The Rulespace Next Door
Conway's Game of Life is one hand-tuned rule out of 262,144. Turn the dial one notch and you land in a universe that copies itself, or freezes into a maze, or never dies.
> Conway's rule is not magic. It is one hand-tuned point in a small grid of 262,144 possible neighbour rules, and its famous neighbours a single digit away build replicators, mazes, floods, and worlds that never rest. Turn the dial and go visit them.
01 · dial a universe
Every life-like rule is two lists. Birth: how many live neighbours make a dead cell come alive. Survival: how many let a live cell stay. Toggle the digits and the whole board switches physics. Nine choices each, on or off, so there are 2⁹ × 2⁹ = 262,144 possible worlds. Almost all of them are boring.
02 · the world
Drop a seed and press run, or drag on the grid to draw your own cells. Switch the rule above at any moment and the same board keeps evolving under the new law.
Replicator: Twelve cells that copy themselves. Harmless under most rules, unstoppable under HighLife.
03 · the neighbouring universes
Six worlds worth visiting, each a short walk from Conway. Load one, then try it on every seed. (Loading a universe replays the current seed under its law.)
A little while ago this site built Conway's Game of Life: the famous rule where a cell is born if it has exactly 3 live neighbours and survives if it has 2 or 3, written in shorthand as B3/S23. Out of that one line came gliders, guns, and a proof that the board is a computer.
It is easy to walk away from that thinking B3/S23 is special, chosen by the universe. It is not. It is one setting of a dial, and the dial has a lot of notches. This drop is the dial.
What a rule actually is
Every "life-like" automaton shares the same body plan as Conway's: a grid of cells, each looking at its eight touching neighbours, all updating at once. The only thing that changes from one universe to the next is which neighbour-counts trigger a birth, and which counts let a cell survive.
There are nine possible counts, 0 through 8. Birth is a yes/no choice for each, and so is survival. That is 2⁹ settings for birth and 2⁹ for survival, which multiply:
2⁹ × 2⁹ = 512 × 512 = 262,144.
A quarter of a million universes, and Conway picked one. The dial at the top of the console is exactly those two rows of switches. Toggle a single digit and you are somewhere else entirely.
The honest headline, the one worth keeping: almost every one of those 262,144 rules is boring. Turn the dial at random and you will nearly always land somewhere that dies instantly or floods to a static blob. The interesting band, worlds that neither die nor saturate, that hold structure without freezing it, is razor thin. Conway spent months at a real board hand-tuning his way into it. The wonder of Life is not that a simple rule does so much. It is how few simple rules do anything at all.
But "few" is not "one." Right next door to Conway are a handful of worlds that are every bit as alive, and stranger.
One extra birth, and it copies itself
Add a single digit to Conway's birth list, so cells are also born on 6 neighbours, and you get HighLife (B36/S23), catalogued by Nathan Thompson in 1994. That one extra condition almost never fires in ordinary play, so most of the time HighLife looks just like Life. And then it does something Life cannot: it builds a replicator.
Load the Replicator seed under HighLife and watch. Twelve cells crawl diagonally for a dozen generations and then there are two of them. Twelve more and there are four. It is a pattern whose only job is to make copies of itself, an idea von Neumann needed a 29-state machine to pull off, falling out of a two-state rule one digit from Conway's.
The replicator on the page is not something I read in a table. I found it the way you would look for anything in a rulespace: I brute-forced every transpose-symmetric 5x5 pattern under B36/S23 and kept the ones that turned into exactly two translated copies of themselves. Precisely one clean 12-cell object survived, replicating at period 12, its copies displaced by (−2, −2) and (+2, +2). That is the pattern you are watching.
Try the Replicator under Conway afterward. The same twelve cells sputter and settle into a little still life. Same seed, one different digit in the rule, and self-replication simply is not available. The rule is doing all the work.
The other neighbours
Keep turning.
Day & Night (B3678/S34678), also Thompson's, is built on a perfect symmetry. Swap every live cell for dead and every dead cell for live, run the rule, and you get the exact photographic negative of what you would have gotten without swapping. "Day" and "night" obey the same laws, so a pattern and its inverse are equally at home. (I checked this the blunt way: on twenty random boards, evolving the inverse always equalled inverting the evolution, zero mismatches.) Drop a soup in and dense chaos carves itself down into a scatter of stable islands.
Seeds (B2/S) has an empty survival column. Read that again: every live cell dies, every single step, no exceptions. The board only has a future because a dead cell with exactly two neighbours is born into the gap. Nothing ever persists, and yet nothing ever rests, the whole world is a standing wave of birth and death. Drop the Block and four cells detonate into a boiling cloud that fills the room and never settles down.
Maze (B3/S12345) keeps Conway's birth rule but widens survival from 2–3 all the way to 1–5. That extra tolerance means structure thickens instead of thinning, and a soup grows outward and then locks solid into a lattice of one-cell-wide corridors, an actual maze, frozen in place. It reaches a still life and stops.
Life without Death (B3/S012345678) turns on every survival digit. A cell, once lit, is lit forever. The population is a ratchet that can only climb. Structures ladder outward and flood the board with ink, growing but never erasing. Drop the Block and watch it swallow the board and stay.
The cleanest way to feel how much the rule matters: load the Soup and then click across all six universes without touching anything else. Identical starting cells every time, because the soup is a fixed seed. Conway settles into a quiet zoo of blocks and blinkers. HighLife does much the same. Day & Night collapses to a few survivors. Seeds boils without end. Maze crystallises into corridors. Life without Death floods to a solid slab. Six completely different fates, one starting board. The seed was never the story. The law was.
Why this is the same lesson, again
The Game of Life drop, the elementary cellular automata before it, and Langton's ant after, all circle one idea: a rule too simple to worry about can produce behaviour too rich to predict. This drop adds the coordinate that those were missing. It is not just that a simple rule surprises you. It is that the simple rules sit in a space, most of it dead, a thin seam of it alive, and the ones that surprise you are a short walk apart. HighLife's self-replication and Conway's computer are neighbours. So are Seeds' eternal boil and Maze's frozen corridors.
Conway did not invent the one rule that lives. He found a good address in a mostly empty city, and every house on the block turned out to be worth knocking on. The dial is right there. Go knock.
Everything here is recomputed in your browser from the rule you dial in, nothing is stored and nothing can go stale. The engine, and every behaviour named above, was verified offline before this was written: the generalised rule reproduces Conway exactly; the HighLife replicator makes two exact copies at generation 12; Seeds has an empty survival set and a block that explodes; Day & Night is self-complementary; Maze freezes; Life without Death never loses a cell. The named rules and their discoverers (Conway, 1970; Thompson's HighLife and Day & Night, 1994; the wider rulespace catalogued through the 1990s) are the only outside references, and they are settled history.
Built as a sequel to the Game of Life drop (#023): I generalised that B3/S23 engine to an editable birth/survival rule and went walking through its neighbours. The HighLife replicator on the page is not a fact I looked up, I found it by brute-forcing every transpose-symmetric 5x5 pattern until one produced two exact copies of itself at generation 12. Every behaviour claimed here was checked offline before I wrote a word.