The Game of Life
The world opens mid-battle: four glider guns fire streams into a block of random soup on a torus of four million cells, every one of them deciding its fate from its eight neighbours alone. Draw your own cells, stamp a gun or a spaceship, and change the numbers in the rule to watch the same universe explode, freeze into a maze or split into day and night.
In 1970 the mathematician John Horton Conway was looking for the simplest rules that could produce something unpredictable. He settled on a grid of square cells, each alive or dead, and four rules applied to every cell at once. A live cell with fewer than two live neighbours dies. One with two or three lives on. One with more than three dies. A dead cell with exactly three live neighbours comes alive. Martin Gardner described it in his Scientific American column that October, and for years afterwards a surprising share of the world's computer time went on running it.
The surprise is how much those four rules contain. Some shapes sit still for ever; some blink. Five cells, the glider, walk across the grid one square every four generations. Within weeks of the column, a group at MIT led by Bill Gosper found a gun that fires a new glider every thirty generations, which settled Conway's question of whether a finite pattern could grow without limit. Five cells arranged as the R-pentomino churn for 1,103 generations before they calm down, and nobody can tell you that without running it — which is exactly what the Methuselah buttons do, counting every cell as they go.
That unpredictability is the deep result. Gliders can carry signals, guns can make them, and collisions can combine them like logic gates, so Life can be wired into a computer that runs any program. It is Turing-complete, and it follows that there is no general shortcut for asking what a pattern will eventually do: in general, the only way to find out is to watch. Change the numbers in the rule and the character of the universe changes with it. HighLife adds one birth condition and gains a pattern that copies itself; Day and Night treats live and dead the same; Seeds never settles; Maze freezes into corridors.
Cellular automata like this one are now used to model forest fires, crystal growth, traffic, and the spread of patterns on seashells, and Stephen Wolfram built a thousand-page book of science around their simplest relatives. Life itself has become a hobby with its own archaeology: its enthusiasts have catalogued millions of objects and built, inside the grid, a clock, a prime-number sieve and a working copy of Life itself.
Two pendulums, one pen and half a minute of friction draw a figure no hand could — and the musical fifths close while the detuned ones never do.