The tile solves the decades-old einstein problem,
which asked whether a single shape could tile a flat surface endlessly without ever repeating its pattern. The name stems from the German phrase “ein stein,” meaning “one stone,” rather than Albert Einstein. While mathematician Roger Penrose demonstrated in the 1970s that two shapes could cover a surface without repetition, geometry enthusiast David Smith and collaborators discovered a single 13-sided shape, nicknamed the “Smith hat,” in 2023.
The Einstein Problem and the Smith Hat
Yuto Moritake and the University of Tokyo
Yuto Moritake, an experimental physicist at the University of Tokyo, encountered the tile in a popular science book in 2024. Specializing in photonic crystals—materials typically built on regular, repeating grids that steer light for lasers and optical sensors—Moritake replaced standard repetition with the hat tile’s never-repeating arrangement.
Electron Beam Lithography and Silicon Nitride
To fabricate the structure, Moritake and his colleagues employed electron beam lithography and etching techniques to punch hundreds of thousands of holes into a thin film of silicon nitride. Each hole measured 100 nanometers in radius, or roughly 500 times thinner than a human hair. When the team directed a laser at the crystal, it yielded a swirling, pinwheel-shaped scattering pattern that varied depending on the spinning direction of the incoming light.
The consistency confirmed a long-range, predictable order typical of a quasicrystal. Because the hat tile lacks mirror symmetry—meaning it differs from its own reflection—the resulting scattering pattern was similarly asymmetrical, a property known as chirality. This asymmetry generated a circular polarization dependence, causing subtle differences in how clockwise- and counterclockwise-spinning light scattered off the material. Ordinary quasicrystals possessing mirror symmetry cannot achieve this effect.
This structure can have some kind of circular polarization dependence,
Moritake said, noting that the outcome was unexpected.
Looking ahead, Moritake aims to utilize the never-repeating pattern to regulate light moving inside a photonic chip rather than light bouncing off its surface, potentially advancing optical communications and optical computing technologies.
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