A generalized neighborhood for cellular automata

A generalized neighborhood for d-dimensional cellular automata is introduced; it spans the range from von Neumann's to Moore's neighborhood using a parameter which represents the dimension of hypercubes connecting neighboring cells. Finite hypercubes and hypertoruses are studied, and the number of neighbors on their boundary and the number of connections between cells are calculated. We come to finite constructs when practically implementing computations of cellular automata. Enumerations of cells, neighbors and connections are considered and implemented in ad-hoc software which generates a canvas of hypercube and hypertorus models in the form of a Petri net. A cell model can be replaced while the underlying canvas of connections remains the same. The generalized neighborhood is extended to include a concept of radius; the number of neighbors is calculated for infinite and finite lattices. For diamond-shaped neighborhoods, a sequence is obtained whose partial sums equal Delannoy numbers.