Media Highlights

Abstract

In the 1960s Robert Berger constructed a set of 20,426 tiles that tile the plane aperiodically but cannot tile it periodically. The race to find smaller sets of such tiles culminated in the 1970s with Roger Penrose’s discovery of a set of just two tiles with this property—his famous kite and dart tiles. Since then, mathematicians have wondered whether there might be one single tile that tiles the plane aperiodically but not periodically, a so-called “einstein,” playing on the German “ein stein” (one stone). Fifty years of searching could not produce an einstein. Does one really exist? In November 2022, David Smith, a retired print technician in northern England, was experimenting with tiles formed by assembling pieces of hexagons in different ways, and found that a 13-sided tile made of four pieces of hexagons, which he called the “hat,” seemed to be able to cover larger and larger areas without repeating a pattern. “It’s a tricky little tile.” Smith enlisted the help of Craig Kaplan, a computer scientist at the University of Waterloo, Chaim Goodman-Strauss at the National Museum of Mathematics and the University of Arkansas, and Joseph Samuel Myers, a software engineer in Cambridge, England, and on March 20, 2023 they announced that the hat did tile the plane, but only aperiodically. It was the long-sought einstein, and so simple, “hiding in plain sight!” Marjorie Senechal called the discovery “Just mind-boggling!” Doris Schattschneider described herself as “flabbergasted.” Meanwhile, Smith announced that he had found another, even simpler, tile he called the “turtle,” which also turned out to be an einstein, and in fact the team discovered that they could morph the hat into the turtle through a continuum of tiles that tile the plane only aperiodically. The Quanta article has a moving graphic showing a hat tiling morphing into a turtle tiling. You should look at the pictures, but the turtle is easy to describe. Think of a hexagon as made up of six equilateral triangles, then add one more equilateral triangle on one edge of the hexagon, and another equilateral triangle on a side of that one. It’s a hexagon with a snout.