A mathematical shape that drew global attention for solving a decades-old puzzle is now revealing an unexpected connection to physics. Researchers have found that structures based on the shape can make light form unusual chiral patterns, pointing to new ways of exploring how geometry can influence optical behavior. In a study published in Nature Communications, researchers from the Institute of Industrial Science, The University of Tokyo, and collaborating institutions built optical structures inspired by the "Smith hat." This unusual shape is known for solving the so-called Einstein problem in mathematics.
When the team illuminated the structures with laser light, they observed diffraction effects unlike those seen in conventional quasicrystals. The Shape That Solved the Einstein Problem The Einstein problem asks whether a single tile shape, known as a "monotile," can cover an entire surface without creating a repeating pattern. Familiar tilings such as checkerboards and honeycombs repeat in a regular way.
An aperiodic monotile, by contrast, can fill a surface without ever settling into a repeating arrangement. In 2023, researchers discovered the first such monotile, the Smith hat. The finding attracted widespread attention because it provided a long-sought solution to the mathematical problem.
"What is especially fascinating about the hat tile is that, although the resulting pattern appears irregular at first glance, it is actually constructed from the honeycomb lattice," says lead author Yuto Moritake. "We wanted to see whether this unique shape could also produce any unexpected physical phenomena." Turning a Mathematical Pattern Into an Optical Structure To test that possibility, the researchers created nanoscale versions of the pattern on silicon nitride films using electron beam lithography. When laser light was directed at the structures, the resulting diffraction patterns formed distinctive pinwheel-like shapes.
These patterns directly revealed the chiral character of the aperiodic structure. Chirality refers to a form of handedness in which a structure and its mirror image cannot be perfectly matched. In this case, the unusual arrangement of the monotile pattern caused the light itself to display a chiral response.
"We found that the diffraction patterns themselves become chiral because the structure lacks mirror symmetry," explains senior author Masaya Notomi. "This kind of optical response is fundamentally different from that observed in conventional quasicrystalline materials." Light Responds to Direction and Polarization The researchers also found that the diffraction pattern changed depending on both the direction and polarization of the incoming light. When the physical structures were mirrored, their optical behavior reversed as well.
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