The Discovery of the Aperiodic Monotile: How David Smith Changed Mathematics

Category Science

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David Smith, a retired print technician from Yorkshire, England, and a team of mathematicians, programmers and tiling enthusiasts, have solved a nearly 60-year-old mathematical problem by discovering a single shape that tiles the infinite plane without repeating and overlapping. This discovery, known as the aperiodic monotile, has delighted mathematicians since its discovery by Robert Berger in the mid-1960s.

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A nearly 60-year-old mathematical problem has finally been solved.The story began last fall when David Smith, a retired print technician from Yorkshire, England, came upon a shape with a tantalizing property. The life-long tiling enthusiast discovered a 13-sided shape—dubbed the hat—that is able to fill the infinite plane without overlaps or gaps in a pattern that not only never repeats but also never can be made to repeat.

It took mathematicians 57 years to find the elusive Einstein after being first discovered by Robert Berger in the mid-1960s

This elusive shape is known to mathematicians as an aperiodic monotile or an einstein, a clever pun that takes its name from the German words ein and stein that mean one stone.

"Dave and I had been in touch over the years and we belong to the same old-fashioned listserv for people interested in tiling, a mix of tiling enthusiasts, programmers and mathematicians," recalls Cheriton School of Computer Science professor Craig S. Kaplan, who collaborated with Smith, software developer Joseph Myers and mathematician Chaim Goodman-Strauss on the paper that has proven that the elusive einstein exists.

The iconic Penrose tiles consist of two pentagonal shapes, which was an important breakthrough at the time

"Dave was on to something big, something historic, but he hit the wall on what he could deduce about this shape by working with paper cut-outs. He knew I had recently published a paper about a related topic for which I developed a piece of software that we could use to understand what his shape was doing. He sent me an email asking, 'Hey, can you run this through your software and see what happens?'" .

The elegant solution discovered by David Smith and the team demonstrates the power of collaboration between mathematicians, programmers and tiling enthusiasts

Mathematicians had been trying to find a shape like David Smith's einstein since the 1960s when American mathematician Robert Berger discovered the first example of aperiodic tiling.

"Berger's aperiodic set of shapes was found in the mid-1960s and that set had 20,426 shapes," Professor Kaplan explained. "It was an elaborate construction with a combinatorial set of features that required a multiplicity of shapes to guarantee that the pattern doesn't repeat. That was an important discovery, but the natural next question for mathematicians is, can we get smaller sets? What's the lowest number of shapes we can do this with?" .

The team used a piece of software to analyze Smith's hat that was developed by Craig S. Kaplan in a related paper published earlier

By 1970, the set of shapes proven to tile aperiodically was down to about 100 and in 1971 mathematician Raphael Robinson got it down to six. Then, in 1974, Sir Roger Penrose discovered the eponymous Penrose tiles, which reduced the number to two.

As shown in this looping animated GIF, the hat is one member of a continuous family of shapes that are all aperiodic, and that all tile the plane in the same way. Credit: arXiv (2023). DOI: 10.48550/arxiv.2303.10798 .

The family of shapes discovered by the team is composed of a continuous loop of shapes that tile the plane without overlaps or gaps

"Those two shapes in Penrose's solution had enough structure that they forbid periodicity. But for almost 50 years mathematicians have been wondering, can we get down to just one shape? Can we do this with a monotile? That's the problem we solved. We found a single shape that does what all these earlier sets of multiple shapes are able to do." .

In mathematics and computer science many problems remain open, but theoreticians have a strong sense what the answer will be even though a formal proof may be decades away.

This breakthrough solves an important puzzle that is widely studied in mathematics and computer science alike

"The famous P vs. NP problem in computer science—a question about how long it takes to execute a particular class of algorithms—is still open, but there's a consensus how that's going to play out," Professor Kaplan said. "Almost every computer scientist thinks that P is no equal to NP, but there's no proof yet. Same here. There was a widespread belief among mathematicians that there should be a single tile, an einstein, that can tile the plane aperiodically, but no one had proven it yet. That's the problem we solved." .

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