Fascinating math!
"... A new shape has been discovered which would make for a pretty unique set of bathroom tiles. Nicknamed “the Hat,” the 13-sided shape can tesselate with itself without ever repeating a pattern, solving a 60-year-old mathematical mystery. ..."
"... This elusive shape is known to mathematicians as an aperiodic monotile or an einstein, a clever pun that takes its name from the German words ein and stein that mean one stone. ..."
From the abstract:
"A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings."
A trick of the hat The story of how a Waterloo computer science professor helped find the elusive einstein tile
An aperiodic monotile (open access)
A tiling of a 13-sided shape called the Hat, the first known example of a shape that can be tiled without gaps or overlaps, and without repeating a pattern across an infinite plane
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