After he made his discovery public last March, these fans of a particular kind printed on t-shirts this new shape, made cookies according to this pattern and even considered tattooing it on their bodies.

This thirteen-sided polygon, called "the hat", is the first pattern that can be assembled infinitely without revealing an overall pattern that repeats itself --for example, a diamond assembled infinitely to other diamonds will at some point produce a large rhombus.

As such, "the hat" is the first "einstein", named after a problem posed 60 years ago and that mathematicians assumed to be insoluble.

David Smith, 64, has done better since then, with "the spectre". Because "the hat" had a small disadvantage: it was necessary to turn the pattern once every seven strokes (or every seven pieces, as for a puzzle) to avoid the appearance of the same shape repeating itself.

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

## Hat, turtle and spectrum

Craig Kaplan, a professor of computer science at Canada's University of Waterloo, said it was "a funny and almost ridiculous story, but wonderful," he told AFP.

Hats and ghosts solve 60-year-old © math problem Julia Han JANICKI, Sabrina BLANCHARD / 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 motive "that did not behave in the way you would expect".

If several copies of this pattern were assembled on a table, no overall pattern appeared. 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 follower of handling 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 reason that does not require to return it periodically. Mission accomplished in less than a week, facing an incredulous Craig Kaplan.

But an analysis confirmed that this new tile was "an einstein without inversion", adds the Canadian computer scientist. And to make sure of this definitively, the amateur and the scientist have even "improved" the shape, so that it cannot be used with an inversion. "The spectre" was born.

## "Fallen from the sky"

Both scientific articles are still being studied in scientific journals before publication, but the math world did not wait to comment on the news.

The discovery is "exciting, surprising and astonishing," Marjorie Senechal, a mathematician at Smith College in Massachusetts, told AFP. Who sees it as 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, a mathematician at Moravian University in Pennsylvania, both forms are "impressive". Even mathematician and 2020 Nobel laureate 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 surprising because "the answer fell from the sky and from the hands of an amateur," says Craig Kaplan. "And in the most beautiful way, thanks to a lover of the subject, who explores it outside of any professional goal".