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
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 . 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
- Moore neighborhood
- Neighbourhood (graph theory)
- Taxicab geometry
- Lattice graph
- Pixel connectivity
- Chain code
References
- ^ Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, p. 60.
- ISBN 9783540688310.
- ISBN 9781420042382.
- ^ ISBN 1-59593-010-8.