# Von Neumann neighborhood

In cellular automata, the **von Neumann neighborhood** (or **4-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*

*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 . 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).

## See also

## References

- Toffoli, Tommaso; Margolus, Norman (1987),
*Cellular Automata Machines: A New Environment for Modeling*, MIT Press, p. 60. - Ben-Menahem, Ari (2009),
*Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1*, Springer, p. 4632, ISBN 9783540688310. - Wilson, Joseph N.; Ritter, Gerhard X. (2000),
*Handbook of Computer Vision Algorithms in Image Algebra*(2nd ed.), CRC Press, p. 177, ISBN 9781420042382. - Breukelaar, R.; Bäck, Th. (2005), "Using a Genetic Algorithm to Evolve Behavior in Multi Dimensional Cellular Automata: Emergence of Behavior",
*Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation (GECCO '05)*, New York, NY, USA: ACM, pp. 107–114, doi:10.1145/1068009.1068024, ISBN 1-59593-010-8.