Earlier this year, big news broke out in the world of geometry. A unique shape was discovered that could tile an infinite plane, creating a geometric pattern that would never repeat.
But that discovery left a problem for geometric purists. The 13-sided "hat-shaped" aperiodic monotile that had been identified could do the job, but required the use of "reflected" tiles to continue the tiling. For the purists, that was like using regular glazed square tiles to cover a floor, but sometimes flipping the grout-side of some tiles up to complete the pattern. They wanted a completely unique and uniform tile shape that would create a pattern that never repeats, but only using the same-shaped tiles whose "glazed"-side would always face upward.
So David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss went back to their drawing board and tweaked their monotile design. At the end of May 2023, they posted a new preprint paper announcing their new discovery. They had defined the shape of a unique monotile capable of fully tiling the infinite plane without ever repeating a geometric pattern. Interesting Engineering reports the development:
Not wanting to leave this problem unsolved, the team kept working. They found a different shape closely related to the hat but could cover the surface without needing to be flipped over. It still did the aperiodic tiling, but now with only a single shape, no mirror images were needed.
“I wasn’t surprised that such a tile existed,” said the co-author Joseph Myers, a software developer in Cambridge, England. “That one existed so closely related to the hat was surprising,” he added.
The new shape, which they called "Spectre," was discovered by tweaking an "equilateral version" of the hat, a shape that didn't initially seem to have the aperiodic tiling ability. By modifying this shape a bit, they found it could do the non-repeating tiling without mirror images.
From here, we'll turn to Ayliean MacDonald's 5-and-a-half minute video featuring the "Spectre" tile discovery, including a brief interview with one of its inventors:
The paper introduces three different variations of the new "strictly chiral" aperiodic monotile, which has 14-sides:
Here's how the geometers describe their new achievement from their paper:
The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.
To really appreciate what they've done, we need to see the new chiral monotiles laid out together on a two-dimensional plane. The following illustration shows the non-repeating pattern that can be created using two of their monotile variations, with the straight-sided polygon on the left-hand side, which transitions into the arc-sided version of the monotile on the right-hand side.
To us, the right-hand side of this figure suggests a devilish application lies ahead for this new geometry. Imagine a jigsaw puzzle where every piece inside the edges uses that exact same shape. *Every* interior piece would fit together regardless of where it might supposed to be within the puzzle!
That may add a whole new level of difficulty to a popular pastime.
References
David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. A Chiral Aperiodic Monotile. [Preprint: PDF Document]. DOI: 10.48550/arXiv.2305.17743. 28 May 2023.
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