An amateur discovers a new mathematical form

(Paris) Looking for an “astonishing” geometric pattern: this is the favorite pastime of David Smith, a peaceful British pensioner, who was engaged in it when he came across a new shape with remarkable properties in November, unleashing enthusiasm of a community of enthusiasts. And the admiration of scholars.

After he made his discovery public last March, these particular enthusiasts have printed this new shape on t-shirts, made cookies with this design and even considered tattooing it on their bodies.

This thirteen-sided polygon, dubbed “the hat”, is the first pattern that can be assembled ad infinitum without showing an overall repeating pattern – for example, a diamond assembled ad infinitum at other diamonds will at some point produce a large diamond.

As such, “the hat” is the first “einstein”, named after a problem posed 60 years ago and which mathematicians assumed was insoluble.

David Smith, 64, has done better since, with “the spectrum”. Because “the hat” had a small drawback: you had to turn the pattern once every seven moves (or every seven pieces, like for a puzzle) to avoid the appearance of the same repeating shape.

The retiree, with the help of three mathematicians, has demonstrated in a forthcoming study that “the spectrum” is a pure “einstein”. This last name is taken from the German “ein Stein” (a stone), and unrelated to that of the famous physicist.

Hat, turtle and spectrum

For Craig Kaplan, professor of computer science at the Canadian University of Waterloo, it is “a funny and almost ridiculous, but wonderful story”, he told AFP.

He says he was contacted in November 2022 by Mr. Smith, a former printing technician in Yorkshire (north of England): he had found a pattern “which did not behave in the way you would expect. “.

If several copies of this pattern were assembled on a table, no overall pattern would appear. A computer program confirmed that it was the first “einstein”, also called in scholarly language an “aperiodic mono-tile”.

Their work was noticed by a proponent of the handling of these tiles, the Japanese researcher Yoshiaki Araki, who created works of art using the “hat” and a variant called “the turtle”.

Encouraged, our British retiree then tries to find a new pattern that does not require returning it periodically. Mission accomplished in less than a week, in the face of an incredulous Craig Kaplan.

But an analysis has confirmed that this new tile was “an einstein without inversion”, adds the Canadian computer scientist. And to make sure, the hobbyist and the scientist even “improved” the form, so that it cannot be used with an inversion. “The Specter” was born.

“Falling from the sky”

Both scientific papers are still being studied in scientific journals before publication, but the world of mathematics has not waited to comment on the news.

This discovery is “exciting, surprising and astonishing”, declared to AFP Marjorie Senechal, mathematician at Smith College (Massachusetts). Who sees in it more than just a beautiful story. The new motif and its variants should “lead to a deeper understanding of order in nature and the nature of order”.

For Doris Schattschneider, mathematician at Moravian University (Pennsylvania), the two forms are “impressive”. Even the mathematician and Nobel Physics 2020 Roger Penrose, a specialist in aperiodic tiles, doubted that such a feat was possible, she notes.

The prestigious University of Oxford is organizing an event in July celebrating this discovery, the Hatfest (hat party), in which Roger Penrose will participate.

This discovery is all the more astonishing as “the answer fell from the sky and from the hands of an amateur”, emphasizes Craig Kaplan. “And in the most beautiful way, thanks to a lover of the subject, who explores it outside of any professional objective”.

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