A tessellation is a way to cover a flat surface with shapes, edge to edge, with no gaps and no overlaps. Every floor tile pattern, every honeycomb, every brickwork wall is a tessellation. Most tessellations are periodic — slide the pattern in some direction and it lines up with itself. Hexagons, squares, and equilateral triangles all tile the plane this way.

In 1974, the British mathematical physicist Roger Penrose found something stranger. He discovered a set of two tile shapes, now known as Penrose tiles, that cover the plane perfectly with no gaps but never produce a repeating pattern. No matter how far the arrangement is slid, it will not align with itself. The pattern is structured but never periodic.

For about fifty years afterward, mathematicians asked the next question. Could a single tile do the same job? A single shape that tiles the entire infinite plane, but only ever aperiodically. The hypothetical object had a nickname — the einstein, from the German ein Stein, meaning "one stone." Many believed it could not exist.

In March 2023, a retired printing-systems engineer in northern England named David Smith was cutting shapes from cardboard at his kitchen table. He noticed that one shape — a thirteen-sided polygon resembling a wide-brimmed hat — seemed to tile in interesting ways.

He sent it to professional mathematicians at the University of Waterloo, including Craig Kaplan. They verified it computationally and, with two collaborators, proved the result formally. Smith's shape, now known as the hat, is the first true einstein. A single tile that covers the infinite plane forever, in any direction, and is mathematically guaranteed never to repeat.

A fifty-year search ended at a hobbyist's kitchen table.