Unsolved math problem uncovers new shape


An Ontario scientist helped discover an “einstein” – a longstanding mathematical problem deemed impossible for more than 60 years – until now.

An “einstein” is a formation of shapes that perfectly interlocks with one another, without gaps or overlaps, just like the bricks that assemble a house. In an einstein, however, the pattern of shapes never repeats.

Until this month, scientists had yet to discover a shape that could create such a pattern.

“The problem we were trying to solve was whether there is a single shape that had this property,” Craig Kaplan, an associate professor in computer science at the University of Waterloo, told CTV News Toronto.

In the math community, this concept is referred to as an “aperiodic monotile,” also the name of a research paper co-authored by Kaplan earlier this month.

“What we found is a single shape that can never make periodic tilings.”

That shape turned out to be what Kaplan and his colleagues have called “the hat” – tied to its resemblance of a fedora. Visually, it resembles eight kites glued together to make a single 13-sided shape.

A tile formation of shapes that perfectly interlock with one another and never repeat (Craig Kaplan). The longstanding belief that an “aperiodic monotile” could not exist was put forward by mathematician Hao Wang in 1961.

Soon after, one of Wang’s students discovered an aperiodic set of 20,426 tiles, and then, a set of 104. In the 1970s, mathematical physicist and Nobel Prize winner Roger Penrose reduced that number down to two.

But over the last several decades, progress hit a standstill.


Kaplan became aware of the existence of “the hat” after receiving an email last November “out of the blue” from David Smith, a retired printing technician in England.

“He emailed me and asked if the work I had done previously might be useful in trying to understand its properties,” Kaplan recalled.

Smith was interested in software Kaplan had previously crafted that had the capability to automate a visualization of hundreds of hats glued together. Smith wanted to use Kaplan’s software to find out if this shape could solve the longstanding problem.

Until this point, Smith had been physically cutting his newfound shape out of cardstock, a pursuit he chronicled in his personal blog, to see if his “hat” could in fact solve the unsolvable.

Within just five months of connecting, Smith, Kaplan and two others, Chaim Goodman-Strauss and Joseph Myers, published a research paper on the previously impossible “aperiodic monotile” in Cornell University’s archive while awaiting a peer-review.

“It settles a longstanding open problem in mathematics,” Kaplan said.

“That sense of discovery is profound.”  

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