For over fifty years mathematicians hunted for a single shape that could cover an endless floor without the pattern ever repeating. They called it an "einstein," German for one stone. Many suspected it couldn't exist. In November 2022, David Smith, a retired printing technician who cut shapes out of card for pleasure, found it: this thirteen-sided tile, "the hat." Below is one. Press grow and watch it conquer the plane, alone.
For the curious kid.
Most tiles that cover a floor forever do it the boring way: stamp, stamp, stamp, the same arrangement over and over. The hat refuses. It covers every floor in the world without gaps and without overlaps, but no matter how far you zoom out, the pattern never settles into a repeat: copy any patch of it, slide the copy anywhere else, and it will never line up perfectly again. And notice the gold hats: they are mirror images of the others, the same shape flipped over, and they are rare: about one hat in eight. Watch the counter as you grow the patch. That ratio is heading somewhere very particular, and it is not a fraction.
Deeper.
The proof that the hat never repeats works like the tiling downstairs in Exhibit Nº3: substitution. Every patch of hats groups into four kinds of clusters (press the skeleton to see them), and those clusters group into bigger clusters of the same four kinds, forever upward. The bookkeeping of that hierarchy forces the ratio of ordinary hats to mirrored hats toward φ⁴ ≈ 6.854102, the fourth power of the golden ratio, an irrational number. A repeating pattern can only produce fractions. No fraction, no repeat. The tiles on this page are placed by the discoverers' own substitution rules, and this exhibit checked the arithmetic before opening: no gaps, no overlaps, ratio marching to φ⁴ exactly.
The history is as good as the mathematics. Smith found the shape by playing (scissors, card, and a good eye), then emailed Craig Kaplan, a computer scientist. With Joseph Myers and Chaim Goodman-Strauss they proved aperiodicity in March 2023. Berger's original 1966 proof that aperiodic tile sets exist needed 20,426 different tiles; Penrose got it down to two in 1974; the hat finished the story at one. Almost: the hat needs its mirror images. Three months later the same team answered that too: a curved-edged cousin called the spectre tiles aperiodically with no reflections at all. Fifty years of pursuit, ended twice in one spring, started by a hobbyist. Some doors are opened by people who simply never stopped playing.
Berger needed 20,426 shapes. Penrose needed two. David Smith, with scissors and patience, needed one.