Mathematicians discover an elusive ‘einstein’ tile, with Fibonacci undertones

Image credit: D. SMITH, J.S. MYERS, C.S. KAPLAN AND C. GOODMAN-STRAUSS (CC BY 4.0)]

Something different here: ‘A 13-sided shape called ‘the hat’ forms a pattern that never repeats’, they say. We note an extra feature: each ‘hat’ contains 8 ‘kites’, and 13 and 8 are Fibonacci numbers.
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After half a century, mathematicians succeed in finding an ‘einstein,’ a shape that forms a tiled pattern that never repeats, says Science News.

A 13-sided shape known as “the hat” has mathematicians tipping their caps.

It’s the first true example of an “einstein,” a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.

“Everybody is astonished and is delighted, both,” says mathematician Marjorie Senechal of Smith College in Northampton, Mass., who was not involved with the discovery. Mathematicians had been searching for such a shape for half a century. “It wasn’t even clear that such a thing could exist,” Senechal says.

Although the name “einstein” conjures up the iconic physicist, it comes from the German ein Stein, meaning “one stone,” referring to the single tile. The einstein sits in a weird purgatory between order and disorder. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can’t form a pattern that repeats.

With a periodic pattern, it’s possible to shift the tiles over and have them match up perfectly with their previous arrangement. An infinite checkerboard, for example, looks just the same if you slide the rows over by two.

While it’s possible to arrange other single tiles in patterns that are not periodic, the hat is special because there’s no way it can create a periodic pattern.

Identified by David Smith, a nonprofessional mathematician who describes himself as an “imaginative tinkerer of shapes,” and reported in a paper posted online March 20 at arXiv.org, the hat is a polykite — a bunch of smaller kite shapes stuck together.

That’s a type of shape that hadn’t been studied closely in the search for einsteins, says Chaim Goodman-Strauss of the National Museum of Mathematics in New York City, one of a group of trained mathematicians and computer scientists Smith teamed up with to study the hat.

It’s a surprisingly simple polygon.

Full article here [includes brief video demo.]
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Talkshop note: If we wanted to push the Fibonacci aspect further, each double kite has 5 sides.