A predicted quasicrystal is based on the ‘einstein’ tile known as the hat

The “hat” wowed mathematicians. Now the shape is shaking up physics.

In 2023, mathematicians reported that the 13-sided tile was the first known “einstein.” That’s a shape that can perfectly cover an infinite plane — no gaps or overlaps — but can do so only without a repeating pattern (SN: 3/24/23).

Now, scientists have predicted the properties for a two-dimensional material based on the hat. It’s a quasicrystal, a material that is orderly like a crystal, but in which the arrangements of atoms don’t repeat. Intriguingly, the hat-based material shares properties with graphene, a crystalline material, the researchers report in a paper to appear in Physical Review Letters.

“It’s got lots of properties that we associate with quasicrystals, but then it acts strangely like crystals,” says physicist Sinéad Griffin of Lawrence Berkeley National Laboratory in California, who was not involved with the research. “It’s a really fun study.”

Previously, mathematicians needed more than one shape to cover an infinite plane in this nonrepeating way, known as an aperiodic tiling. Some earlier aperiodic tilings have connections to real-world materials. Penrose tilings, based on sets of two tiles discovered in the 1970s by mathematician Roger Penrose, look like a 2-D slice through a quasicrystal. Such quasicrystals have been found in meteorites and atomic bomb test debris, in addition to being made in the lab (SN: 5/17/21).

So scientists wanted to know what a material based on the hat tiling might be like. Physicist Adolfo Grushin and colleagues calculated the properties of electrons in a 2-D material in which atoms sit at the hats’ vertices.

To characterize a material, scientists can look at the relationship between the energies of its electrons and their wavelengths. (According to quantum physics, electrons travel through materials as waves; the wavelength denotes the size of those waves.) In this energy–wavelength relationship, the researchers found striking similarities between the hat quasicrystal and graphene, a 2-D crystal of carbon.

That’s because many of the vertices of the hat tiling fall along a hexagonal grid like that of graphene, says Grushin, of Institut Néel of CNRS in Grenoble, France.

The fact that the hat tiling is made up of a single tile shape, rather than multiple shapes, also helps explain how it straddles the worlds of crystals and quasicrystals. The use of a single tile means it’s closer to being periodic than other aperiodic tilings, without actually repeating.

Unlike graphene, however, the hat material is chiral, which means that the electrons would behave differently if you were to flip the material as if reflected in a mirror. In a real material, that chiral property might affect how light interacts with the substance, for example, by rotating the light’s polarization, the orientation of its electromagnetic waves.

More interesting features popped up when the researchers investigated what would happen if the material were placed in a magnetic field. In the hat tiling, a fraction of the hat tiles are mirror images of the others. Electrons, specifically those with zero energy, became trapped around the flipped hats at certain values of magnetic field. “We found it quite beautiful that this happens,” Grushin says.

Although the material is entirely theoretical for now, the researchers proposed some ways that the material could be brought to reality. For example, scientists could manually place molecules on a surface in a pattern matching the hat tiling. That would be the ultimate hat trick.