Systems
Still, pulsing, moving: the three species of Life
Conway's Game of Life has four rules and no plan, yet it fills up with recognisable creatures. Still lifes never change, oscillators repeat, spaceships travel. Learn the three species and a chaotic board becomes readable.
You do not read a Life board cell by cell. You read it by species: what stays, what repeats, and what moves.
Four rules, no plan
John Conway devised the Game of Life in 1970, and Martin Gardner introduced it to the world in his Scientific American column that October. The board is a grid of cells, each alive or dead, and every cell looks at its eight neighbours.
There are four rules. A live cell with fewer than two live neighbours dies. A live cell with two or three survives. A live cell with more than three dies. A dead cell with exactly three becomes alive. Nothing in those rules mentions shapes, yet the board fills with shapes that behave like species, and naming them is how you read the board.
Still lifes: nothing changes
A still life is a pattern whose next generation is exactly the same as the current one: no cell dies and no cell is born. For that to happen, every live cell needs two or three live neighbours, and no dead cell may have exactly three.
The block is the simplest: a 2 by 2 square. Each of its four cells has three live neighbours, so all survive, and every empty cell around it touches at most two, so nothing is born. The beehive is six cells in a stretched hexagon. Each live cell has exactly two neighbours; the two empty cells inside it have five, the empty cells outside have at most two, none have three, so not a single cell changes. The loaf, the boat and the tub work the same way. Still lifes are the debris a chaotic board leaves behind once it settles.
Oscillators: a heartbeat in place
An oscillator returns to its starting shape after a fixed number of generations, its period, without moving anywhere. The blinker is a row of three cells. The two end cells have only one neighbour each and die; the centre has two and survives; the cells directly above and below the centre each see three and are born. The row becomes a column, and one generation later the column becomes a row again.
Period two is only the start: the toad and the beacon also repeat every two generations, the pulsar every three, and the pentadecathlon every fifteen. Oscillators give the board a rhythm. A frozen picture can tell you where cells are; it cannot tell you which ones are pulsing.
Spaceships: patterns that travel
A spaceship returns to its own shape, but somewhere else. The glider is five cells; after four generations it has the same shape, shifted one cell diagonally. The lightweight spaceship moves along a row instead, twice as fast. A spaceship is how information crosses a Life board: an arrangement of cells carried from one place to another.
Conway offered fifty dollars in 1970 to anyone who could prove or disprove that a pattern can grow without limit. That November a team led by Bill Gosper at MIT won it with the glider gun, a pattern that fires a new glider every thirty generations forever. Endless growth turned out to need nothing more than a machine for making spaceships.
From soup to ash, and to computation
Fill a board at random and the first generations are a boil of births and deaths. Then it burns down. In one simple run, a half-filled 256 by 256 board with wrapped edges kept only about three percent of its cells alive after 4,000 generations, and the most common leftovers were blocks, beehives and blinkers, with gliders flying off. You can watch the same thing happen on the Life board.
The same parts, arranged on purpose, do far more. In Winning Ways (1982), Berlekamp, Conway and Guy showed how streams of gliders, guns and carefully placed still lifes can act as wires and logic gates, which makes Life capable of universal computation. The cost of that power is that, in general, you cannot predict a pattern's future without running it. The one-dimensional version of this idea, with 256 rules that fit in a byte, is in 256 rules, one byte each.