# A Mathematical Tile Pattern Creates Unexpected Light Effects in Laboratory Tests
A shape that emerged from pure mathematics in 2023 has now demonstrated surprising physical behavior in the laboratory. Researchers from the Institute of Industrial Science at the University of Tokyo report that optical structures based on the "Smith hat" tile can twist light into unusual chiral patterns when illuminated with lasers, according to a study published in Nature Communications.
The Smith hat solved what mathematicians call the Einstein problem—a question that asked whether a single tile shape, or "monotile," could cover an entire surface without ever settling into a repeating arrangement. Familiar patterns like checkerboards and honeycombs repeat regularly. The Smith hat, by contrast, can fill a surface while remaining aperiodic, creating a pattern that never repeats. The discovery in 2023 attracted attention as the first solution to the half-century-old puzzle.
"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."
To investigate that possibility, the research team created nanoscale versions of the tile pattern on silicon nitride films using electron beam lithography. The team created nanoscale versions of the tile pattern using electron beam lithography. When laser light struck these 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. 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."
The researchers observed numerous sharp diffraction peaks, indicating long-range order in the structure. When they changed the position where the laser hit the sample, the positions of these peaks remained unchanged—a key signature of quasicrystalline order. While a quasicrystal shows perfect rotational symmetry only around specific points, its characteristic diffraction pattern can be observed regardless of illumination position.
The team also discovered that the diffraction pattern changed depending on both the direction and polarization of incoming light. When they fabricated mirror-image versions of the structures, the optical behavior reversed accordingly. This showed that the response of the light was directly tied to the symmetry of the underlying pattern.
Particularly striking was the response to circularly polarized light. The researchers measured how left-handed and right-handed circularly polarized light diffracted differently from the structures. The diffraction patterns showed a clear dependence on the helicity of the incident circular polarization—a phenomenon never observed in conventional quasicrystals with mirror symmetry.
The researchers traced the origin of the chiral pinwheel structure back to a geometric property of the hat tiling itself. As the mathematical inflation rules are applied to generate larger versions of the pattern, the metatiles—groups of individual hat tiles—gradually tilt away from the underlying honeycomb framework. The team found that this twisting angle follows the Fibonacci sequence and can be calculated analytically, yielding a value of approximately 15.52 degrees. The diffraction peaks in the patterns align precisely with this geometrically determined angle.
"These results open a new direction of research on the fusion of quasiperiodic order and chirality," remarks Moritake. "Monotile patterns provide a platform for exploring optical phenomena that emerge from the interplay of symmetry, chirality, and aperiodicity."
The researchers suggest that structures inspired by monotiles could eventually contribute to technologies designed to manipulate light, control polarization, and support advanced optical devices. The findings demonstrate how a discovery that began as an abstract mathematical puzzle can lead to unexpected physical effects when realized in the laboratory.
**Sources:**
Institute of Industrial Science, The University of Tokyo. "The shape behind the Einstein problem just revealed strange new physics." ScienceDaily, September 15, 2026. https://www.sciencedaily.com/releases/2026/09/260914102432.htm
Moritake, Y., Takiguchi, M., Aihara, T., & Notomi, M. "Chiral diffraction from aperiodic monotile structure." Nature Communications, volume 17, article 6085 (2026). https://doi.org/10.1038/s41467-026-75023-7