One of the oldest and simplest problems in geometry has caught mathematicians off guard—and not for the first time. Since antiquity, artists and geometers have wondered how shapes can tile the entire ...

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 ...

A geometry problem that has been puzzling scientists for 60 years has likely just been solved by an amateur mathematician with a newly discovered 13-sided shape. Called “The hat” because it vaguely ...

The surprisingly simple tile is the first single, connected tile that can fill the entire plane in a pattern that never repeats — and can’t be made to fill it in a repeating way. In mid-November of ...

Many repeating, or periodic, tilings can be tweaked to form non-repeating ones. Consider, say, an infinite grid of squares, aligned like a chessboard. If you shift each row so that it’s offset by a ...

Mathematicians predicted that if they imposed enough restrictions on how a shape might tile space, they could force a periodic pattern to emerge. But they were wrong. One of the oldest and simplest ...