Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host. And now, on to the math.

With the end of 2023 approaching, I spent some time browsing the internet for interesting math news from this year, and realized there was one big unexpected breakthrough by an amateur that really deserves discussion: the aperiodic monotile. While that term sounds exotic, it actually covers a very simple concept. We’ve talked about planar tilings before: basically, this just means a shape of tile that can cover an infinite version of your bathroom floor. Your current bathroom probably has a bunch of square or hexagonal tiles, which completely fill the available space. Well, maybe there are a few practical issues at the edges, but if you imagine the floor to be infinite, those shapes could fill it forever. If you want to fill your floor with a set of more interesting shapes, you could imagine drawing arbitrary squiggly lines in a hexagon to divide it into several smaller tiles of arbitrary irregularity, and thus these smaller tiles would cover your floor as well. But all these coverings are periodic: they consist of an infinite repetition of some small pattern at known intervals. In contrast, an aperiodic tiling would still cover your floor with copies of a few basic shapes, but the pattern would not show this kind of regular repetition.

When the idea first came up in the early 1960s, logician Hao Wang had hypothesized that no aperiodic tiling existed. But he was proven wrong within a few years by his student Robert Berger, who discovered a set of tiles that would indeed cover the plane aperiodically. His initial set was very complicated, consisting of over 20,000 tiles. But this led to a surge of interest in the topic, and by 1974 Roger Penrose had published a simple example of a pair of 4-sided tiles, known as the “kite” and “dart”, that could cover the plane aperiodically with just their two shapes. These were shown to be examples of an infinite variety of 2-tile sets that enabled such tilings. And that is essentially where the problem stood, for almost 50 years, with nobody knowing whether a single tile could cover a plane aperiodically. Bizarrely, during this long wait, chemists discovered that this mathematical game actually had an application in the real world, where these aperiodic tilings formed the basis of “quasicrystals”, real crystalline structures. Quasicrystals have been used for applications like nonstick cookware and reinforcing steel, and led to the 2011 Nobel Prize in Chemistry awarded to Israeli scientist Dan Shechtman.

Throughout this half century, many mathematicians wondered if there might be a single tile, rather than a set, that could cover the plane aperiodically. The problem was nicknamed the “Einstein” problem, not because of any relation to the famous physicist, but as a pun, with the words “Ein Stein” being German for “One Stone”. In 2010 Socolar and Taylor got close, publishing a single tile that could solve this problem— but their tile was unconnected, consisting of a central bumpy hexagon and six floating double-square shapes that orbited it in known positions. A nice breakthrough, but arguably not a single tile, depending on how you define the concept. Other researchers found a different solution that was contiguous, but only worked if some small overlaps were allowed; also an interesting result, but not really a satisfying solution to the Einstein problem.

The real solution finally came earlier this year, published by Smith, Myers, Kaplan, and Goodman-Strauss: a single tile that would actually cover the plane without regularly repeating a pattern. It was formed by stitching together eight kite shapes into something that looked like an odd-shaped hat: not trivial, but surprisingly simple for something that had evaded discovery for almost fifty years. They also proved that this was just a representative of an infinite family of shapes with this property. You can see it illustrated in some of the articles linked in the show notes at mathmutation.com.

The most surprising thing about this discovery was the fact that it was discovered by an amateur— it didn’t stem from efforts by a professional mathematician. David Smith is a retired print technician who enjoys jigsaw puzzles and playing with shapes. He was using a commonly available software package called the PolyForm puzzle solver, seeing what kind of interesting patterns he could make with different shaped tiles. When he saw that the patterns with his experimental “hat” tile looked especially unusual, he decided to cut out a bunch of this tile out of cardboard and start experimenting by hand. Convincing himself that he had found something significant, he contacted a professor he knew, Craig Kaplan of the University of Waterloo. Together, they investigated further, and after Kaplan recruited two more colleagues to assist with the proof, they confirmed that Smith really had, after all these years, found an Einstein tile.

As a result, if you’ve been thinking of remodeling your bathroom, there is now a nice new floor decoration option you should seriously consider.

And this has been your math mutation for today.

References:

https://cs.uwaterloo.ca/~csk/hat/

https://aperiodical.com/2023/03/an-aperiodic-monotile-exists/

https://hedraweb.wordpress.com/2023/03/23/its-a-shape-jim-but-not-as-we-know-it/

https://en.wikipedia.org/wiki/Aperiodic_tiling

https://en.wikipedia.org/wiki/Penrose_tiling

https://en.wikipedia.org/wiki/Quasicrystal

https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/

https://maxwelldemon.com/2010/04/01/socolar_taylor_aperiodic_tile/