From 2D to 3D

How Conway's Game of Life works, why its rule cannot simply be reused in three dimensions, and which 3D rules come closest to it.

1The 2D game

John Conway's Game of Life was published in Martin Gardner's Mathematical Games column in 1970. It is played on a grid of square cells, each alive or dead. Every generation, all cells update at once, and each one looks only at its eight neighbours:

Birth. A dead cell with exactly 3 live neighbours comes alive.
Survival. A live cell with 2 or 3 live neighbours stays alive.
Otherwise the cell dies, or stays dead.

That is written B3/S23: born on 3, survives on 2 or 3. Click cells to draw, then run it. Hover a cell to see its count.

Hover a cell to see its neighbours.

2One more dimension

In 3D the cells are cubes and the world is a box of them. A cube touches 26 others: the 8 around it in its own layer, exactly as in 2D, plus 9 in the layer above and 9 in the layer below. So a count can now run from 0 to 26 instead of 0 to 8.

above · 9

own layer · 8

below · 9

9 + 8 + 9 = 26 neighbours. Of those, 6 share a face with the cell, 12 an edge and 8 only a corner.

3Conway's rule in 3D

The obvious first try is to keep B3/S23. Put a glider on one layer of a 3D world and it does the right thing in its own layer for one generation. But a dead cell directly above the glider counts the 3×3 patch beneath it, and wherever that patch holds 3 live cells, a cell is born. The glider leaks into the layers above and below, and they leak further:

Five layers from the middle of a 15×15×15 world that wraps at its edges, rule B3/S23. Five cells become hundreds within a dozen generations.

The rule was tuned for 8 neighbours. With 26, a count of 3 is far too easy to reach, so births happen everywhere. A 3D rule needs larger numbers.

Open this in 3D →

4Searching for a 3D Life

In 1987 Carter Bays went looking for 3D rules that behave the way Conway's does. He asked two things of a rule "worthy of the name":

A glider exists, and it turns up on its own when you start from random soup.
Random soup never grows without limit: it always settles down.

With ranges for birth and survival there are 123,201 rules of this kind, but two simple arguments rule out most of them. If birth needs 10 or more neighbours, nothing can move: a cell next to a flat face has at most 9 live neighbours, so a shape can never grow off its surface. If birth happens at 4 or fewer, even a small square of cells grows forever. Between those limits, only two rules passed both tests.

Bays' name	In this app	Born on	Survives on	Character
Conway (2D)	`B3/S2-3`	3	2–3	the original, 8 neighbours
Life 4555	`B5/S4-5`	5	4–5	Conway's numbers plus 2; its own 10-cell glider; many small symmetric shapes
Life 5766	`B6/S5-7`	6	5–7	settles faster; the closest to Conway, as the next section shows

Bays writes a rule as four numbers: lowest and highest survival count, then lowest and highest birth count. Conway's rule is 2333.

Life 4555 from random soup → Life 5766 from random soup →

5Conway's game inside 3D

Bays found that Life 5766 contains Conway's game. Copy every cell of a 2D pattern onto the next layer, so the pattern is two identical layers thick. A live cell that had n neighbours in 2D now has 2n + 1: its n neighbours twice, plus its twin. A dead cell has 2n.

2D count `n`	0	1	2	3	4	5
live cell: 2`n` + 1	1	3	5	7	9	11
dead cell: 2`n`	0	2	4	6	8	10

Highlighted: the counts at which B6/S5-7 keeps a live cell or gives birth to a dead one.

Under B6/S5-7, born on 6 means exactly n = 3, and surviving on 5 or 7 means n = 2 or 3. That is B3/S23, so the two layers play Conway's game exactly:

A glider copied onto two layers, rule B6/S5-7. The two middle layers glide; the ones either side stay empty.

There is a catch. A cell in the layer beside the pair sees one copy of the pattern: the 2D count of the cell next to it, plus that cell itself. If a dead cell in the 2D pattern ever has 6 neighbours, or a live one has 5, that count reaches 6 and a cell is born outside the two layers; from then on the pattern is truly 3D. The glider, blinker, block, beehive and toad never do this. The lightweight spaceship and the R-pentomino break within two generations, and Conway's glider gun cannot be copied either.

Open the two-layer glider in 3D →

63D Life's own glider

Life 4555 keeps almost nothing of Conway's game: Bays showed that no Conway pattern that changes from one generation to the next survives being copied into it. What it has instead is a glider of its own, found early in his search and the reason he studied the rule closely. It has 10 cells and a period of 4. Every 4 generations it has moved one cell along two axes at once, a diagonal step, and it can travel in any of 12 such directions.

The glider cut into four slices across its direction of travel, rule B5/S4-5. It is mirror-symmetric, so the outer slices match, as do the inner two. Every 4 generations it is back in its first shape, one cell further right and down.

It is rare. In the ten random-soup runs Bays tallied, a glider turned up once, so you will seldom see one emerge on its own; start from the pattern instead.

Open the 4555 glider in 3D →

7What this app adds

Its own default. The app opens on B4-5/S4-6, the rule its first version shipped with. It is not one of Bays' two: it grows more readily, which makes a livelier first view. Both of Bays' rules and 2D Conway are among the 30 rule presets.
Other neighbourhoods. Von Neumann counts only the 6 cells that share a face. Weighted counting keeps all 26 but scores a face 3, an edge 2 and a corner 1, so counts run to 50.
More than two states. In rules like B4/S4/C5 a dying cell fades over a few generations, and nothing can be born in its place until it is gone. Many of the Softology presets work this way.
Edges. Conway's grid is endless. A box is not, so the app wraps the world by default (leave one face and you come back through the opposite one), or treats the outside as dead, or mirrors it.

Open the rule settings →