A quartet of mathematicians from the universities of Yorkshire, Cambridge, Waterloo and Arkansas have discovered a 2D geometric shape that does not repeat when tiled.
David Smith, Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss have written an article describing how they discovered the unique shape and possible uses for it. His full article is available on the arXiv preprint server.
When people tile their floors, they tend to use simple geometric shapes that lend themselves to repeating patterns, like squares or triangles. Sometimes though, people want patterns that don’t repeat themselves but are challenging if the same types of shapes are used. In this new effort, the research team has discovered a single geometric shape that, if used for mosaics, will not produce repeating patterns.
Under their scenario, the researchers noted that tiling refers to joining shapes together so that there are no overlaps or gaps. Tiling that does not have repeating patterns is known as aperiodic tiling and is usually achieved through the use of multiple tiling shapes. For many years, mathematicians have been studying the idea of creating shapes that could be used to create an infinite variety of patterns when tiled.
One of the first attempts resulted in a set of 20,426 tiles. That was followed by the development of the Penrose mosaics, back in 1974, which come in sets of two differently shaped diamonds. Since then, mathematicians have continued to search for what is known as the “einstein” form, a unique form that could be used on its own for aperiodic tilings.
Notably, the name comes from the phrase “a stone” in German, not the famous physicist. In this new effort, the research group claims to have found Einstein’s elusive form and has proven it mathematically, Phys.org reports.
The shape has 13 sides and is referred to simply as “the hat” by the team. They found it by first narrowing down the possibilities using a computer and then studying the resulting smaller sets by hand. Once they had what they thought was a good possibility, they proved it using a combinatorial software program, and then proved that the shape was aperiodic using an argument from geometric incommensurability. The researchers conclude by suggesting that the hat’s most likely application is in the arts.