(Paris) In search of an ‘amazing’ geometric pattern: it’s the favorite pastime of David Smith, a peaceful British pensioner, who was looking into it when he stumbled upon a new shape with remarkable properties and sparked a community’s excitement in November enthusiasts. And the admiration of scholars.
Posted at 8:08 am
Daniel Lawler Agence France-Presse
After making his discovery public last March, these particular enthusiasts have printed this new shape on t-shirts, baked 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 composited to infinity without showing an overall repeating pattern — for example, a diamond composited to infinity from other diamonds will eventually yield a large diamond.
Thus, “the hat” is the first “Einstein,” named after a problem posed 60 years ago that mathematicians thought was unsolvable.
David Smith, 64, has since done better with “the spectrum.” Because “the hat” had a small disadvantage: You had to turn the pattern over once every seven moves (or every seven pieces, like in a puzzle) to avoid the appearance of the same repeating shape.
The retiree, with the help of three mathematicians, has proven in a forthcoming study that “the spectrum” is pure “Einstein.” This surname derives from the German “ein Stein” (a stone) and has nothing to do with that of the famous physicist.
Hat, Turtle and Spectrum
For Craig Kaplan, a professor of computer science at Canada’s University of Waterloo, it’s “a funny and almost ridiculous but wonderful story,” he told AFP.
He says he was contacted in November 2022 by Mr Smith, a former print engineer in Yorkshire (northern England): He found a pattern “that was not behaving as you would expect.”
If multiple copies of this pattern were placed together on a table, no overall pattern would appear. A computer program confirmed that it was the first “Einstein”, also known as the “aperiodic monotile” in technical jargon.
Her work was noticed by a proponent of handling these tiles, the Japanese explorer Yoshiaki Araki, who created artwork featuring the “hat” and a variant called “the turtle”.
Encouraged, our British pensioner then tries to find a new pattern that doesn’t require regular returns. Mission accomplished in less than a week in the face of an incredulous Craig Kaplan.
But an analysis has confirmed that this new tile is “an Einstein without inversion,” adds the Canadian computer scientist. And just to be safe, the hobbyist and scientist even “improved” the shape so that it can’t be used with an inversion. “The Ghost” was born.
“Falling from the Sky”
Both scientific papers are still being scrutinized in scientific journals ahead of their publication, but the world of mathematics hasn’t waited to comment on the news.
This discovery is “exciting, surprising and amazing,” Marjorie Senechal, a mathematician at Smith College (Massachusetts), told AFP. Who sees in it more than just a beautiful story. The new motif and its variants should “lead to a deeper understanding of the order in nature and the essence of order”.
For Doris Schattschneider, a mathematician at Moravian University (Pennsylvania), the two forms are “impressive”. Even mathematician and 2020 Nobel Prize in Physics Roger Penrose, a specialist in aperiodic tiling, doubted that such a feat was possible, she notes.
The prestigious University of Oxford is organizing an event in July to celebrate this discovery: the Hatfest (hat party), which Roger Penrose will attend.
This discovery is all the more amazing because “the answer fell from heaven and from the hands of an amateur,” points out Craig Kaplan. “And in the most beautiful way, thanks to a lover of the subject who deals with it independently of professional goals.”