In search of an ‘amazing’ geometric pattern: it’s the favorite pastime of David Smith, a British pensioner, who indulged in it when he stumbled upon a new shape with remarkable properties in November, sparking the excitement of a community of enthusiasts. And the admiration of scholars.

After making his discovery public last March, these particular enthusiasts 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 creating an overall repeating pattern — for example, a diamond composited to infinity from other diamonds will eventually make one large diamond. This makes “the hat” the first “Einstein”, named after a problem that was posed 60 years ago and which mathematicians assumed 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 and ghost

Craig Kaplan, a professor of computer science at Canada’s University of Waterloo, says he was contacted by David Smith in November 2022: he found a pattern “that was not behaving as one 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.

Encouraged, the British pensioner then tries to find a new reason that doesn’t require regular return. Mission accomplished in less than a week. However, an analysis has confirmed that this new tile is “an Einstein without inversion”, adds the 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.

An “exciting, surprising and amazing” discovery

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.