What can perfectly cover a 14-sided, curved surface with no gaps or overlaps? This is not a riddle – this is “Vampire Einstein”.
In March, David Smith, a retired printing technician, stumbled upon a remarkable discovery in the world. Mathematics. He found a The 13-sided shape can be completely tiled over the surface without ever repeating. Nicknamed the “hat” for its elusive fedora shape, the shape is the culmination of a decades-long hunt by mathematicians around the world.
Since 1961 Mathematicians were surprised If such a form exists. First, the mathematicians found a set of 20,426 shapes that create a pattern that never repeats (unlike kitchen floor tiles, which create a repeating pattern). Eventually, mathematicians found a set of 104 shapes that could create tilings that never repeat.
Later in the 1970s, physicist and Nobel laureate Roger Penrose discovered a pair of motifs that together created a non-repeating tiling. For decades, mathematicians have wondered whether the same trick could be done with just a single shape. Formally known as the aperiodic monotile, that semi-mythical figure became known as “Einstein”, which means “a stone” in German.
But for all the celebration surrounding Smith’s discovery of the Einstein tile, there was a small fly in the ointment. To create a non-repeating tiling, the “hat” needs to work with its mirror image. Technically it’s the same shape, just flipped over, but some have argued that Smith didn’t actually find an Einstein.
Now, however, Smith and his colleagues have put an end to those objections: they’ve discovered a shape that can be tiled over a surface without repeating or flipping it. They described the new look in a paper published May 28 in the Preprint database arXivAlthough it has not yet been peer reviewed.
The team named their form “Spectre” in homage to vampires who can’t see their own reflections and don’t need mirrors.
“In plane tiling, reflecting tiles is fairly standard; however, some people are unhappy that the aperiodic hat monotile requires reflections for tiling the plane,” wrote co-author Joseph Samuel Meyers. Mastodon. “In our new preprint, we present Spectra, the first example of a vampire Einstein: an aperiodic monotile that tiles the plane without reflections.”
To find the ghost figure, the team started with the original “hat” shape and added an extra facet to it. That new shape still required its mirror image to be completely tiled, but by changing the straight edges of the 14-sided shape into curved ones, the researchers found that they could avoid mirror images and work with just one shape.