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Mathematicians at the Universities of Yorkshire, Cambridge, Waterloo, and Arkansas have perfected themselves by finding a close relative of the “hat,” a unique geometric shape that does not repeat when tiled, that is, a true chirality aperiodic monolith. David Smith, Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss have published an article outlining their new findings on the arXiv preprint server.
Just three months ago, four mathematicians announced what is known in the field as the Einstein form, the only form that can be used alone for a non-periodic tiling. They call it a “hat”.
The discovery appears to be the latest step in a 60-year search for form. Previous efforts resulted in multi-block results, which were only reduced to two in the mid-1970s. But since then, attempts to find the shape of Einstein have been unsuccessful – until March, when the team working on a new project announced this.
But others point out that technically the shape the command describes is not a single aperiodic tile—it and its mirror image are two unique tiles, each responsible for creating the shape the command describes. Seemingly agreeing with their colleagues’ assessment, the four mathematicians revised their form and found that after a slight modification, the mirror was no longer needed and indeed represented Einstein’s true form.
It is worth noting that the name used to describe the shape is not a tribute to the famous physicist, but comes from the German phrase meaning “stone”. The team calls the new uniform simply a close relative of the hat. They also noted that changing the edges of newly discovered polygons in a certain way led to the creation of a whole set of shapes called Spectra, all of which are strictly chiral aperiodic monoliths.
Further information: David Smith et al., Chiral Aperiodic Monotile, arXiv (2023). DOI: 10.48550/arxiv.2305.17743
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Post time: Jun-03-2023