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 ...

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 ...