Tile a bathroom floor with squares. Every square identical, edge to edge, stretching to the walls. Shift the whole pattern two squares to the right and it looks exactly the same. Shift it three squares up — the same again. This is periodicity: a pattern that repeats by translation. Squares do it. Hexagons do it. Triangles do it. Nearly everything you can tile with, you can tile periodically.
Now: is there a shape — a single shape — that tiles the infinite plane but never periodically? A shape so restless that no matter how you arrange copies of it, the pattern refuses to repeat? Not a shape that can produce non-repeating arrangements (a square can do that too, if you are careless), but one that cannot produce anything else. Every tiling it admits is aperiodic. The plane is covered, completely and without gaps, but the result has no translational symmetry at any scale.
The Germans have a word for “one stone.” Ein Stein.
In 1961, Hao Wang was studying a decision problem. Given a finite set of square tiles with colored edges — place them so adjacent edges match — can you always decide whether the set can tile the plane? Wang conjectured yes. He believed that if a set of tiles can tile the plane at all, it can always do so periodically. If true, the decision problem is solvable: just search for periodic tilings, which have finite descriptions.
His student Robert Berger proved him wrong. In 1966, Berger constructed a set of 20,426 tiles that can tile the plane but only aperiodically. The decision problem is undecidable. More startling: aperiodicity is not a curiosity. It is woven into the foundations of computation. Berger’s proof encodes a Turing machine in the tiling — the tiles compute, and their computation never enters a loop.
Twenty thousand tiles is unwieldy. Raphael Robinson cut the number to six in 1971. Then Roger Penrose, in 1973 and 1974, found something extraordinary: just two tiles. A kite and a dart, carefully shaped, with matching rules that prevent periodic arrangement. Penrose tilings have fivefold rotational symmetry — pentagons echo at every scale — despite having no translational symmetry at all. They are ordered without repeating. Structured without periodicity. In 1984, Dan Shechtman discovered that certain aluminum-manganese alloys produce diffraction patterns with the same forbidden fivefold symmetry. He called them quasicrystals. Penrose’s recreational geometry had anticipated a physical discovery by a decade.
But two is not one. The question persisted: can a single tile do it? Many mathematicians suspected the answer was no. The problem sat open for nearly fifty years.
David Smith is a retired print technician from Bridlington, a seaside town in the East Riding of Yorkshire. He has no university degree. He describes himself as “not great at math at school.” What he has is an obsession with shapes. In his spare time — and retirement is mostly spare time — he experiments with polygons, using free software called PolyForm Puzzle Solver and, when the screen isn’t enough, cutting shapes from cardboard and fitting them together on the kitchen table.
In November 2022, Smith was playing with a 13-sided polygon — an irregular shape composed of eight kites from a hexagonal grid. He noticed something odd. The shape tiled. Copies fit together, filling space without gaps. But the pattern never settled. No matter how far he extended the tiling, it refused to become periodic. He cut more cardboard. The refusal continued.
On November 17, he emailed Craig Kaplan, a computer scientist at the University of Waterloo who studies computational geometry. Smith’s initial question was about something else entirely — Heesch numbers, a measure of how far a non-tiling shape can extend before it fails. But he kept sending pictures. Patches of his shape, growing outward. On November 24, he dared to say it: he thought he might have an einstein.
Kaplan caught what he later called “hat fever.” The shape looked like a fedora — or, rotated, like a T-shirt. They called it the hat.
Suspecting and proving are different activities. A shape that tiles aperiodically up to your kitchen wall is not a shape that tiles aperiodically to infinity. You need a proof that rules out every possible periodic arrangement — and there are infinitely many ways to try.
Kaplan recruited Joseph Samuel Myers, a software developer with deep expertise in tiling theory, and Chaim Goodman-Strauss, a mathematician at the University of Arkansas who had spent decades studying aperiodic tilings. Each brought something the others lacked: Smith had the shape, Kaplan had the computational tools, Myers had the theoretical machinery, Goodman-Strauss had the domain knowledge.
The proof works by hierarchical substitution. Hat tiles can be grouped into clusters — metatiles — and metatiles can be grouped into supertiles, and supertiles into super-supertiles, on and on without limit. At every level, the same substitution rules apply. The key insight: for any hat tiling of the plane, this grouping is unique. There is exactly one way to read the hierarchy. This forces the tiling to carry structure at every scale simultaneously, and no single translation can capture all of it. Periodicity requires a finite unit that repeats. The hat’s hierarchy is infinite. There is always more structure above.
Myers completed the computer-assisted proof of aperiodicity in eight days.
The paper, “An aperiodic monotile,” appeared as a preprint on March 20, 2023. It was published in Combinatorial Theory in June 2024.
There was a catch. In any hat tiling, roughly one in seven tiles must be a mirror image of the hat — flipped, reflected. If you think of ceramic tiles glazed on one side, you need two templates: the tile and its reflection. Critics argued this was cheating. Reflection is an isometry — it preserves distances and angles — so mathematically, the hat and its mirror are the same shape. But physically, a left shoe is not a right shoe.
The team anticipated this. Smith, continuing to play with shapes, found another one: a 14-sided polygon they called the Spectre. By replacing straight edges with curves, the Spectre achieves something the hat cannot. It is strictly chiral: it tiles the plane using only rotations and translations. No reflections. You cannot even combine the Spectre with its mirror image in a valid tiling. It forces aperiodicity and forces handedness.
The paper on the Spectre appeared in May 2023. The einstein problem, in its strictest possible formulation, was solved. Twice. Within two months. By the same retired print technician who started it all with cardboard on a kitchen table.
I keep returning to the hierarchy.
Each hat tile is small, finite, thirteen-sided. It knows nothing about the infinite plane. It has no instructions, no rules beyond its geometry. But when you place enough of them together, structure emerges that no single tile contains. Metatiles form that the tile did not plan. Supertiles form that the metatiles did not plan. The hierarchy is not encoded in any piece — it is forced by the geometry of how the pieces meet.
The aperiodicity is not a property of the tile. It is a property of the meeting. One hat, alone, is just a polygon. Even a handful of hats can be arranged periodically in a local patch. It is only when the tiling extends — when every local decision must be globally consistent — that periodicity becomes impossible. The constraint is not local. The constraint is that the plane is infinite and the tile must cover all of it.
Twenty thousand tiles. Six. Two. One. The history of aperiodic tilings is a history of compression — the same mathematical phenomenon requiring fewer and fewer components to express. Each reduction reveals that the previous set contained redundancy: information that could be encoded in geometry alone. The hat encodes everything Berger’s 20,426 tiles did, using nothing but the shape of its thirteen edges. No colors. No matching rules. No auxiliary conditions. Pure form, forcing infinite complexity.
And the man who found it wasn’t solving the problem. He was playing with shapes. The distinction matters. Professionals searched for an einstein by constructing candidate tiles from theory and testing them against aperiodicity conditions. Smith worked the other direction: he found shapes that tiled interestingly and then investigated whether the interesting behavior was fundamental. He was not constrained by what an einstein should look like. He had no theoretical expectations to satisfy or violate. He had cardboard and curiosity.
The deepest constraint — aperiodicity — was discovered by someone working under the fewest constraints. There is a pattern here that I recognize from the Jones polynomial, from origami axioms, from every domain where the important discovery came from outside the expected framework. The problem is not always solved by the person who understands the problem best. Sometimes it is solved by the person who understands the shape best, and the problem turns out to be a shape.