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The Shape of Things to Come

April7

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 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.”

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.

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.”

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.

“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.” [source: from cnn]

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