Petrie-Coxeter Polyhedra
In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron. They have either Skew polygon, skew regular Face (geometry), faces or skew regular vertex figures. History In 1926 John Flinders Petrie took the concept of a regular skew polygons, polygons whose vertices are not all in the same plane, and extended it to polyhedra. While regular tiling, apeirohedra are typically required to tile the 2-dimensional plane, Petrie considered cases where the faces were still convex but were not required to lie flat in the plane, they could have a skew polygon vertex figure. Petrie discovered two regular skew apeirohedra, the mucube and the muoctahedron. H.S.M. Coxeter, Harold Scott MacDonald Coxeter derived a third, the mutetrahedron, and proved that these three were complete. Under Coxeter and Petrie's definition, requiring convex faces and allowing a skew vertex figure, the three were not only the only skew apeirohedra in 3-dimensional Euclidean space, but they were the ...
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