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.[1] 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[2] Garner investigated regular skew apeirohedra in hyperbolic 3-space with Petrie and Coxeters definition, discovering 31[note 1] regular skew apeirohedra with compact or paracompact symmetry.
In 1977[3][1] Grünbaum generalized skew polyhedra to allow for skew faces as well. Grünbaum discovered an additional 23[note 2] skew apeirohedra in 3-dimensional Euclidean space and 3 in 2-dimensional space which are skew by virtue of their faces. 12 of Grünbaum's polyhedra were formed using the blending operation on 2-dimensional apeirohedra, and the other 11 were pure, i.e. could not be formed by a non-trivial blend. Grünbaum conjectured that this new list was complete for the parameters considered.
In 1985[4][1] Dress found an additional pure regular skew apeirohedron in 3-space, and proved that with this additional skew apeirohedron the list was complete.