A ‘vampire einstein’ tile outdoes mathematicians’ latest feat

Einstein, meet “vampire einstein.”

It’s been just months since researchers reported the first  “einstein” — a single tile that can cover an infinite plane, but only with a pattern that never repeats (SN: 3/24/23). Now, the same team has found a shape that’s even more special.

The original einstein, nicknamed “the hat,” made a pattern that involved both the hat and its mirror image. The new tile also makes a pattern that never repeats, but without such reflections, the researchers report May 28 at arXiv.org. Because the shape isn’t accompanied by its reflection, you might call it a “vampire einstein,” the researchers point out. (The “einstein” part of the name comes from the German for “one stone,” not from the name of the famous physicist.) The shape is part of a family of vampire einsteins that the researchers found, which they called “spectres.”

Describing how tiles cover an infinite plane without any overlaps or gaps between them is a time-honored fascination for mathematicians. While certain other tiles can be arranged so that they don’t form a repeating pattern, einsteins are special because that’s the only way they can tile. Previously, mathematicians knew of sets of tiles that could tile the plane only with non-repeating patterns. But until this year, they didn’t know of a single tile that would do it.

After finding the first einstein, the researchers wondered if they could find a tile that would make a non-repeating pattern without any reflected versions of the tile. Starting from a shape related to the hat and curving its edges in such a way that the tile’s reflection no longer fit together with itself, the researchers created the vampire einstein tile.

“I would never have predicted that we’d stumble upon a shape that solves this [vampire einstein] subproblem so quickly,” says computer scientist Craig Kaplan of the University of Waterloo in Canada.