= Life glider
{c}
{title2=$F^4(G)=G+(1,1)$}
{wiki=Glider_(Conway's_Game_of_Life)}
= Glider in Conway's Game of Life
{c}
{synonym}
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 <set>
$$
G_0=\{(1,0),(2,1),(0,2),(1,2),(2,2)\}.
$$
Every cell outside $G_0$ starts dead. Applying the <Conway's Game of Life> rule, rather than moving the drawing by hand, gives:
\Image[/conway-life-glider-frames.png]
{title=Four synchronous <Conway's Game of Life> steps translate the <Life glider> by (1,1)}
{description=Original computed toy configuration, not measured data. All five panels use fixed coordinates. Orange dashed outlines in the last panel mark the original live cells; blue cells show the computed fourth generation. The displayed window is part of an infinite board, not a finite boundary.}
For an exact check of every frame, the intermediate live <sets> are
$$
\begin{aligned}
G_1&=\{(0,1),(2,1),(1,2),(2,2),(1,3)\},\\
G_2&=\{(2,1),(0,2),(2,2),(1,3),(2,3)\},\\
G_3&=\{(1,1),(2,2),(3,2),(1,3),(2,3)\},\\
G_4&=\{(2,1),(3,2),(1,3),(2,3),(3,3)\}=G_0+(1,1).
\end{aligned}
$$
Here $G_0+(1,1)$ 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
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