Life oscillator ($F^p(A)=A$)

A Life oscillator is a nonconstant periodic orbit of the Conway's Game of Life update function $F$. Its period is the least positive integer $p$ for which $F^p(A)=A$, comparing the entire live-cell set at the same coordinates. A configuration that never changes is instead a fixed point; a Life glider is not a Life oscillator on the infinite board because it returns at displaced coordinates. The Life blinker has period two.^[ref]

The Life blinker alternates between horizontal and vertical lines of three live cells. Start with $B_0=\{(-1,0),(0,0),(1,0)\}$ and every other cell dead. The two endpoints each have one live neighbor and die; the center has two and survives. The dead cells $(0,-1)$ and $(0,1)$ each have three live neighbors and are born. No other dead cell has three. ThereforeThe same neighbor-count argument with horizontal and vertical exchanged gives the second step. Since $B_1\ne B_0$, the period is exactly two, rather than merely a divisor of two.

The Life blinker is a minimal hands-on check of synchronous updating. Mark both endpoints for death and both births before changing anything. Removing an endpoint early would make a prospective birth see only two neighbors; adding a birth early would alter other counts. A correct implementation reproduces both sets above exactly, rather than merely keeping the live population equal to three.