I saw someone comment on a YouTube video with a weird result they had gotten, and then replicated that result myself. It is the following:
Consider the following modification on Conway's Game of Life:
On even steps, nothing changes: regular Game of Life rules hold. But on odd steps, cells can't die. So the rule for creating an alive cell (exactly $3$ alive neighbors) still holds, but no alive cell dies.
When I programmed this in Python in bounded grids (the largest I tried was $10000\times1000$), something peculiar seemed to happen: the ratio alive cells/all cells seemed to, after a certain amount of steps, oscillate between $\frac1\pi$ (on odd steps) and $\frac2{3\pi}$ (on even steps).
So I wonder: with this version of Conway's Game of Life on an infinite grid, are there (necessarily never-repeating) patterns of which the ratio of alive cells to all cells are $\frac1\pi$ and $\frac2{3\pi}$ such that they map to each other under these rules? And moreover, why do other patterns seem to converge to these patterns?