The geometric shape of the tile that solves the problem was dubbed ‘the Specter’ and can completely cover a surface without repeating itself.
An international team of mathematicians discovered a geometric figure dubbed ‘the Spectre’ that solves the plane geometry problem of covering a surface with a single tile (tesserae) in an aperiodic manner. The problem, that is known as Einstein monotessellation., made the theoretical case of finding an asymmetric tile that can be combined with each other in a non-periodic pattern to cover an area. This mathematical unknown has its origins in the 1960s.
Einstein’s monotessellation is a pun derived from the German expression ‘Ein Stein’, which means ‘one piece’. Last March, the same research team discovered the Sombrero, a geometric shape that tried to solve this theoretical challenge. However, the Hat had a limitation, as it required the original tile to be mirror-reflected to give rise to two mirror tiles that were symmetrical to each other.
The limits of the Hat
In this sense, the Hat is what is known as a Penrose tessellation, which is a set of tiles (usually two) that can cover a surface, without gaps or overlap, in a periodic manner. These are different tiles that can be combined with each other to form a pattern in which each displaced piece never exactly matches the original piece. This detail meant that many mathematicians did not consider it a satisfactory solution to the monotessellation problem.
“In the tessellation of planes it is normal that some tiles are reflected,” explained Joseph Samuel Meyers, from the University of Cambridge (United Kingdom) and co-author of the discovery. “However, some were not satisfied with a monotessellation that requires mirroring. In our new study we introduced the Spectrum, the first example of vampire Einstein, an aperiodic monotessellation covering a plane without reflections“he added. The adjective vampire refers to the characteristic of these beings from popular mythology whose image cannot be reflected in mirrors.
The Specter is a 14-sided tile, with a slightly ghostly look, originally designed with straight faces. However, when mathematicians replaced the straight faces with other curves, they were able to obtain a mirror image to form a Penrose tessellation. The Spectrum constitutes a strict chiral aperiodic monotessellation. The chiral character alludes to the property of molecules that makes it impossible to superimpose them on their mirror image no matter how much they rotate in space. The results of the investigation were published last Sunday on the arXiv preprint server.