![]() Such constructions are much harder to visualize, but computationally it's not that much more difficult to prove should we be so lucky as to find a three-dimensional shape-a polyform-that like the hat tiles only aperiodically." It would be nice to have a shape that repeats non-periodically in three dimensions. "The most interesting, for me at least, is whether this can be done in 3D. ![]() "We can pose many variations of the problem," Kaplan says. The obvious question for mathematicians and tiling enthusiasts is what's next? But if Tile(1,1) is modified by replacing its straight edges with curves, it becomes a vampire einstein-a single shape that without reflection tiles the infinite plane in a pattern that can never be made to repeat. As Kaplan explains, if you use reflections of Tile(1,1) the pattern does repeat. What if we tile with this shape but without reflections? As he did that, Dave found that he could build tilings progressively outward in a pattern that didn't stop and didn't repeat."īut then this shape came with a different quibble. "Yoshi had turned Tile(1,1) into turtles and it's a bit hard to see the other reflected turtle in that picture. One more reflected turtle is hidden in the tiling. In the tiling, it is said that around 12.7% of tiles are reflected. Yoshiaki posted an intriguing question on Twitter: "An aperiodic turtle tessellation based on new aperiodic monotile Tile(1, 1.1). "Yoshiaki Araki, a Japanese mathematician and well-known artist whose work is in the spirit of MC Escher, had posted pictures of Tile(1,1) that got Dave interested in looking at it further." "Dave emailed us a couple of days after our hat paper went online to say that he had been playing around with a related shape that seemed to be behaving strangely," Kaplan says. The discovery of the vampire einstein began with the musings of David Smith, a retired print technician and self-described shape hobbyist from Yorkshire, England, whose curiosity months earlier led to the original einstein discovery. "If you tiled a large bathroom floor aperiodically with hat-shaped tiles that had been glazed on one side you would need hats and mirror images of hats," Kaplan says.īut it was not this quibble that motivated the recent discovery. You can't, for example, wear a right-handed glove on your left hand. To mathematicians, the hat and its mirror image are a single shape, but in the physical world left-handed and right-handed shapes can behave differently. It is a non-reflective or vampire einstein. However, if you force it to tile without reflections, it tiles only aperiodically. But one member of this infinite set of shapes - known technically as Tile(1,1) - is not an aperiodic monotile. "It was our good fortune that we found a shape that not only solves this subproblem, but also solved it so soon after the first paper." As shown in this looping animation, the hat is one member of a morphing continuum of shapes created by increasing and decreasing edges to produce a tiling that requires reflected shapes to ensure the pattern never repeats. "Our first paper solved the einstein problem, but as the shape required reflection to tile aperiodically people raised a legitimate question: Is there a shape that can do what the hat does but without reflection," Kaplan explains. ![]() For this reason, it has also been called a "vampire einstein"-a shape that tiles aperiodically without requiring its reflection. Dubbed the "specter," the new shape tiles a plane in a pattern that never repeats without the use of mirror images of the shape. They found another shape, related to the first, that meets an even stricter definition. The shape, a 13-sided polygon they called "the hat," is known to mathematicians as an aperiodic monotile or an "einstein," the German words that mean "one stone."īut the team's most recent discovery has raised the bar once again. The discovery mesmerized mathematicians, tiling enthusiasts and the public alike.
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