Two rules, ten thousand steps, a highway

An ant stands on a white square of an infinite grid. Rule: on white, flip the colour, turn right, step forward. On black, flip the colour, turn left, step forward. For about ten thousand steps the trail looks like chaos. Then — without warning — the ant starts laying down a perfectly periodic 104-step pattern that drifts off to infinity. Nobody has proven the highway always appears, only that it always has so far. Two rules, an unsolved emergent miracle.

Rule

Steps / frame

800

Cell size (px)

4px

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Foldscape · An atlas of mathematical curiosities

Curated by Simon Franzen · zauberware

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