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 ...

Those wondrously intricate tile mosaics that adorn medieval Islamic architecture may contain a mastery of geometry not matched in the West for hundreds of years. Historians have long assumed that ...

Surprisingly, the answer turns out to be yes—if you’re allowed to shift, rotate, and reflect the tile, and if the tile is disconnected, meaning that it has gaps. Those gaps get filled by other ...