In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron. They have either skew regular faces or skew regular vertex figures.
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 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. Harold Scott MacDonald Coxeter derived a third, the mutetrahedron, and proved that the 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 only skew polyhedra in 3-space as there Coxeter showed there were no finite cases.
In 1967 Garner investigated regular skew apeirohedra in hyperbolic 3-space with Petrie and Coxeters definition, discovering 31[1]