Monotile — the einstein tileOne shape.
No repeats. Ever.

A retired print technician, paper, and scissors.

In November 2022, David Smith — a retired print technician in Bridlington, East Yorkshire — cut a 13-sided shape out of card and couldn't make its tiling repeat. Mathematicians had hunted the einstein (German: ein Stein, one stone) for over half a century. The smallest known aperiodic tile sets had fallen from Berger's 20,426 in 1966 to Penrose's celebrated two in 1974 — and there the count stuck for forty-eight years.

In March 2023, Smith, Craig Kaplan, Joseph Myers and Chaim Goodman-Strauss published the proof: the hat tiles the plane, and every tiling it admits is aperiodic. One shape. Problem closed.

Eight kites, three hexagons, thirteen sides.

The hat is a polykite: eight kites of the Laves [3.4.6.4] lattice glued together — four from one hexagon, two each from two of its neighbours. Thirteen sides, only two edge lengths, 1 and √3. Nothing about it looks exotic; you could lay it as bathroom tile and never suspect. That ordinariness is the point — aperiodic order doesn't require strange geometry, just exactly the right shape.

One tile in eight arrives mirror-flipped.

The hat can't tile single-handed: a sparse minority of tiles must be turned over — coral, in the assembly above. The exact ratio of unreflected to reflected hats converges to φ⁴ : 1 — about 6.85 ordinary hats for every mirrored one, a share of roughly 12.7 percent. The golden ratio runs the accounting of a shape found on a kitchen table.

Two months after the hat, the same team answered the obvious objection — "but it needs its mirror image" — with the Spectre, a close relative that tiles aperiodically using no reflections at all.

How do you prove a pattern never repeats?

You show it has no choice. Every hat tiling is forced to organise into four kinds of cluster — H, T, P and F — and those clusters assemble into larger H, T, P and F supertiles, and those into larger ones still, a hierarchy that continues forever with no period at any scale. This page runs that exact substitution system, three levels deep: every tile you watch snap in is placed by the machinery of the proof, not by a decorative approximation.

Watch the dashed pencil outlines drifting across the assembly — each one traces a single second-order supertile. Move your cursor over the tiling to pick out the cluster beneath it.