Four Rules Create Anything | Conway's Game Of Life

Added: 2026-05-15

[00:00:00]This is Conway's Game of Life, a grid of cells. Each cell is either alive or dead. Every step, every cell looks at its eight neighbors and decides whether to live or die. Four rules total. We will get to them in a minute. From those four rules, this is what comes out. That structure in the upper left is called a Gosper glider gun. It is a stable configuration of cells that emits a small five-cell shape called a glider once every 30 steps forever. The gliders stream diagonally out across the grid.

[00:00:36]They never stop. They never collide with each other. They never decay. That little oscillating cross in the middle is called a pulsar. It cycles through three shapes, period three, forever until something disturbs it. These four heavy travelers in the corners are spaceships, lightweight, middleweight, and heavyweight. Each one moves across the grid at its own fixed speed, leaving no trace. Everything you are seeing came from four rules and an initial pattern of dots. Nobody designed the gun. Nobody designed the spaceships. Nobody designed the pulsar. They were discovered. People sat with the rules and noticed, "Oh, this configuration does this. Oh, that configuration does that." The whole thing is what mathematicians call an emergent system. The behavior is not in the rules. The behavior is in what the rules end up doing when you run them on a grid for long enough. Here are the four rules. They are the entire specification of Conway's Game of Life.

[00:01:38]Take one cell, count its live neighbors.

[00:01:41]There are eight neighbors, the eight cells around it. Rule one, if the cell is alive and has fewer than two live neighbors, it dies, underpopulation.

[00:01:52]Rule two, if the cell is alive and has two or three live neighbors, it survives. Rule three, if the cell is alive and has more than three live neighbors, it dies, overcrowding. Rule four, if the cell is dead and has exactly three live neighbors, it becomes alive, reproduction. That is it, four rules. Apply them to every cell simultaneously on every step. Watch a single isolated dot, zero neighbors, it dies on the next step. Watch a horizontal row of three dots. The middle dot has two live neighbors, it survives.

[00:02:31]The end dots have one neighbor each, they die. But the cells above and below the middle now have three live neighbors, so they come alive. The result is a vertical row of three. Next step, by symmetry, it becomes horizontal again, period two, the simplest oscillator. It is called a blinker. John Horton Conway wrote down those four rules in 1969 at the University of Cambridge. He was looking for the simplest rule set that could produce unbounded interesting behavior. Four rules turned out to be enough. The first non-trivial discovery was the glider, five live cells arranged in this shape.

[00:03:12]Apply the rules. After one step, the configuration has changed slightly, but the shape is recognizable. After two steps, it has changed again. After three steps, again. After four steps, the original shape is back, identical to where it started, except it has moved, one cell down, one cell to the right.

[00:03:34]The glider moves diagonally across the grid, one cell per four steps. It moves forever. It does not decay. It does not change shape. It is an indestructible traveler. The glider was found by Richard Guy at Cambridge in 1970, by hand on graph paper. He was watching configurations evolve and noticed that this one came back to itself after a few steps in a new position. He showed it to Conway. Conway named it. Within a few years, it became the unofficial symbol of the entire computer hacking culture.

[00:04:09]The glider is the most basic mover in Game of Life. From it, every more complex moving structure is built.

[00:04:16]Spaceships are gliders' larger cousins.

[00:04:19]Glider guns shoot gliders. Glider eaters absorb them. In a way, every interesting thing in Conway's universe is downstream of this one five-cell shape. People started cataloging what they found. The catalog has grown for 50 years. Still life, configurations that never change.

[00:04:39]The block, the beehive, the loaf, the boat. They sit there indefinitely doing nothing. Oscillators, configurations that cycle. The blinker, period two. The toad, period two. The beacon, period two. The pulsar, period three. The pentadecathlon, period 15. The Cox's galaxy, period eight. Spaceships, configurations that move and return to themselves. The glider, the lightweight spaceship, the middleweight spaceship, the heavyweight spaceship, the Shick engine, the weekender, the crab, the caterpillar, 12 million cells, the largest known spaceship built by hand by David Bell in 2004. Methuselahs, small configurations that survive an unreasonably long time before stabilizing. The R-pentomino, five cells, runs for 1,103 generations before settling into a quiet state of blocks and gliders. The acorn, seven cells, runs for 5,206 generations. Bunnies, nine cells, runs for 17,156.

[00:05:51]Each one is a tiny seed that explodes into a chaotic transient and then eventually freezes into something stable plus a few wandering gliders. The catalog today runs into hundreds of thousands of named patterns. There is a wiki, Life Wiki, that lists every pattern anyone has bothered to publish.

[00:06:11]There are software tools. Golly is the standard one that run patterns trillions of generations into the future at billions of cells per second. People have been doing this for 55 years on and off and they are still finding new things. For about a year, everyone assumed Game of Life was bounded. Every starting pattern, given enough time, would eventually settle. Conway himself believed this and offered $50 to anyone who could prove him wrong. In November 1970, a researcher named Bill Gosper at MIT, 26 years old, working in the AI lab, found this. A configuration of about 30 live cells. It oscillates, period 30. Every 30 generations, it spits out one glider and returns to its original shape. Then it spits out another glider and another forever. The number of live cells in the universe grows linearly. The configuration is unbounded. Conway paid the $50. He sent the check. The glider gun is the proof that Conway's Game of Life can grow without bound. It is the first example of a self-sustaining factory in a cellular automaton. From it, you can build a glider gun that fires faster, a gun that fires in a different direction, a gun that fires complicated wave patterns of gliders. From two guns aimed at each other, you can build a logic gate. A glider arriving means a one, no glider means a zero. Two streams colliding can be made to and, or, or not each other. From logic gates, you can build a flip-flop. From flip-flops, a register. From registers, a CPU. Gosper himself never claimed any of this. He just found the gun. The implications took everyone else a few decades to fully unpack. The gun was the door. Once you know the door is unlocked, you start trying to walk through it. In 2010, a game of life enthusiast named Adam Goucher published a complete pattern called the OTCA metapixel. An OTCA metapixel is an enormous life configuration, a square block 2,000 cells on a side, 4 million cells total, that simulates one single cell of a different cellular automaton. You arrange these blocks in a grid. The blocks talk to each other through gliders. The result, when you zoom out, is a cellular automaton being run by a cellular automaton. You can program the inner automaton to be Conway's Game of Life. You then have Conway's Game of Life running slowly inside Conway's Game of Life. You can zoom out from one OTCA metapixel and see, very slowly, a glider made of glider guns gliding across a grid made of glider guns. The construction was not invented for amusement. It was a public answer to an old question, is Conway's Game of Life Turing complete? Yes. The OTCA metapixel is one explicit way to demonstrate it.

[00:09:18]Paul Chapman built a Turing machine in Life in 2002. Adam Goucher built a programmable computer in 2010. There is now a full implementation of Tetris in Life, 11 million cells running on a custom-built Life CPU. Conway's four rules are universal. Whatever a normal computer can compute, a sufficiently large Game of Life pattern can also compute, given enough generations. That is a remarkable statement. The four rules John Conway scribbled at a coffee table in 1969 are, in a precise mathematical sense, as powerful as every programming language ever written. They can simulate themselves, simulate every other Turing-complete system, and run every algorithm that has ever been or ever will be coded. Conway's Game of Life is a two-dimensional cellular automaton with eight neighbors. Stephen Wolfram, in the early 1980s, asked, "How simple can you make a cellular automaton and still get interesting behavior?" The simplest possible setup is one-dimensional, a single row of cells.

[00:10:25]Each cell looks at itself and its two immediate neighbors. Each cell is alive or dead. That gives eight possible local configurations of three cells. For each of those eight configurations, you decide whether the center cell becomes alive or dead on the next step. That is one rule. Eight binary outcomes means there are two to the eighth, or 256 possible rules. Wolfram numbered them zero through 255.

[00:10:54]Most are boring. Rule zero turns everything off. Rule 255 turns everything on. Most rules quickly settle into a stripe pattern, a fixed point, a small oscillation. Rule 30 does not settle. Start with a single live cell on an otherwise empty row, apply rule 30 over and over, drawing each new row underneath the previous one. The result is a pattern that looks half ordered, half random. There is a clear left edge structure that grows linearly, and there is a chaotic-looking interior that, on careful inspection, has no obvious period and no obvious symmetry. The center column of rule 30, read top to bottom, passes every standard statistical test for randomness.

[00:11:40]Mathematica uses it as the default random number generator. That is a one-dimensional cellular automaton, eight bits of rule defined on a grid of two states. From those eight bits, statistical grade randomness emerges.

[00:11:56]Wolfram has spent 40 years pointing out how unlikely this should have been, given how simple the rule is.

[00:12:03]Wolfram looked at all 256 rules. He noticed that despite their variety, they fell into exactly four classes of long-term behavior. Class one, almost any starting pattern dies out. The world goes blank. Class two, patterns settle into stable structures or simple oscillations. Stripes, blocks, repeating periodic motifs. Class three, patterns produce chaos, apparently random output with no obvious large-scale structure.

[00:12:36]Rule 30 is the prototype. Class four, patterns produce complex, organized, locally interacting structures.

[00:12:44]Travelers, stable structures, collisions, computation. Rule 110 is the prototype, and Matthew Cook proved in 2004 that rule 110 is Turing complete.

[00:12:56]Wolfram conjectured that this four-class taxonomy applies far beyond cellular automata. He thinks it describes the long-term behavior of essentially every computational system. Whether you agree with that or not, it remains the best description we have of what very simple rules can do. Game of Life and Rule 30 share one important fact about reality.

[00:13:20]Complexity does not require complicated rules. The configurations you have been watching for the past 15 minutes are not the result of careful design. They are the result of four rules applied trillions of times to a starting grid.

[00:13:36]Glider guns, spaceships, oscillators, Turing complete computers, all of it falls out of underpopulation, survival, overcrowding, and reproduction. Four sentences. Rule 30 takes it further.

[00:13:51]Three input cells, eight possible patterns, one binary outcome per pattern. That is the entire specification. Eight bits of information, and it produces a stream of randomness indistinguishable from quantum noise. This is not how anyone in the 18th or 19th century would have predicted complexity to work. Everyone assumed that complicated outcomes required complicated machinery. Cellular automata are an existence proof that this is wrong. The implication is that the complexity we see in physics, in biology, in social systems, might emerge from local rules at least this simple.

[00:14:31]We do not know that for certain, but the existence of Game of Life makes it plausible. Conway, when asked late in his life what he thought of all this, said he had grown to dislike the game.

[00:14:42]He felt it had overshadowed his real mathematical work, knot theory, surreal numbers, group theory, monstrous moonshine, all of it serious results.

[00:14:54]The Game of Life was a distraction he could never quite escape. Four rules, a grid of cells, apply forever. From this, you get gliders. You get oscillators that beat in three-step rhythms. You get spaceships that travel forever without changing shape. You get a configuration that produces gliders one at a time on a 30-step clock indefinitely. From the gun, you get logic gates. From logic gates, you get a CPU. From a CPU, you get Tetris running on 11 million cells simulated step-by-step on a grid that is following the same four rules a single blinker follows. Every interesting thing in Conway's universe, every gun, every spaceship, every Turing machine, every game of life running inside Game of Life is downstream of underpopulation, survival, overcrowding, and reproduction. The source code for this video is available through the link in the description.