These are called von Neumann neighborhoods of range or extent r.

His construction uses the von Neumann neighborhood, and cells with large numbers of states.

In most cases, the von Neumann neighborhood (four adjacent cells) is considered.

A circle of radius 1 (using this distance) is the von Neumann neighborhood of its center.

It is one of the two most commonly used neighborhood types, the other one being the 4-cell von Neumann neighborhood.

The two most common types of neighborhoods are the von Neumann neighborhood and the Moore neighborhood.

The latter includes the von Neumann neighborhood as well as the four remaining cells surrounding the cell whose state is to be calculated.

Another common neighborhood type is the extended von Neumann neighborhood, which includes the two closest cells in each orthogonal direction, for a total of eight.

On a grid (such as a chessboard), the points at a Lee distance of 1 constitute the von Neumann neighborhood of that point.

Codd's CA has eight states determined by a von Neumann neighborhood with rotational symmetry.

In cellular automata, the von Neumann neighborhood comprises the four cells orthogonally surrounding a central cell on a two-dimensional square lattice.

Langton's Loops run in a CA that has 8 states, and uses the von Neumann neighborhood with rotational symmetry.

CoDi uses a von Neumann neighborhood modified for a three-dimensional space; each cell looks at the states of its six orthogonal neighbors and its own state.

Almost surely, every cell of the automaton eventually enters a repeating cycle of states, where the period of the repetition is either n or (for automata with n odd and the von Neumann neighborhood) n + 1.

In two dimensions, with no threshold and the von Neumann neighborhood or Moore neighborhood, this cellular automaton generates three general types of patterns sequentially, from random initial conditions on sufficiently large grids, regardless of n. At first, the field is purely random.