to higher dimensions, for example forming a 6-cell octahedral neighborhood for a cubic cellular automaton in three dimensions.[4]

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 2-dimensional von Neumann neighborhood of range r can be expressed as ${\displaystyle 1+2r(r+1)}$. The number of cells in a d-dimensional von Neumann neighborhood of range r is the Delannoy number D(d,r).[4] The number of cells on a surface of a d-dimensional von Neumann neighborhood of range r is the Zaitsev number (sequence A266213 in the OEIS).