In 2022, David Smith, an amateur mathematician and former print technician living in the United Kingdom, discovered a strange shape while cutting and pasting cardboard at his kitchen table. This 13-sided tile, resembling a hat, could tile a plane without any gaps, yet its pattern never repeated in exactly the same way twice. This was the moment when the "einstein problem"—a 50-year-old unsolved question in mathematics asking whether a single type of tile could tile a plane aperiodically—was finally solved.
The "einstein" in the einstein problem does not refer to the famous physicist Albert Einstein, but rather comes from the German phrase "ein Stein," meaning "one stone." The solution to this pure geometric puzzle set mathematics enthusiasts around the world abuzz. But this discovery has now moved beyond the realm of shape puzzles. A research team led by Associate Professor Yuto Moritake of the University of Tokyo and Professor Masaya Notomi of Institute of Science Tokyo brought this arrangement of "hats" into physical space and illuminated it with laser light. What they found was that this tiling produced a pinwheel-like optical vortex—something that ordinary crystals or quasicrystals could never generate.
A Quasi-Lattice of "Hats" Lacking Mirror Symmetry
Crystals found in nature owe their properties to the regular arrangement of atoms. However, structures can possess overall regular order without having a periodic structure like a crystal. These are known as "quasicrystals" or "quasiperiodic structures."
The tiling problem, first proposed by mathematician Hao Wang in the 1960s, began as the question of whether a plane could be tiled aperiodically using only a given set of tiles. Initially, Wang himself conjectured that no such set of tiles existed. Later, in the 1970s, mathematician Roger Penrose narrowed this aperiodic tiling problem down to just two types of tiles—the kite and the dart. These later became widely known as "Penrose tiles," and discoveries related to this aperiodic structural order eventually led to the 2011 Nobel Prize in Chemistry. However, finding a single tile that could accomplish this feat alone remained impossible for nearly half a century.
Conventional quasicrystals and quasiperiodic structures shared one major characteristic: they often possessed mirror symmetry or inversion symmetry, meaning that flipping them left-to-right would produce a pattern identical to the original. What the research team focused on this time was the inherent "chirality" (handedness) of the hat-shaped monotile. Like the human right and left hands, its mirror image never coincides with the original shape.
In a theoretical paper published in 2023, physicist Joshua Socolar had already predicted that this hat-shaped tiling would produce chiral diffraction patterns. The team designed a new point array—a "monotile quasi-lattice"—by placing a point at the centroid of each tiled hat-shaped monotile. This lattice possesses "three-fold rotational symmetry," meaning that rotating it by one-third of a full turn (120 degrees) around specific points produces a pattern that exactly matches the original. However, when reflected left-to-right as in a mirror, it never returns to its original arrangement. The research team set out to experimentally verify how this mathematical symmetry breaking would manifest when it actually interacted with light.
Nanoscale Holes Draw a Pinwheel of Light
To translate this abstract mathematical concept into a physical light experiment, the research team employed NTT's cutting-edge semiconductor nanofabrication technology. This experiment demanded extremely high precision in order to transform a theoretical point array into an actual optical device. Using electron-beam lithography and dry etching on an ultra-thin silicon nitride film just 350 nanometers (nm) thick, the team precisely arranged 372,100 circular holes—each with a radius of only 100 nm—across an area measuring 0.5 millimeters square, without a single error. Even the slightest distortion or defect at this scale can easily destroy the interference pattern of light. This vast arrangement of holes was the physical embodiment of the monotile quasi-lattice in two-dimensional space.
When a green laser beam was shone onto the surface of the thin film, a vivid diffraction pattern emerged on the wall behind it. Despite the lack of periodicity, numerous bright points appeared arranged in a regular pattern known as "Bragg peaks." Bragg peaks typically only appear in structures with clear long-range order. The observation of these peaks despite the aperiodic nature of the structure is unmistakable evidence that the entire structure maintains genuine order as a quasicrystal.
Even more striking was the shape of the diffraction pattern itself. The cluster of points formed a pinwheel-like shape lacking mirror symmetry, with the entire pattern twisted in a consistent direction. The angle of this twist was approximately 15.5 degrees. According to the research team, this angle is not arbitrary but is derived directly from the golden ratio and the Fibonacci sequence, which govern the geometry of the tile. Furthermore, when light was shone on an inverted structure fabricated using the mirror image of the tile (a flipped hat), the pinwheel's twist neatly reversed direction.
Circularly Polarized Light Reveals New Behavior in Quasiperiodic Structures
Having confirmed the twist in the pattern, the team went a step further and investigated the response to the rotation of the light wave itself. Light has a property in which the plane of oscillation of its electric field can spiral either clockwise or counterclockwise along its direction of travel—a property known as the helicity of circular polarization.
Associate Professor Yuto Moritake, who led the research team, approached this experiment with the reasoning that "since this is a chiral structure, dependence on circular polarization should not be forbidden." Although the initial attempt failed to detect any difference, by introducing a more sensitive camera, the team finally succeeded in capturing this subtle distinction. The brightness of specific diffraction peaks clearly differed depending on whether left-circularly polarized or right-circularly polarized light was incident on the structure.
What is particularly noteworthy here is the quantitative contrast with conventional representative structures. In fully periodic structures such as the honeycomb lattice typified by graphene, no intensity difference arises from the rotational direction of the incident light. Moreover, many conventional quasicrystal structures, such as Penrose tiling, inherently contain mirror symmetry and therefore do not exhibit a pronounced asymmetric response to circularly polarized light. However, in this monotile quasi-lattice, it was confirmed that the intensity distribution of the diffraction peaks completely swaps depending on the helicity of the circularly polarized light, when compared with its mirror-image structure. In conventional materials science, such optical responses have primarily been discussed in the context of molecules with three-dimensional twisting, such as the DNA double helix. The innovation of this research lies in the fact that this effect was generated purely through the geometric "order of a point array" on a two-dimensional plane.
A Mathematical Game Driving Nanophotonics
This discovery brings a new design principle to the fields of "metasurfaces" and "nanophotonics," which control the direction and polarization of light using sub-wavelength microstructures. Conventional nanophotonics has frequently relied on periodic structures, typified by the honeycomb lattice, due to their ease of handling and computational simplicity. However, the monotile quasi-lattice—aperiodic yet possessing chiral long-range order—has the potential to become a hitherto unexplored "platform for manipulating light."
Of course, this research is still at the stage of basic physics demonstration, and it is not something that can be immediately incorporated into any optical device. Whether two-dimensional chirality can serve a practical function as a mechanism for advanced polarization control in future optical communications, or for novel photon manipulation in quantum cryptography technology, remains unverified.
Nevertheless, one fact remains unshaken: a shape-puzzle game that began with cardboard on a kitchen table has, within just a few years, connected with cutting-edge nanotechnology to reveal an unknown optical phenomenon. The history of mathematics providing physics with unexpected tools continues to be renewed, even now, by Einstein's "hat."