MATHEMATICS

Cellular Automata can be one-dimensional, two-dimensional or of any dimension. in cellular automata, cells can have many types of neighbours. for example, in two-dimensional cellular automata, cells can have 4 neighbours which are exactly adjacent to it or cells can have 8 neighbour in which 4 are exactly adjacent to it and other 4 are diagonally adjacent to it.

The theory of cellular automata is immensely rich, with simple rules and structures being capable of producing a great variety of unexpected behaviors. For example, there exist universal cellular automata that are capable of simulating the behavior of any other cellular automaton or Turing machine. It has even been proved by Gacs (2001) that there exist fault-tolerant universal cellular automata, whose ability to simulate other cellular automata is not hindered by random perturbations provided that such perturbations are sufficiently sparse.