The Universe Might Be Running on Four If-Statements

In March 1970, a 12-page typed letter arrived at the desk of Martin Gardner, a columnist at Scientific American. The sender was John Horton Conway, a 32-year-old Cambridge mathematician with a reputation for being simultaneously the most brilliant and most chaotic person in any room he walked into.

The letter described a game. Not a game of strategy or chance, but something stranger: a universe simulator built from four rules that could fit on a napkin.

Gardner read it. Then he published it in his October 1970 column.

What happened next was, at the time, the largest reader response in Scientific American's history.

Meet Conway

Before we get to the game, you need to understand the man who made it.

John Conway was, by most accounts, a force of nature. He was a self-described "terribly introverted adolescent" who had deliberately reinvented himself as an extrovert when he arrived at Cambridge. He gave the impression of constantly playing, constantly bouncing between ideas, picking up objects, making noise. His office at Princeton was eventually declared a fire hazard by the fire marshals because of the scattered toys, geometric models dangling from the ceiling, and towers of paper threatening to collapse on visitors.

He worked nearly every waking hour. He just didn't work in any way that looked like working.

Conway had produced genuinely significant mathematics: surreal numbers (a wild new number system that extended arithmetic to infinities and infinitesimals), three sporadic groups in abstract algebra, and what he would later call the Free Will Theorem (a proof that if humans have free will, so do elementary particles). He was proud of all of it.

But to the world, he would become known for a game.

The Four Rules

The Game of Life runs on a grid. Each cell on the grid is either alive or dead. At every tick of the clock, the grid updates according to four rules, applied to every cell simultaneously:

That's it. No exceptions. No other rules. You set up an initial pattern, you apply these four conditions over and over, and you watch what happens.

Conway spent nearly two years experimenting with different rule combinations before landing on these specific numbers. He was looking for a "sweet spot": rules that were neither too restrictive (everything dies quickly) nor too permissive (everything explodes into chaos). The rules had to be right on the edge.

He found it.

What Grows

Press Play and watch what a simple starting configuration can do.

A few things worth trying:

Glider: The simplest moving object. Five cells, traveling diagonally at one cell every four generations. Richard K. Guy stumbled on it in 1969 while studying a different pattern. It's the first thing most people discover, and it never stops being satisfying to watch.

Pulsar: A period-3 oscillator, one of the most beautiful patterns in the game. Every three generations it returns to its original shape, pulsing like a living thing.

R-pentomino: Five cells arranged in an L-shape. It looks harmless. It runs for 1,103 generations before stabilizing, eventually producing 116 cells across multiple moving and stationary structures. Conway himself studied it for weeks without being able to predict where it would end up without actually running the simulation. Patterns like these are called methuselahs.

Gosper Gun: The pattern that won Conway's $50 prize. Bill Gosper and his team at MIT discovered it in November 1970, proving that some patterns grow without limit. The gun fires a new glider every 30 generations, forever.

The key insight: none of this is written into the rules. The rules know nothing about gliders or guns or oscillators. These structures are purely emergent. They arise because the rules, applied repeatedly, create stable local configurations that interact with each other in ways that produce complexity.

You can't look at five cells and predict what they'll become in a thousand generations. You have to run it.

The Turing Bombshell

Here's where things get genuinely strange.

In 1982, Conway proved something that should not be true on its face: the Game of Life is Turing-complete. That is, it has the computational power of a universal Turing machine. Given enough space and time, you can build any computation inside it.

How? Gliders act as signals, carrying information across the grid. When two glider streams collide at the right angle, they annihilate in a way that implements logical operations: AND gates, OR gates, NOT gates. Chain enough of those together and you have a CPU.

Researcher Paul Rendell actually built a working Turing machine entirely within the Game of Life. In 2010, a group built a copy of itself that constructed another copy of itself inside the simulation. In 2016, someone ran Conway's Game of Life inside Conway's Game of Life.

Think about what this means. A grid of cells following four neighborhood rules can compute anything a modern computer can compute. There is no algorithm, no program, no calculation that lies beyond what these four if-statements can eventually produce given enough cells and enough generations.

Wolfram's Discovery

Conway's game was two-dimensional. Around 1983, physicist-turned-mathematician Stephen Wolfram started thinking smaller: one row of cells, one dimension.

He called these elementary cellular automata. Each cell has two states (alive or dead). Each cell's next state depends only on itself and its two immediate neighbors. Since there are $2^3 = 8$ possible neighbor combinations, and each can produce either 0 or 1, there are exactly $2^8 = 256$ possible one-dimensional rules. Wolfram numbered them 0 to 255.

Most are boring. Rule 0 turns everything off. Rule 255 turns everything on. But some are extraordinary.

Rule 30 is one of them. Starting from a single alive cell, it produces this:

That pattern has never repeated. It passes every statistical test for randomness that has been applied to it. It looks completely disordered. But it comes from the same single cell, deterministically, every time.

Stephen Wolfram thought this was so remarkable that he embedded Rule 30 as the default random number generator in Mathematica, the mathematical computing system used by researchers worldwide. When Mathematica generates a random number, it is reading off the center column of Rule 30.

In 2019, Wolfram offered $30,000 in prizes for answers to open questions about Rule 30's mathematical properties. Three basic questions remain unanswered: does the center column always produce equal numbers of 0s and 1s? Does any column ever become periodic? Is there a closed-form expression that predicts the center column without running the simulation?

Nobody knows. A rule you can write in one line produces behavior that mathematics, so far, cannot explain.

It Was Already in Nature

Here is the part that should stop you cold.

In 1952, Alan Turing published a paper on how biological organisms develop their patterns: zebra stripes, leopard spots, the arrangement of fingers on a hand. He showed that simple chemical reactions, diffusing and inhibiting each other across tissue, could spontaneously produce regular patterns from uniform starting conditions. He called this reaction-diffusion.

Turing died in 1954. He never saw the connection to cellular automata.

Decades later, researchers noticed something: the shells of certain sea snails are covered in patterns that look exactly like the output of elementary cellular automata.

The Conus textile snail produces a shell pattern that, when examined computationally, follows rules equivalent to Rule 30. Pigment cells along the lip of the shell secrete color based on the activity of their immediate neighbors, obeying local rules, producing a global pattern that looks designed but is entirely emergent.

The snail did not invent cellular automata. It stumbled into the same math because the math is deep enough to show up independently in biological chemistry and in Stephen Wolfram's notebook.

Zebra stripes, the branching of blood vessels, the arrangement of leaves around a stem: all of these appear to follow similar principles. Simple local rules. Complex global order.

Conway Hated It

By the mid-1970s, John Conway had a problem. He had become famous.

Not for surreal numbers. Not for his work on sporadic groups. Not for the Free Will Theorem. For a game.

Fan letters poured in from hobbyists and programmers and curious teenagers who had run Game of Life on their home computers. When Conway was introduced at lectures, the introduction often went: "John Conway, creator of the Game of Life."

He called his letter to Gardner "the fatal letter."

He would bellow "I hate Life!" when the subject arose in conversation, in the dramatic, exasperated way of a man who had been asked the same question one too many times. The bitterness was real. He had spent decades producing what he considered far deeper work, and the world remembered him for the thing he made in an afternoon.

His perspective softened eventually. After being introduced at a late-career lecture as "John Conway, Creator of Life," he paused, then admitted: "That's actually quite a nice way to be known." But the ambivalence never fully went away.

This is the strange fate of elegant ideas. They escape their creators. They become larger than their origins. The Game of Life stopped belonging to Conway the moment Gardner published that column.

April 2020

On April 8, 2020, Conway developed symptoms of a respiratory illness at his care facility in New Brunswick, New Jersey.

On April 11, 2020, he died of complications from COVID-19. He was 82. He was the first Princeton faculty member known to have been killed by the virus.

The world was in its first month of lockdown. A pandemic was reshaping human behavior in real time, forcing populations to make decisions based on invisible local interactions: who to contact, how far to travel, whether staying isolated would protect others. The aggregate behavior of billions of individual decisions producing emergent global patterns.

Conway had built a model of exactly this, in a different medium, fifty years earlier. Four rules. A grid. Local interactions. Emergent complexity.

He did not get to see what came of it.

What came of it is this: the Game of Life is now used to teach computation, emergence, and complex systems to students in mathematics, biology, physics, and computer science. Researchers use cellular automata to model neural activity, tumor growth, traffic flow, and the formation of galaxies. Wolfram spent a decade writing a 1,200-page book arguing that all of physics might reduce to cellular automaton rules.

Whether or not he is right, the question is legitimate. And it is legitimate because of Conway.

The man who hated his greatest invention turned out to have built something the world wasn't done with yet.