Langton's Ant
An ant walks on a grid following two simple rules: turn right on white squares (flip to black), turn left on black squares (flip to white). After ~10,000 steps of apparent chaos, the ant suddenly builds a regular "highway" that repeats forever.
White cell: turn 90° right, flip to black, move forward | Black cell: turn 90° left, flip to white, move forward
Langton's ant was introduced by Christopher Langton in 1986. Despite only two rules, the system exhibits complex emergent behavior: the first ~10,000 steps appear chaotic and pseudo-random, then the ant spontaneously organizes into a periodic "highway" pattern that repeats every 104 steps and grows indefinitely.
This highway has never been proven to emerge from all starting configurations — it has just been observed to happen. The long-term behavior of multiple ants or modified rules (more than 2 colors) remains largely unpredictable and is the subject of ongoing research.
Multi-state variants (e.g., "LLRR", "LRRL") produce wildly different patterns — from expanding triangles to symmetric snowflakes to pure randomness — demonstrating how small changes in rules can produce qualitatively different emergent behavior.