With J: the Hodge Podge machine

TLDR: The Game of Life is incredibly intriguing, giving rise to complex behavior that is visually stimulating, mathematically interesting and, moreover, it is known to be capable of universal computation.

Abstract: Cellular automata are collections of cells arranged in some manner such that each cell contains a value that is updated from generation to generation according to a local rule. The Game of Life [1-5,7,15] is perhaps the best known cellular automaton. It is based upon a rectangular arrangement of cells that are either 0 or 1, along with simple rules of evolution: if a cell is 0 and has exactly 3 immediate neighbors then it becomes a 1; if a cell is 0 and has exactly 2 or 3 immediate neighbors that are 1, then the cell remains 1; otherwise, the cell becomes or remains 0. The Game of Life is incredibly intriguing, giving rise to complex behavior that is visually stimulating, mathematically interesting and, moreover, it is known to be capable of universal computation. Despite the fact that at a basic level it was designed to model alive and dead cells, it is primarily a toy model in the sense that it does not model any physical behavior well.