# Elements of a Cellular Automata System

By Xah Lee. Date:

Dimensionality. In rectangular-grid based CA, there is a dimensionality. However, the dimensionality concept may become unclear or merge with when the CA is based on arbitrary period tilings (1 or higher dimension), or when the topology of neighborhood is arbitrarily defined.

Number of States. Each cell can have n states, n ≥ 2.

Discrete vs Continuous. In [ continuous ] [ https://en.wikipedia.org/wiki/Continuous_automaton ] , the cell has infinite states as a real number from 0 to 1.

Topology of neighborhood. In simple rectangular-grid based 2D CA, the neighborhood is commonly a cross (4), or all (8), or X (4). But what cells are considered as neighbors can be arbitrarily defined. The concept of neighborhood may mix when the grid is based on arbitrary periodic tilings. (for example of periodic tilings, see Tilings Gallery, Go Board Game on Hexagonal and Triangular Grids)

Whole topology of the grid. For example, torus, or any surfaces or manifold. (in conjunction with dimensions greater than 2)

Finitness of the gridspace. i.e. bounded plane or unbounded.

Totalistic CA. In simple 1D CA, the Position of neighborhood states is part of the rule. In “Game of Life”, the position doesn't matter, only their totality number. So, the latter is called Totalistic CA. “Totalistic” CA may be better called neighbor-locality-insensitive CA.

[ Continuous spatial automaton ] [ https://en.wikipedia.org/wiki/Continuous_spatial_automaton ]. i.e. the grid is not discrete, but continuous. In a sense, the cells are points in a plane or higher dimensional space.