For some people, tiles are usually associated with home renovations, but for mathematicians, they present a confusing challenge that they find difficult to solve. In the field of mathematics, a tiling problem asks experts to cover an area using a given set of tiles without having any overlap.
In March 2022, a group of mathematicians led by retired printing systems engineer David Smith from Yorkshire, England, discovered a 13-sided shape that forms a pattern that covers an infinite plane but does not repeat. This discovery allowed the experts to prove the existence of a kind of pattern that was only predicted to be possible theoretically.
This long-sought shape was called the "einstein" from the German word, which means "one stone." Just like a bathroom floor tile, it has the ability to cover an entire surface without gaps. The Einstein pattern was also called "the hat" because its shape is composed of a hat and its mirror image.
Mathematicians have long been fascinated by the ability of tiles to cover an infinite plane without creating overlaps or gaps between them. Other tiles can be positioned in a way that they will not form a repeating pattern, but einsteins are special because there is only one way that they can be tiled.
An Upgraded Version of Einstein Tile
Just a few months after their discovery, the same team of researchers has found another shape that is even more special. The new tile discovered makes the same non-repeating pattern but without such reflections. Since the tile is not accompanied by its own reflection, it was called the "vampire einstein." The shape discovered by Smith and his team is part of a family of vampire Einsteins, which are called "spectres."
In the past, mathematicians were aware of the sets of shapes that could tile the plane using only non-repeating patterns. Until the discovery of the vampire Einstein, scientists never thought that a single tile could have this ability.
When they found the first Einstein, the researchers sought a tile that could make a non-repeating pattern without the need for reflected versions. Smith and his team began with a shape related to the hat and curved its edges so that the reflection would not fit together with the original tile. This led to the successful creation of the vampire einstein tile. This ability of the specter to tile a surface without repetition is referred to as a "strictly chiral aperiodic monotile."
Computer scientist Craig Kaplan from the University of Waterloo in Canada said, "I would never have predicted that we'd stumble upon a shape that solves this [vampire einstein] subproblem so quickly."
The Mathematics of Tiling
In the field of mathematics, tiling refers to a plane-filling arrangement of figures or shapes. It is a collection of disjoint open sets whose closure can potentially cover a given plane. Experts predict that if a set of shapes can tile a plane, then they can always be arranged to do it periodically in a notion known as Wang's conjecture.
It was during the nineteenth century when the systematic study of the theory of tilings started. It was mainly motivated by physical experiments and theories concerning the structure of solid matter.
Tiling started as a recreational mathematics, but it has undergone a transformation into a challenging area of research. Today, tiling challenges have become famous and have appeared in some problems, with many of the solutions rooted in high-level mathematics.
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