Conway's Game of Life

In 1970, a Cambridge mathematician named John Horton Conway penned a simple set of rules for changes in population location that can be easily represented graphically. This makes Conway's game of life, also known as the Game of Life or simply the Game, sound arcane and boring when it's actually accessible and exciting. You can read more about it here:
Conway's Game of Life

To really get a feel for the game, it's best to play around with it for a while. When the game of life was conceived iteration was a time-consuming pain, but the proliferation of computing power in modern times means anyone with a decent internet connection can explore Conway's work to hir heart's content. One decent simulator is here.

As an aside, I like that Wikipedia describes Dr. Conway as an expert on things like knot theory and combinatorial game theory. Those statements aren't very meaningful to me since I have only a vague idea of exactly what knot theory entails and not the slightest idea of what happens in the combinatorial branch of game theory, but it sounds exciting to me that things with names like these exist. At least knot theory is covered by topology, the results of which are beautiful even if the nuts and bolts behind them are mind-numbing to behold.

The basic idea is this. The game board is a grid of arbitrary size composed of square cells. At the beginning of the game, the player selects an arbitrary shape of cells as "live," the rest being "dead." Any live cell with fewer than two live neighbors dies of loneliness, any live cell with more than three live neighbors dies from overcrowding, and live cells with two or three live neighbors live on. Dead cells with three live neighbors are made fertile by their companions, and are revived in the next iteration. It's a simple set of rules, but it leads to a wild menagerie of forms, patterns, and narratives.

Random arrangements typically evolve into a static pattern of stills and high-frequency oscillators, but some careful planning can create beings that reproduce indefinitely or march across the board endlessly. That such complexity could be found in a simple, easily-understood and totally-defining set of rules is the kind of thing that excites mathematicians and other easily-amused people. If nothings else, it's fun to watch the shapes rising and rambling like surf on a beach while the rules and iterations tick along.