Joseph Samuel Myers<\/a>, a software engineer in Cambridge, England, with a doctorate in combinatorics. Myers started devoting all his spare time to the hat tile, and in just over a week, he had proved that it is aperiodic. \u201cWe were all pretty blown away by how quickly he nailed this all down,\u201d Kaplan said.<\/p>\nThe proof adapted an approach originated by Berger in the 1960s, which involves piecing together tiles into larger versions of themselves, creating a hierarchical structure. Myers began by identifying four intermediate shapes built of hats, which he called H<\/em>, T<\/em>, P<\/em> and F<\/em>. An H<\/em>, for example, is made of four hats, joined together to form a shape roughly like a triangle with its tips chopped off. Myers showed that you can put together combinations of the four shapes to make bigger ones. You can make a bigger H<\/em>, for instance, by surrounding one T<\/em> with three H<\/em>\u2019s, then surrounding this object with a combination of P<\/em>\u2019s and F<\/em>\u2019s.<\/span><\/p>\nThis provided a way to make bigger and bigger hat tilings. You can start with an H<\/em>, increase its size, then fill it with the above combination of the four shapes. Next, you can inflate this whole assembly and fill all the shapes inside the (now huge) H<\/em> with assortments of H<\/em>, T<\/em>, P<\/em> and F<\/em> shapes. You can repeat these steps indefinitely, building an increasingly large hierarchy of shapes within shapes. At the bottom level of the hierarchy is the hat.<\/p>\nThe researchers proved that the tilings created by these hierarchical constructions are never periodic. They also proved that hierarchical constructions are the only way to make hat tilings. Therefore, a hat tiling can never be periodic. \u201cIt\u2019s a very cool result,\u201d Socolar said.<\/p>\n