‘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 plane without gaps or overlaps. And yet, “not a lot has been known until fairly recent times,” said Alex Iosevich, a mathematician at the University of Rochester.

The most obvious tilings repeat: It’s easy to cover a floor with copies of squares, triangles or hexagons. In the 1960s, mathematicians found strange sets of tiles that can completely cover the plane, but only in ways that never repeat.

“You want to understand the

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