 Life glider

The Life glider is a five-cell pattern that recreates its shape after four generations, shifted one cell along each coordinate. Use $x$ increasing rightward and $y$ increasing downward, as in the figure, and start from the live setEvery cell outside $G_0$ starts dead. Applying the Conway's Game of Life rule, rather than moving the drawing by hand, gives:

For an exact check of every frame, the intermediate live sets areHere $G_0+(1,1)$ means add $(1,1)$ to each ordered pair in $G_0$. In the first step, the live cells $(1,0)$ and $(0,2)$ have respectively one and one live neighbors, so die; the other three have respectively three, three and two and survive. Exactly $(0,1)$ and $(1,3)$ are born, with three live neighbors each. This derives $G_1$ and explains how motion can occur while individual cells merely switch states.

Because the rule uses the same relative neighbors at every position, applying the global update function and applying a translation commute. Thus $G_{t+4}=G_t+(1,1)$ for every generation $t$; the four-step calculation establishes continued motion on the infinite board. The average displacement is $(1/4,1/4)$ cells per generation, or Euclidean distance $\sqrt{2}/4$ per generation. The live-cell count happens to stay five in these frames, but the rule does not conserve live-cell count in general: a single isolated live cell dies immediately. The Life glider persists indefinitely in an otherwise empty infinite board; encounters with other patterns or a finite implementation's edge can change that conclusion.

// Destination: computer-science.bigb