Science

# Hat trick: Newly discovered shape solves 60-year mathematical mystery

Hat trick: Newly discovered shape solves 60-year mathematical mystery
A tiling of a 13-sided shape called the Hat, the first known example of a shape that can be tiled without gaps or overlaps, and without repeating a pattern even across an infinite plane
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A tiling of a 13-sided shape called the Hat, the first known example of a shape that can be tiled without gaps or overlaps, and without repeating a pattern even across an infinite plane
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An animation of the Hat cycling through a family of related shapes that can also be classed as "aperiodic monotiles"

A new shape has been discovered which would make for a pretty unique set of bathroom tiles. Nicknamed “the Hat,” the 13-sided shape can tesselate with itself without ever repeating a pattern, solving a 60-year-old mathematical mystery.

This very niche milestone was made possible by David Smith, a self-proclaimed tiling enthusiast from Yorkshire, England. Along with many others in the community, Smith had been hunting for a specific kind of shape called an aperiodic monotile or an “einstein.” That name has nothing to do with the famous physicist but is instead a play on the German words ein (meaning one) and stein (meaning stone).

This “one stone” is a formerly hypothetical shape that could tile a plane without any overlaps or gaps, and even if it was stretched out over an infinite space would never repeat the same pattern – and physically can’t be made to repeat. Smith’s Hat is the first such shape found to fit the bill.

It might not sound like a big deal for many people, but mathematicians have been searching for a puzzle piece with these properties since the mid-1960s. That’s when the first set of shapes to exhibit aperiodic (non-repeating) tiling was discovered, but the toolbox required over 20,000 different shapes to ensure the pattern never repeated. Further work over the next decade whittled that number down smaller and smaller, until Sir Roger Penrose got it right down to just two, forming arrangements now known as Penrose tilings.

“Those two shapes in Penrose’s solution had enough structure that they forbid periodicity,” said Professor Craig Kaplan, a researcher on a study describing the new shape. “But for almost 50 years mathematicians have been wondering, can we get down to just one shape? Can we do this with a monotile? That’s the problem we solved. We found a single shape that does what all these earlier sets of multiple shapes are able to do.”

Smith had originally been experimenting with paper cutouts of shapes, but there’s only so much you can test with a finite plane. To confirm the Hat’s special abilities, he got in contact with Kaplan, who had recently developed software that could check a given shape on larger scales. For example, it can identify all the different ways the shape can be arranged into small groupings or “neighborhoods,” and then work out if these can exist in a larger tiling without breaking the rules.

The Hat might not be the only possible aperiodic monotile, either. The team says that technically, it’s part of a family of very similar shapes with slight tweaks that still follow the same rules.

“The more interesting question is are there fundamentally different aperiodic monotiles?” said Kaplan. “My answer is that there’s no reason to suspect otherwise and every reason to suspect there ought to be others.”

Research describing the new shape is available on the preprint server ArXiv.

Source: University of Waterloo

vinnyviking
Define repeating!

If not looking at the colours, I already in the first image can see the patterns repeating itself.
Also, where is the description of the 60-year mathematical mystery?

First thought; it looks more like a t-shirt than a hat. Second thought; The video show a bunch of different shapes can maintain the same layout. And lastly, I find it a dubious claim to affirm there is no repeating pattern ever. There might be a repeating pattern at an incredibly large number such as a billion by a billion.
joerauh
Penrose tiles. If this is a single shape, it's a variation of Penrose tiles. Found also in a Mosque over 400 years old.
FoFu
Wait a minute. Am I crazy, or do you thing you can so easily gaslight us all with "math" thus "you'll never understand"? If you remove the thick black lines and colors, it's nothing but a bunch of hexagons. The "new shape" is just a 3 hexagon cut out, 4 segments from one and 2 from the other 2. And how are they not repeating patterns? You mean in their orientations? They are in fact All repeating patterns, including orientations. What kind of dopey ass article is this? And what kind of junk math are these people wasting time on?
McDesign