The hat: mathematicians invent the “einstein” tile that never repeats

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A geometry problem that has baffled scientists for 60 years has probably just been solved by an amateur mathematician with a newly discovered 13-sided shape.

Called “The Hat” because it vaguely resembles a fedora, the elusive shape is an “einstein” (from German “ein stein” or “a stone”). This means that it can completely cover a surface without ever creating a repeating pattern, which had not been achieved before with a single tile.

“I’m always on the lookout for an interesting shape, and this one was more than that,” David Smith, its designer and retired print technician from the north of England, said in an interview. telephone. Shortly after discovering the shape in November 2022, he contacted a math professor and later, along with two other academics, published a self-published scientific paper about it.

“To be honest, I’m not really into math — I did it in school, but I didn’t excel at it,” Smith said. That’s why I got these other guys involved, because I couldn’t have done this without them. I discovered the form, it was a bit of luck, but it was also my perseverance.

Most wallpapers or tiles in the real world are periodic, which means you can identify a small group that constantly repeats to cover the entire surface. “The Hat”, however, is an aperiodic tile, meaning it can always completely cover an area without any gaps, but you can never identify a periodically repeating group to do so.

Fascinated by the idea that such sets of aperiodic forms could exist, mathematicians first pondered the problem in the early 1960s, but initially believed the forms to be impossible. This turned out to be wrong, because within a few years a set of 20,426 tiles was created that when used together could do the job. This number was quickly reduced to just over 100, then to six.

In the 1970s, the work of British physicist and Nobel laureate Roger Penrose further reduced the number of shapes from six to two in a system known since as the Penrose tiling. And that’s where things got stuck for decades.

Smith became interested in the issue in 2016, when he started a blog on the subject. Six years later, in late 2022, he thought he had beaten Penrose to find the Einstein, so he got in touch with Craig Kaplan, a professor at the School of Computer Science at the University of Waterloo in Canada.

“From my perspective, it all started with an out-of-the-box email,” Kaplan said in a phone interview. “David knew I had recently published an article describing software that might help him figure out what was going on with the tile.”

With the help of the software, the two realized they were onto something.

There’s nothing inherently magical about “The Hat,” according to Kaplan.

“It’s really a very simple polygon to describe. It doesn’t have any weird, irrational angles, it’s just something you get by cutting out hexagons. For this reason, he adds, it may have been “discovered” in the past by other mathematicians creating similar shapes, but they just didn’t think to check its tiling properties.

The discovery has created quite a stir since its publication in late March. As Kaplan points out, he inspired artistic renderingsknitted quilts, cookie cutters, TikTok explainers, and even a joke in one of Jimmy Kimmel’s opening monologues.

“I think it might open a few doors,” Smith said, “I have a feeling we’ll have a different way of looking at finding these kinds of anomalies, if you want.”

The shape is publicly available, even for 3D printing, and it will not be copyrighted.

“We’re not trying to protect him in any way,” Kaplan said. “It belongs to everyone, and I hope people will use it in all kinds of decorative, architectural and artistic content.”

And the bathroom tiles? “I can only hope we see a lot of bathrooms decorated with this, but it’s going to be a little tricky,” he added. “One of the reasons we use periodic tiling in places like bathrooms is that the laying rule is quite simple. With that you have a different challenge – you could potentially start laying it and sneak into a corner where you’ve created a space that you can’t fill with more hats.

Far from being satisfied with having rewritten the history of mathematics, Smith has already discovered a “sequel” to “The Hat”. Called “The turtle“, the new form is also an einstein, but it is made up of 10 kites, or sections, instead of eight, and therefore larger than “The Hat”.

“It’s a bit of an addiction,” Smith confessed of her constant quest for new forms.

The science paper on “The Hat,” co-authored with Joseph Myers, a software developer, and Chaim Goodman-Strauss, a mathematician at the University of Arkansas, has yet to be reviewed. peer review – the process of verification by other scientists that is standard in scientific publications – but will do so over the next few months.

“I’m really looking forward to seeing what comes out of this process,” Kaplan said, acknowledging that it could mean the findings could be challenged. “I strongly believe in the importance of peer review as a way of doing science. So until that happens, I agree there should be some reason to be unsure yet. But based on the evidence we’ve accumulated, it’s hard to imagine how we could be wrong.

The finding, once confirmed, could be important in other areas of research, according to Rafe Mazzeo, a professor in the department of mathematics at Stanford University, who was not involved in the study.

“Tilings have many applications in physics, chemistry and beyond, for example in the study of crystals,” he said in an email. “The discovery of aperiodic tilings, many years ago now, caused a sensation, because their existence was so unexpected.

“This new discovery is a surprisingly simple example. There are no known standard techniques for finding new aperiodic tiles, so this involved a really novel idea. It’s always exciting,” he added.

Mazzeo said it was also nice to hear about a mathematical discovery that was so easy to visualize and explain: “It shows that mathematics is still a growing subject, with many problems that have not yet been resolved.”

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