“A 13-sided shape known as “the hat” has mathematicians tipping their caps. It’s the first true example of an ‘einstein,’ a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats. ‘Everybody is astonished and is delighted, both,’ says mathematician Marjorie Senechal of Smith College in Northampton, Mass., who was not involved with the discovery. Mathematicians had been searching for such a shape for half a century. ‘It wasn’t even clear that such a thing could exist'”

“Although the name ‘einstein’ conjures up the iconic physicist, it comes from the German ein Stein, meaning ‘one stone,’ referring to the single tile. The einstein sits in a weird purgatory between order and disorder. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can’t form a pattern that repeats.”—”Mathematicians have finally discovered an elusive ‘einstein’ tile. A 13-sided shape called ‘the hat’ forms a pattern that never repeats.”

“An aperiodic monotile. A longstanding open problem asks for an aperiodic monotile, also known as an ‘einstein’: a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the ‘hat’ polykite, can form clusters called ‘metatiles’, for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical — and hence aperiodic — tilings.”