‘Nasty’ Geometry Breaks Decades-Old Tiling Conjecture | Quanta Magazine
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.
Hasnain says:
““Our proof is constructive, so everything is explicit and computable,” Greenfeld said. “But because it’s very, very far from being optimal, we just didn’t check it.”
Indeed, the mathematicians think they can find aperiodic tiles in much lower dimensions. That’s because some of the more technical parts of their construction involved working in special spaces that are conceptually “very close to being two-dimensional,” Greenfeld said. She doesn’t think they’ll find a three-dimensional tile, but she says it’s feasible that a 4D one could exist.
And so, Iosevich said, they didn’t just disprove the periodic tiling conjecture: “They did this in the most humiliating fashion possible.””
Posted on 2022-12-25T06:42:23+0000