What has 14 sides, is full of curves, and can cover a surface without gaps or overlaps? It’s no mystery – it’s “Einstein’s vampire”.
In March, a retired printmaker named David Smith made a remarkable discovery in a scientist mathematics. is found A 13-sided figure that can coat the entire surface without repeating. Nicknamed “The Hat” because of its vaguely fedora-like shape, it was the culmination of decades of hunting by mathematicians around the world.
Since 1961 mathematicians wondered If such a format can exist. First, mathematicians found a set of 20,426 shapes that could be strung together while creating a pattern that never repeats (unlike the tiles on the kitchen floor, which create a repeating pattern). In the end, mathematicians have found a set of 104 shapes that can create a tiling that never repeats.
Then in 1970 physicist and Nobel Prize winner Roger Penrose found a pair of shapes that together created a non-repeating tiling. And for decades since then, mathematicians have continued to wonder if the same trick could be done with just one shape. This semi-legendary shape, formally known as a non-periodic monolayer, became known as “Einstein,” which means “one stone” in German.
But for all the celebration around Smith’s discovery of the Einstein piece, there was a small fly in the ointment. In order to create a non-repeating tiling, the “hat” had to work with its mirror image. Technically, it’s the same shape, just flipped, but some have argued that Smith never really found a real Einstein.
Now, however, Smith and his colleagues have put those objections to rest: They’ve found a shape that can coat a surface without looping or flipping. They described the new shape on May 28 in a paper published in a preprint database arXivalthough it has not yet been peer-reviewed.
The team named their form “Spectre”, in honor of vampires who can’t see their reflections, and therefore don’t need a mirror.
Co-author Joseph Samuel Myers writes: “In plane tiling it is quite usual for tiles to reflect; however, some people have been dissatisfied with the fact that a non-periodic monolayer hat requires reflections of plane tiles.” mastodon. “In our new printing, we present Spectre, the first example of an Einstein vampire: a non-periodic plane-smearing monolayer without reflections.”
To find the ghostly figure, the team started with the original “hat” figure and added an extra aspect to it. This new shape still required a fully mirrored image, but the researchers discovered that by turning the straight edges of the fourteen-sided shape into curved edges, they could do without the mirror images and work with just one shape.
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