Moore Neighborhood
Moore Neighborhood in Cellular Automata
In the study of cellular automata, the Moore neighborhood refers to a group of cells surrounding a central cell in a two-dimensional square lattice. It consists of the central cell itself and the eight surrounding cells that are at a Chebyshev distance of one from the central cell. This configuration forms a 3x3 grid structure and is contrasted with the von Neumann neighborhood, which includes only the four orthogonally adjacent cells.
Key Characteristics
The Moore neighborhood is named after Edward F. Moore, a computer scientist who contributed significantly to the field of automata theory. This neighborhood configuration is widely used in cellular automata because it provides a rich set of interactions possible between a cell and its neighbors, enabling complex patterns to emerge.
Applications
One of the most famous applications of the Moore neighborhood is in Conway's Game of Life, a zero-player game where the evolution is determined by its initial state. Each cell's future state is determined by the number of live cells in its Moore neighborhood. This game demonstrates how simple rules in cellular automata can lead to complex behaviors and is a classic example of a life-like cellular automaton.
The Moore neighborhood can also be extended to three dimensions, forming a cubic neighborhood with 26 surrounding cells. This capability is employed in three-dimensional cellular automata such as 3D Life.
Mathematical Representation
Mathematically, if a cell is located at position (i, j) in a grid, its Moore neighborhood includes all cells at positions (i + x, j + y) where x and y can take values -1, 0, or 1, excluding configurations where (x, y) = (0, 0) unless the central cell is to be included.
Comparison with Von Neumann Neighborhood
While the Moore neighborhood includes both adjacent and diagonal cells, the von Neumann neighborhood, named after John von Neumann, includes only the four directly adjacent cells (north, south, east, and west). This distinction illustrates different neighborhood compositions used in two-dimensional cellular automata models.
The diversity in neighborhood structures like Moore and von Neumann neighborhoods enables researchers to study various phenomena in computational and physical systems. These models can simulate a wide range of behaviors, from simple osmosis to complex biological and ecological processes.