Chip
April 4th, 2023, 10:20 PM
‘The miracle that disrupts order’: mathematicians invent new ‘einstein’ shape
https://i.guim.co.uk/img/media/c8b7d1cc84d28a578a069f6f3f79d82a5c221f91/0_18_1200_720/master/1200.jpg?width=1140&quality=85&dpr=1&s=none
Called ‘the hat’, the 13-sided shape can be arranged in a tile formation such that it never forms a repeating grid
Matthew Cantor
4 Apr 2023
One of mathematics’ most intriguing visual mysteries has finally been solved – thanks to a hobbyist in England.
The conundrum: is there a shape that can be arranged in a tile formation, interlocking with itself ad infinitum, without the resulting pattern repeating over and over again?
In nature and on our bathroom walls, we typically see tile patterns that repeat in “a very predictable, regular way”, says Dr Craig Kaplan, an associate professor of computer science at the University of Waterloo in Ontario. What mathematicians were interested in were shapes that “guaranteed non-periodicity” – in other words, there was no way to tile them so that the overall pattern created a repeating grid.
Such a shape would be known as an aperiodic monotile, or “einstein” shape, meaning, in roughly translated German, “one shape” (and conveniently echoing the name of a certain theoretical physicist).
“There’s been a thread of beautiful mathematics over the last 60 years or so searching for ever smaller sets of shapes that do this,” Kaplan says. “The first example of an aperiodic set of shapes had over 20,000 shapes in it. And of course, mathematicians worked to get that number down over time. And the furthest we got was in the 1970s,” when the Nobel-prize winning physicist Roger Penrose found pairs of shapes that fit the bill.
Now, mathematicians appear to have found what they were looking for: a 13-sided shape they call “the hat”. The discovery was largely the work of David Smith of the East Riding of Yorkshire, who had a longstanding interest in the question and investigated the problem using an online geometry platform. Once he’d found an intriguing shape, he told the New York Times, he would cut it out of cardstock and see how he could fit the first 32 pieces together.
“I am quite persistent but I suppose I did have a bit of luck,” Smith told the Guardian in an email.
https://www.theguardian.com/science/2023/apr/03/new-einstein-shape-aperiodic-monotile?CMP=Share_iOSApp_Other
https://i.guim.co.uk/img/media/c8b7d1cc84d28a578a069f6f3f79d82a5c221f91/0_18_1200_720/master/1200.jpg?width=1140&quality=85&dpr=1&s=none
Called ‘the hat’, the 13-sided shape can be arranged in a tile formation such that it never forms a repeating grid
Matthew Cantor
4 Apr 2023
One of mathematics’ most intriguing visual mysteries has finally been solved – thanks to a hobbyist in England.
The conundrum: is there a shape that can be arranged in a tile formation, interlocking with itself ad infinitum, without the resulting pattern repeating over and over again?
In nature and on our bathroom walls, we typically see tile patterns that repeat in “a very predictable, regular way”, says Dr Craig Kaplan, an associate professor of computer science at the University of Waterloo in Ontario. What mathematicians were interested in were shapes that “guaranteed non-periodicity” – in other words, there was no way to tile them so that the overall pattern created a repeating grid.
Such a shape would be known as an aperiodic monotile, or “einstein” shape, meaning, in roughly translated German, “one shape” (and conveniently echoing the name of a certain theoretical physicist).
“There’s been a thread of beautiful mathematics over the last 60 years or so searching for ever smaller sets of shapes that do this,” Kaplan says. “The first example of an aperiodic set of shapes had over 20,000 shapes in it. And of course, mathematicians worked to get that number down over time. And the furthest we got was in the 1970s,” when the Nobel-prize winning physicist Roger Penrose found pairs of shapes that fit the bill.
Now, mathematicians appear to have found what they were looking for: a 13-sided shape they call “the hat”. The discovery was largely the work of David Smith of the East Riding of Yorkshire, who had a longstanding interest in the question and investigated the problem using an online geometry platform. Once he’d found an intriguing shape, he told the New York Times, he would cut it out of cardstock and see how he could fit the first 32 pieces together.
“I am quite persistent but I suppose I did have a bit of luck,” Smith told the Guardian in an email.
https://www.theguardian.com/science/2023/apr/03/new-einstein-shape-aperiodic-monotile?CMP=Share_iOSApp_Other