Consider the tiles on a bathroom floor or wall; they’re often arranged in a repeating pattern. But is there a single shape that tiles such a surface — an infinite one — in a pattern that never repeats ...

Have you ever admired how the slats of a hardwood floor fit together so cleanly, or how the hexagons underneath your bathroom rug perfectly meet up? These are examples of geometric tilings, ...

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 ...

The discovery earlier this year of the “hat” tile marked the culmination of hundreds of years of work into tiles and their symmetries. Every day we see examples of repeating motifs. This symmetry and ...

The recently discovered “hat” aperiodic monotile admits tilings of the plane, but none that are periodic [SMKGS23]. This polygon settles the question of whether a single shape—a closed topological ...