Occurrence:

The cellular automaton, developed by John von Neumann, first appeared in the 1940s. Von Neumann introduced the concept of a self-replicating machine, which consisted of simple rules for the interaction between individual elements (cells) in space. He used this concept to investigate the processes of self-reproduction and evolution.

Principle of operation:

In a von Neumann cell automaton, each cell (if it is not a marginal cell) has 6 neighbors, since only the cell that shares a side with the grain (the original cell) is considered a neighbor. The future state of the cell itself depends on the state of the grain and the condition of its neighbors:

y’[k][i][j] = f(y[k][i][j], y[k + 1][i][j], y[k][i + 1][j], y[k][i][j - 1], y[k][i][j + 1], y[k][i - 1][j], y[k - 1][i][j])