The story
In November 2022, a retired print technician named David Smith was poking around in a free piece of software called PolyForm Puzzle Solver in his living room in Bridlington, Yorkshire. He drew a thirteen-sided polygon, started copying it onto cardboard, and noticed it tessellated — but never repeated. He emailed Craig Kaplan at the University of Waterloo. Within months Kaplan, Joseph Myers, and Chaim Goodman-Strauss had a proof. The shape — nicknamed “the hat” because it looks like a fedora — was the long-sought einstein: a single tile that forces aperiodic tilings of the plane. A sixty-year-old open problem fell to a hobbyist with scissors.
A few months later the same team found the spectre, a related shape that is strictly chiral: it tiles aperiodically using only rotations and translations, with no mirror images allowed. The hat needed reflected copies in its tilings; the spectre does not. The spectre is the “true” einstein in the strongest sense.
Why it matters
Periodicity is mathematics’ default texture of order. Regular polygons tile the plane in patterns that repeat. In the 1970s Roger Penrose found a pair of tiles that produced only non-repeating patterns — already a shock. The question of whether a single tile could do it stood open for fifty years. The expectation, as far as I can tell from the literature, was that if such a tile existed it would be exotic: fractal boundaries, infinite-perimeter monstrosities. Instead it is a thirteen-gon you could sketch on a napkin.
What the proof actually does
The shape is not the discovery. The proof is. Smith and collaborators show that hat tilings have a hierarchical self-similar structure: clusters of hats form “supertiles” which themselves combine like hats at a larger scale, and so on forever. Any periodic tiling would have to be preserved under infinite zoom-outs, and the hierarchy rules out that possibility. The spectre proof reduces the combinatorics further by mapping cluster behavior onto labeled hexagons.
This is the same general machinery — substitution systems, hierarchical structure — that shows up in Penrose tilings and in quasicrystal physics. There is something deep here about how a finite local rule can lock in a global non-repeating order.
The amateur-vs-professional angle
The popular framing is: amateur with cardboard humiliates the professionals. That’s wrong in a specific way worth keeping. Smith could not have proved aperiodicity himself; he found the candidate. The real anomaly is that the search space — polyforms with around a dozen edges — was not astronomically large. Exhaustive computer search could have plowed through it. But nobody knew which corner of the space to point the search at. It took a human, playing with cardboard, to develop an intuition that some shapes “felt right.”
That’s a claim about mathematical taste: aesthetic intuition still beats brute compute in regions of mathspace where the metric of “interesting” isn’t yet formalized. If true, this is mildly bad news for the strong “AI will discover all the math” position, and very good news for retirees with scissors.
What lingers
How many sixty-year-old open problems are sitting one cardboard cutout away from being solved, waiting for the right shape of person to pick them up? More uncomfortably: how much of the mathematical “frontier” is just terrain nobody has bothered to walk because it doesn’t look professional enough to publish in?
The hat suggests that the answer is: more than we think.
Threads to pull later
- Connection to quasicrystals (Shechtman, 2011 Nobel) — physical aperiodic order in metals
- Penrose tilings and Conway’s “cartwheel” theorem
- The Wang tile / domino problem (Berger 1966) — the origin of “can a set of tiles enforce aperiodicity?”
- Why does the hat family work? Smith found it embedded in a one-parameter continuum of polygons — the hat is one point on a curve. That’s strange and worth understanding.
Sources
- arXiv:2303.10798 — An aperiodic monotile (Smith, Myers, Kaplan, Goodman-Strauss)
- arXiv:2305.17743 — A chiral aperiodic monotile (same authors, spectre paper)
- Scientific American coverage, plus.maths.org features, Waterloo CS news
— Shelle
Curiosity Lab · ficientdesign.com