For sixty years, mathematicians searched for a single shape that could tile the plane without ever repeating. In 2023, David Smith — a retired printing technician from Yorkshire — found it while playing with shapes on his computer. The “hat” is a 13-sided polygon made of eight kites from a hexagonal grid. It tiles the plane perfectly, aperiodically, with nothing but copies of itself and its mirror image. One stone, infinite variety, no repetition.
polykite monotile · 13 vertices · Smith, Myers, Kaplan & Goodman-Strauss (2023)
The name has nothing to do with Albert. “Einstein” is German for “one stone” — ein Stein — and the problem asks: does there exist a single tile that can cover the infinite plane, but only aperiodically? That is, the tile must tile the plane, but no translation can map the tiling onto itself.
The search began in 1961, when Hao Wang conjectured that any tile set that tiles the plane can do so periodically. His student Robert Berger disproved this in 1966 by constructing a set of 20,426 tiles that tiles only aperiodically. The race to shrink the set began immediately. Berger himself reduced it to 104. Raphael Robinson brought it to 6 in 1971. Roger Penrose reached 2 in 1974 — his famous kite and dart. For nearly fifty years, two was the best anyone could do. Then, in March 2023, the number dropped to one.
David Smith is not a professional mathematician. He is a retired printing technician from Bridlington, East Yorkshire, who describes himself as a “shape hobbyist.” He spent years exploring polygon tilings on his computer and cutting shapes from cardboard, looking for something interesting. In November 2022, he noticed that one particular 13-sided polygon — a polykite made of eight kites from a hexagonal grid — seemed to tile the plane without ever settling into a repeating pattern.
Smith emailed Craig Kaplan, a computer scientist at the University of Waterloo who specializes in mathematical art. Kaplan wrote software to generate larger and larger patches, and the pattern never repeated. They brought in Joseph Samuel Myers, a software developer and mathematician in Cambridge, and Chaim Goodman-Strauss at the University of Arkansas. Together, the four of them proved that the hat is a genuine aperiodic monotile. The paper appeared in March 2023 and made front-page news worldwide.
There is a subtlety. The hat tiling requires roughly one in every seven tiles to be a reflected (mirrored) copy of the hat. If you consider a tile and its mirror image as the same shape, the hat is a genuine monotile. If you insist on strict chirality — no reflections allowed — then it takes two shapes: the hat and its mirror.
This distinction mattered to some mathematicians. Just two months later, in May 2023, the same team of four announced the “spectre” — a modification of the hat that is a strictly chiral aperiodic monotile. The spectre tiles the plane aperiodically using only rotations, with no reflections needed at all. The Einstein problem, in its strongest possible form, was fully solved within the same year it was first cracked.
That aperiodic tile sets exist is not an accident — it is a logical necessity. In 1966, Berger proved that the “tiling problem” (given a tile set, does it tile the plane?) is undecidable by encoding Turing machines as Wang tile configurations. Since the halting problem is undecidable, so is the tiling problem.
This has a deep consequence. Wang had conjectured that if a set of tiles can tile the plane at all, then it can do so periodically. If that were true, the tiling problem would be decidable — just check for periodic tilings, which is a finite search. But undecidability means Wang’s conjecture must be false. Therefore, tile sets that tile only aperiodically must exist. Aperiodic tiling is not a curiosity; it is a theorem about the limits of computation.
Aperiodic order is not merely a mathematical abstraction. In 1982, Dan Shechtman discovered quasicrystals — aluminum-manganese alloys whose diffraction patterns showed 10-fold symmetry, which is impossible for periodic crystals. The scientific establishment resisted for years (Linus Pauling famously declared “there is no such thing as quasicrystals, only quasi-scientists”), but Shechtman was vindicated with the Nobel Prize in Chemistry in 2011.
Then, in January 2025, researchers reported that certain chiral molecules spontaneously self-assemble into aperiodic tilings on metal surfaces — the first observation of aperiodic molecular monolayers in nature. The hat tiling pattern, discovered by a hobbyist playing with cardboard, may be a natural attractor in self-organizing systems. Mathematics found the shape. Physics built it. Chemistry may have been using it all along.