Tuesday, March 7, 2017

Ulam's two step cellular automata

Above are some of the nice images generated by a cellular automata described in one of Martin Gardner's essays about Conway's Game of Life (you can find the essays here). Cells have four neighbours (north, south, east, west), and follow only two rules that are applied at each step: if a cell has one live neighbour it turns on, and if a cell is on it turns off after two steps. The images above start happening around step 100 after turning on a single cell at the centre of a 61 by 61 grid.

You can play with these here. Eventually, these will start to repeat or disappear completely (I suspect they will oscillate, but have not found out when yet). On a 5 by 5 board, a single central cell will lead to a pattern that dies out in 10 generations; once you get to the uniform checkerboard state, the next is an empty board.