Pentagon Tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular pentagons, regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn (geometry), turn. However, regular pentagons can tile the hyperbolic plane with Order-4 pentagonal tiling, four pentagons around each vertex (Order-5 pentagonal tiling, or more) and sphere with Spherical dodecahedron, three pentagons; the latter produces a tiling topologically equivalent to the regular dodecahedron, dodecahedron. Monohedral convex pentagonal tilings Fifteen types of convex pentagons are known to tile the plane monohedral tiling, monohedrally (i.e. with one type of tile). The most recent one was discovered in 2015. This list has been shown to be complete by (result subject to peer-review). showed that there are only eigh ...
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