Wired

‘Nasty’ Geometry Breaks a Decades-Old Tiling Conjecture

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 ...
Quanta Magazine

Hobbyist Finds Math’s Elusive ‘Einstein’ Tile

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 ...
Wired

‘Nasty’ Geometry Breaks a Decades-Old Tiling Conjecture

Proving the “discrete” periodic tiling conjecture for high-dimensional lattices is a slightly different problem than proving the continuous version of the conjecture, as there are tilings that are ...
Quanta Magazine

‘Nasty’ Geometry Breaks Decades-Old Tiling Conjecture

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 ...