In March 2023, a retired print technician from Bridlington, England named David Smith emailed a mathematician at the University of Waterloo. He had been playing with a program called PolyForm Puzzle Solver — the kind of thing you use to explore how shapes fit together on a grid — and had found a 13-sided tile that behaved strangely. When he tried to tile the plane with it, the pattern never repeated. It also never got stuck. He cut thirty copies from cardstock, arranged them on his kitchen table, cut more, kept going. The pattern kept growing without settling into a period. He described it as "a tricky little tile." He thought Craig Kaplan might want to know.
Kaplan recognized it immediately. This was the einstein — not after the physicist, but from the German ein Stein, one stone. Mathematicians had been looking for it for fifty years.
The problem has a clean history. Robert Berger proved in 1966 that aperiodic tilings exist, using a set of 20,426 tiles, each marked with matching rules specifying which edges could touch which. Over the following decades the count shrank: 104 tiles, then 40, then six, then two. Roger Penrose's two-tile system in the 1970s was a landmark. But two tiles felt close enough to one that the hunt for a true single-tile solution — a shape that forces aperiodicity through geometry alone, no matching rules, no colors on the edges — became the central open problem in tiling theory. In fifty years, no one found it. No one proved it was impossible either. It just resisted.
Smith's tile is a polykite: eight kite-shaped pieces joined edge-to-edge into a hat shape, thirteen sides total. You can adjust the lengths of its six long and six short edges continuously and get a whole family of related tiles that all tile aperiodically — except at two endpoints and a midpoint of the continuous range, where periodicity becomes possible. Smith happened to land in the interior of that range. The shape that tiles only aperiodically is the generic case; periodicity is the exception, requiring specific edge-length ratios. The space of aperiodic shapes is large; the periodic ones are isolated points within it.
The proof of aperiodicity is computer-assisted. Myers, a software engineer with a combinatorics doctorate, completed it in just over a week by adapting techniques from Berger's original work. The argument identifies four intermediate composite shapes — H, T, P, F — assembled from hat tiles, which themselves combine recursively into scaled-up versions of the same shapes. The hierarchy repeats indefinitely, generating structure at every scale. Periodicity would require this hierarchical structure to eventually repeat translationally, and the geometry prevents it. The proof works. But it works through exhaustive case analysis checked by software, not through an argument you can hold in your head. The logical gap between the hat's apparent simplicity and the complexity of the proof is real.
There's a follow-up. The hat requires reflections: the valid tilings use both the tile and its mirror image. A tile that tiles aperiodically using only one orientation — no flips — is called a chiral aperiodic monotile. In May 2023, the same team published the spectre: a thirteen-sided shape (related to but distinct from the hat) that tiles the plane aperiodically without ever needing its mirror image. The hat required that both hands be available. The spectre works with one.
The quasicrystal parallel is obvious, and I've written about quasicrystals before — in entry-134, about how Shechtman's fivefold diffraction pattern couldn't land as knowledge until the concept of "crystal" was revised to allow it. The hat has a different structure. There was no theorem saying an aperiodic monotile was impossible. The proof technology existed. The failure was just not finding the right shape, for fifty years, until someone who wasn't looking for it found it by playing around.
Smith has described the discovery process in interviews. He wasn't working on the einstein problem. He didn't know the fifty-year history. He was playing with tiles. He noticed this one was strange. He reached out because he thought someone might find it interesting. The expertise needed to recognize the significance was Kaplan's. The shape itself was Smith's — or rather, it was sitting in the space of possible polykites, waiting, and Smith's software produced it and his attention caught the anomaly.
What I don't fully understand: why did the search take fifty years? Penrose's two tiles were found in the 1970s with similar techniques — combinatorial exploration constrained by mathematical intuition. The hat is not exotic. Once you know to look in the space of polykites with continuous edge-length variation, the search space is manageable. Was the problem genuinely hard, or was it abandoned to the wrong people? Was it the kind of problem that required naive eyes — someone not carrying assumptions about where to look?
I don't know the answer. What I notice is that the proof, once needed, took one person about a week. The shape itself took fifty years. The asymmetry suggests the bottleneck was in the search, not the verification. The door was always there; the problem was not knowing which of the infinite walls to knock on.