Mathematicians say they have invented an 'impossible' tile that never repeats

Called "The hat" because it vaguely resembles a fedora, the elusive shape is now publicly available. (David Smith/Joseph Samuel Myers/Craig S. Kaplan/Chaim Goodman-Strauss)

A geometry problem that has been puzzling scientists for 60 years has likely 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 the German "ein stein," or "one stone"). That means it can completely cover a surface without ever creating a repeated pattern — something that had not yet been achieved with a single tile.

"I'm always looking for an interesting shape, and this one was more than that," said David Smith, its creator and a retired printing technician from northern England, in a phone interview. Soon after discovering the shape in November 2022, he contacted a math professor and later, with two other academics, they released a self-published scientific paper about it.

"I'm not really into math, to be honest — I did it at school, but I didn't excel in it," Smith said. That's why I got these other guys involved, because there's no way I could have done this without them. I discovered the shape, which was a bit of luck, but it was also me being persistent."

Going from 20,426 to one

Most wallpapers or tiles in the real world are periodic, meaning you can identify a small cluster that's just constantly repeated to cover the whole surface. "The hat," however, is an aperiodic tile, meaning it can still completely cover a surface without any gaps, but you can never identify any cluster that periodically repeats itself to do so.

Fascinated by the idea that such aperiodic sets of shapes could exist, mathematicians first mulled the problem in the early 1960s, but they initially believed the shapes were impossible. That turned out to be wrong, because within years a set of 20,426 tiles that — when used together — could do the job was created. That number was soon reduced to just over 100, and then down to six.

In the 1970s, the work of British physicist and Nobel Prize winner Roger Penrose further reduced the number of shapes from six down to two in a system that has since been known as Penrose tiling. And that's where things were stuck for decades.

Smith became interested in the problem in 2016, when he launched a blog on the subject. Six years later, in late 2022, he thought he had bested Penrose in finding the einstein, so he got in touch with Craig Kaplan, a professor in the School of Computer Science at the University of Waterloo in Canada.

"From my perspective, it started with an email out of blue," Kaplan said in a phone interview. "David knew that I had recently published a paper describing a piece of software that could help him understand what was going on with the tile."

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

How 'The hat' works

There's nothing inherently magical about "The hat," according to Kaplan.

"It's really a very simple polygon to describe. It doesn't have weird, irrational angles, it's basically just something you get by cutting up hexagons." For that reason, he adds, it might have been "discovered" in the past by other mathematicians creating similar shapes, but they just did not think about checking its tiling properties.

The finding has created quite a stir since its release in late March. As Kaplan points out, it has inspired artistic renditions, knitted 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've got a feeling we'll have a different way of looking at how to find these sorts of anomalies, if you like."

The shape is publicly available, even for 3D printing, and it's not going to be copyrighted.

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

What about bathroom tiling? "I can only hope we'll see lots of bathrooms decorated with it, but it's going to be a little bit tricky," he added. "One of the reasons we use periodic tilings in places like bathrooms is because the rule for how to lay them is pretty simple. With this, you have a different challenge — you could potentially start laying it down out and hack yourself into a corner where you've created a space that you can't fill with more hats."

Peer review ahead

Far from being content with having rewritten math history, Smith has already discovered a "sequel" to "The hat." Called "The turtle," the new shape is also an einstein, but it's made of 10 kites, or sections, rather than eight, and therefore bigger than "The hat."

"It's a bit of an addiction," Smith confessed about his constant quest for new shapes.

The scientific paper on "The hat," coauthored with Joseph Myers, a software developer, and Chaim Goodman-Strauss, a mathematician at the University of Arkansas, has not yet gone through 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 really look forward to seeing what comes out of that process," Kaplan said, acknowledging that it could mean that the findings could be disputed. "I believe strongly in the importance of peer review as a way of conducting science. So, until that happens, I would agree that there should be reason to not be certain yet. But based on the evidence that we accumulated, it's hard to imagine a way that we could be wrong."

The discovery, once confirmed, could be significant across other fields 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, now many years ago, created a stir, since their existence was so unexpected.

"This new discovery is a strikingly simple example. There are no standard techniques known for finding new aperiodic tiles, so this involved a really new idea. That is always exciting," he added.

Mazzeo said it's also nice to hear of a mathematical discovery that is so easy to visualize and explain: "This illustrates that mathematics is still a growing subject, with many problems that have not yet been solved."