The aperiodic hat

Is there a single tile whose copies cover the whole plane, but only in a pattern that never repeats? For half a century this question, the einstein problem (a pun on the German ein Stein, “one stone”), stayed open. The answer came in 2023, and it looks surprisingly ordinary: a thirteen-sided shape resembling a hat, or a T-shirt.

How it works

The hat is glued from eight identical kites with angles of 60°, 90°, 120° and 90°, the kites into which the hexagons of an ordinary honeycomb grid can be cut. So every hat sits on that grid, turned by a multiple of 60°, but some must be flipped over: without mirror images the tiling cannot be completed.

Aperiodicity rests on a hierarchy. Hats gather into four metatiles: H (four hats, one of them reflected), T (one), P and F (two each). The metatiles assemble into the same H, T, P and F one level up, about φ² ≈ 2.6 times larger, and so on without end. The generator climbs as many levels as the frame needs, then descends to single hats.

The hat counts are squares of Fibonacci numbers, and the ratio of neighbouring counts tends to φ⁴ ≈ 6.854. Unreflected hats outnumber reflected ones in the same ratio, φ⁴ : 1. That this ratio is irrational is the key to aperiodicity: in a periodic tiling the proportions of any tiles would be rational. The hard part of the proof is showing that every hat tiling is forced to assemble into metatiles.

A bit of history

Aperiodic sets date from 1966, when Robert Berger built one of 20,426 Wang tiles. In 1971 Raphael Robinson cut it to six tiles, and in the 1970s Roger Penrose to two. Then progress stalled: the 2010 Socolar–Taylor tile was a single shape but needed rules linking tiles that do not touch. In November 2022 the English amateur mathematician David Smith, experimenting with shapes, came across the hat, and in March 2023 he, Craig Kaplan, Joseph Myers and Chaim Goodman-Strauss posted a proof. One objection remained: the mirror images. By May the same four had found the “spectre”, which needs no reflections.

What to tweak

• “Colouring” set to “classic” paints the reflected hats darkest: try to spot them, roughly one hat in eight.
• “Supertiles” at 1 outlines the metatiles H, T, P and F; 2 and 3 show the same shapes a level up.
• “Shift” moves the frame elsewhere in the endless tiling. Any patch in view recurs infinitely often, but the picture as a whole never does.
• Put the “Kaleidoscope” effect on top: it forces strict symmetry onto a pattern that has not even got a period.