The Game of Life

The "glider" is a pattern that moves 1 cell diagonally every 4 generations.

The Game of Life is a mind-bogglingly complex universe-simulator invented by the mathematician John Conway. Based around an infinite grid and a few simple rules, it gives a new meaning to emergent complexity. Each tile on the grid can exist in one of two states: the default "dead" state, or "live". Given a blank, dead grid, a player then seeds the canvas with little colonies of "live" squares. A live square that is neighbored by fewer than two, or more than three, other live squares dies. A dead square that is bordered by exactly three live squares becomes live. A website which allows you to tinker around with Life patterns can be found here.

Not to be confused with Milton Bradley's board game.

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References in fictional works:

• Tea With The Black Dragon by R. A. MacAvoy has a metaphorically-significant scene in which the protagonist is introduced to the game.
• Kingdom of Loathing has a mushroom farming minigame that follows the same rules (although with more than one kind of mushroom), but the board is only 4x4, making it impossible to create any particularly interesting patterns.
• Square Root of Minus Garfield's "Garfield, Infinite Canvas, and The Game of Life", in which a Garfield comic about not having a life is used as the starting state for the Game of Life.
• Glory Season by David Brin features this game with the variation of being played as a contest, with each side getting part of the board to use, with the goal being eliminating the other side.
• One of the intro cinematics for Darwinia is a simulation of Life, with the added caveat that each Darwinian will die after a set number of years no matter what. As it's incredibly difficult not to get attached to the little guys while playing Darwinia, seeing the last "block" and "spaceship" formations settle, flicker, and die can be very haunting.
• Revival "Game of Life" is a spellcard used by Eirin Yagokoro in Touhou 8: Imperishable Night.
• An element in the Boulder Dash clone Rocks N Diamonds.
• In Lyndon Hardy's 1988 novel Riddle of the Seven Realms, a character creates a dimension that operates under these rules, even calling it "the realm of the conways."