An amateur discovers a new mathematical form and astounds the greatest scientists

Searching for an “incredible” geometric pattern: such is the favorite pastime of David Smith, a pacifist British pensioner, that he indulged in when he encountered an unprecedented shape with remarkable properties in November, unleashing the enthusiasm of a community of enthusiasts. and the admiration of scholars. After he announced his discovery last March, these particular enthusiasts printed this new shape on T-shirts, made cookies with this design and even thought of drawing it on their bodies.

This thirteen-sided polygon, dubbed a “hat”, is the first pattern that can be stacked infinity without showing an overall repeating pattern – for example, a diamond that has been stacked to infinity will produce other diamonds in At some point a great deal. As such, the “hat” is the first “Einstein,” named after a problem posed 60 years ago that mathematicians assumed was unsolvable. David Smith, 64, has done better since then, however, with “Spectrum.” Because the “hat” had a small drawback: you had to turn the pattern over once every seven moves (or every seven pieces, like a puzzle) to avoid the same repeated shape appearing.

Retired, with the help of three mathematicians, showed in a forthcoming study that the “spectrum” was pure “Einstein”. This last name is from the German “ein Stein” (stone), and has nothing to do with the famous physicist’s name.

Hat, turtle and cute

For Craig Kaplan, a professor of computer science at Canada’s University of Waterloo, this is “a funny, almost funny, but fascinating story.” He says he was contacted in November 2022 by David Smith, a former printing technician in Yorkshire (northern England): he found a pattern that “didn’t behave the way one might expect”. If several copies of this pattern are stacked on a table, no general pattern will appear.

A computer program confirmed that this is the first “Einstein”, which in scientific language is also called a “non-periodic monad”. Their work was noted by a proponent of this tile handling, the Japanese scholar Yoshiaki Araki, who created works of art using the “hat” and a variant called the “tortoise”. Encouraged, our British pensioner is trying to find a new style that doesn’t require periodic reinstatement.

Mission accomplished in less than a week, in the face of the amazing Craig Kaplan. The Canadian computer scientist adds that an analysis confirmed that this new square was an “Einstein without a reflection.” To be sure, the hobbyist and scientist “optimized” the figure, so that it couldn’t be used with a reflection. The “Ghost” was born.

“falls from the sky”

Both scientific papers are still being considered in scientific journals before publication, but the mathematician did not wait to comment on the news. The discovery is “exciting, amazing, amazing,” says Marjorie Senechal, a mathematician at Smith College (Massachusetts). Who sees in it more than just a beautiful story.

The new design and its variants should lead to “a deeper understanding of the system in nature and the nature of the system”. For Doris Schachneider, a mathematician at Moravian University (Pennsylvania), the two figures are “impressive.” Even 2020 Nobel mathematician and physicist Roger Penrose, who specializes in aperiodic tiles, doubted that such a feat was possible, she noted. The prestigious University of Oxford is organizing an event in July to celebrate this discovery, the Hatfest (hat party), in which Roger Penrose will be taking part.

The discovery is so amazing that “the answer fell from the sky and from the hands of an amateur,” Craig Kaplan asserts. “And in the nicest way, thanks to a lover of the subject he explores beyond any professional goal.”