A trick of the hat

April 13, 2023 The story of how a Waterloo computer science professor helped find the elusive einstein tile By Joe Petrik Cheriton School of Computer Science 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. 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 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. "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.