On an infinite square lattice, the B36/S125 cellular automaton proceeds similarly to Conway’s Game of Life (B3/S23), but with different birth and survival conditions. Specifically, a dead cell becomes live if surrounded by 3 or 6 live neighbours, and a live cell survives if surrounded by 1, 2 or 5 live neighbours. As usual (c.f. question 9), each cell dies unless specified otherwise.
The dynamics are not as interesting as the Game of Life, although there is a naturally-occurring glider and related infinite-growth pattern (discovered by Nathaniel Johnston during a massive search with random initial conditions):
One of the more interesting properties of the cellular automaton is that if the universe is composed entirely of 2 × 2 blocks in a square lattice arrangement, this will be the case for all time. We can think of B36/S125 as emulating a simpler cellular automaton:
This cellular automaton is slightly unusual in that the cells are in different positions in odd generations and even generations. A spacetime diagram of this gives truncated octahedral cells in a configuration known as the body-centred cubic lattice:
For want of a better term (block cellular automaton is slightly ambiguous, and can refer to the similar — albeit distinct — concept of a Margolus neighbourhood), I shall henceforth refer to these as BCC automata. Nathaniel Johnston investigated rectangular oscillators in the BCC automaton arising from B36/S125, showing how they in turn emulate an even simpler (one-dimensional) cellular automaton, namely Wolfram’s Rule 90. It transpires that this behaviour was investigated earlier by Dean Hickerson on the notorious Usenet group, comp.theory.cell-automata.
Emulation can go in the opposite direction. For a cellular automaton such as Life, we can arbitrarily group the cells into 2 × 2 blocks, thus enabling it to be emulated in a 16-state BCC automaton. This is used as the base case for Bill Gosper’s algorithm HashLife, which uses hashed quadtrees to simulate patterns frighteningly quickly.