Einstein, meet “vampire einstein.”
, but only with a pattern that never repeats . 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, 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 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.
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