Toroidal Life
Conway’s Game of Life on a torus — the donut-shaped surface where the grid wraps in both directions. Every implementation of Life uses toroidal boundary conditions, but here you can see the wrapping. Watch a glider leave one edge and reappear on the opposite side, tracing a path across the surface of a rotating torus.
B3/S23 x(u,v) = (R + r·cos v)cos u y(u,v) = (R + r·cos v)sin u z(u,v) = r·sin v
Conway’s Game of Life
The Game of Life is a cellular automaton devised by John Conway in 1970. On a 2D grid, each cell is either alive or dead. At each generation, four rules apply: (1) a live cell with fewer than 2 live neighbors dies (underpopulation), (2) a live cell with 2 or 3 neighbors survives, (3) a live cell with more than 3 neighbors dies (overcrowding), (4) a dead cell with exactly 3 neighbors becomes alive (reproduction). These four rules produce staggering complexity.
Why a torus?
In most Life implementations, the grid edges wrap around — the top connects to the bottom, and the left connects to the right. Mathematically, this is a torus: gluing opposite edges of a rectangle gives you a donut shape. This experiment renders the grid on an actual 3D torus so you can see this topology directly. A glider that exits the right edge reappears on the left — and on the torus, you can see it was never “teleporting” at all. It was always moving smoothly on a surface.
The torus surface
A torus is parametrized by two angles: u (around the ring) and v (around the tube). The grid rows map to u and columns to v, so each cell occupies a small patch on the donut surface. The major radius R controls the ring size, and the minor radius r controls the tube thickness.
Notable patterns
Glider: a 5-cell pattern that moves diagonally one cell every 4 generations. Gosper glider gun: discovered by Bill Gosper in 1970, it emits a new glider every 30 generations. Pulsar: a period-3 oscillator, one of the most common. R-pentomino: a small pattern that takes 1103 generations to stabilize.