Physicists Carved the Famous 'Hat' Tile Into a Chip and Got Pinwheels of Light

The "einstein problem" has nothing to do with Albert Einstein. The name is German for "one stone," and the question it poses is deceptively simple: can a single tile shape cover an infinite surface without the pattern ever repeating?

Mathematicians chased that shape for decades. In 2023 they found it — a 13-sided figure nicknamed the "hat," discovered by a team including retired print technician David Smith. It tiles the plane forever and never repeats, the first single shape ever proven to do so.

Now physicists at the University of Tokyo's Institute of Industrial Science have taken the hat out of the realm of pure mathematics and etched it into silicon. Writing in Nature Communications on July 29, a group led by Yuto Moritake with senior author Masaya Notomi fabricated nanoscale optical structures laid out according to the hat tiling and fired laser light at them.

What came back had not been seen before. Instead of the familiar sharp spots of a periodic crystal or the symmetric rosettes of a conventional quasicrystal, the light scattered into distinctive pinwheel-shaped diffraction patterns — patterns that spiral in a specific rotational direction and cannot be superimposed on their own mirror image.

That property is called chirality, and it is the reason your left hand does not fit into a right-handed glove. The hat tiling is chiral because generating a nonrepeating pattern from one shape requires using both the tile and its mirror image in a particular ratio, which leaves the finished arrangement without mirror symmetry.

"The diffraction patterns themselves become chiral because the structure lacks mirror symmetry," Notomi said, noting that this is a fundamental departure from how conventional quasicrystalline materials behave.

Quasicrystals — ordered but nonrepeating solids — have been studied since Dan Shechtman's Nobel-winning discovery in 1982, and photonic quasicrystals are already used to control how light propagates through engineered materials. But those structures are built from multiple tile shapes and are typically mirror-symmetric. A single-tile aperiodic structure that is inherently handed is a genuinely new object in optics, and its behavior was not predictable from the existing quasicrystal literature.

The practical hook is that chiral optical structures interact differently with left- and right-circularly polarized light. That asymmetry is the working principle behind circular dichroism spectroscopy, the technique chemists use to distinguish mirror-image molecules — a distinction that matters enormously in pharmaceuticals, where one handedness of a drug can be therapeutic and the other useless or harmful. Building the asymmetry directly into a chip's geometry, rather than into a material's chemistry, opens a different route to the same effect.

For now the result is a demonstration, not a device. But it settles something worth settling: a shape that existed for three years as a curiosity in tiling theory turns out to have physics attached to it.