Never Repeats

A bathroom floor repeats. Pick any tile, slide the whole floor sideways by the right amount, and it lands exactly on itself. For a long time it seemed that any set of shapes able to cover a floor with no gaps could also be made to cover it in a repeating way. In 1974 Roger Penrose found two that cannot: a kite and a dart, cut from a rhombus by the golden ratio and marked so they may only meet in certain ways. They cover the plane forever, and they never repeat.

The surprising part is how local and how ordered that refusal is. Pick any patch of the floor — a tile and its ring of neighbours, or two rings — and it turns up again and again, within a short walk of anywhere you stand. Tap a tile here and every copy lights up. Yet the copies never line up into a lattice: the counter beneath the floor tries every shift that carries your patch onto a copy of itself and checks the whole floor, and none of them works. The floor even has five-fold symmetry — turn it 72° about the centre and every tile lands on an identical tile — which no repeating pattern can have; a lattice can only turn by a half, a third, a quarter or a sixth.

Count the tiles as the floor grows and the golden ratio appears on its own. Each deflation cuts every tile into smaller kites and darts by a fixed rule, so the counts step through the Fibonacci numbers and their ratio closes in on φ ≈ 1.618. That irrational number is the deeper reason nothing repeats: a repeating floor would need a ratio of whole numbers.

Penrose's tilings stopped being a curiosity in 1982, when Dan Shechtman found a metal alloy whose diffraction pattern had ten-fold symmetry — forbidden for a crystal. These quasicrystals are ordered like a Penrose floor rather than like a lattice, and the discovery won the 2011 Nobel Prize in Chemistry. Then in 2022 an amateur, David Smith, found the hat: a single thirteen-sided tile that only tiles without repeating, needing no marks at all; it was announced with three mathematicians in 2023. The last tiling here is built from it.