Regular skew apeirohedron

[1] Harold Scott MacDonald Coxeter derived a third, the mutetrahedron, and proved that the these three were complete.

John Conway named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron.

[5] Coxeter gives these regular skew apeirohedra {2q,2r|p} with extended chiral symmetry [[(p,q,p,r)]+] which he says is isomorphic to his abstract group (2q,2r|2,p).

As with the finite skew polyhedra of full rank, all three of these can be obtained by applying the Petrie dual to planar polytopes, in this case the three regular tilings.

Equivalently the blend can be obtained by positioning P and Q in orthogonal spaces and taking composing their generating mirrors pairwise.

Since the skeleton of the square tiling is bipartite, two of these blends, {4, 4}#{} and {4, 4}π#{}, are combinatrially equivalent to their non-blended counterparts.

[14] These represent 14 compact and 17[note 1] paracompact regular skew polyhedra in hyperbolic space, constructed from the symmetry of a subset of linear and cyclic Coxeter groups graphs of the form [[(p,q,p,r)]], These define regular skew polyhedra {2q,2r|p} and dual {2r,2q|p}.

All of these exist as a subset of faces of the convex uniform honeycombs in hyperbolic space.

The mucube is a regular skew apeirohedron.
{3,6}#{} with the edges of one face highlighted in red
Skewed muoctahedron Petrial mucube Muoctahedron Mucube Petrial muoctahedron Halved mucbe Petrial halved mucube Skewed Petrial muoctahedron Mutetrahedron Petrial mutetrahedron Trihelical square tiling Tetrahelical triangular tiling Rectification Rectification Petrie dual Petrie dual Petrie dual Petrie dual Petrie dual Petrie dual Petrie dual Dual polyhedron Dual polyhedron Second-order facetting Second-order facetting Second-order facetting Second-order facetting Second-order facetting Second-order facetting Petrial cube Petrial tetrahedron Tetrahedron Cube
Some relationships between the 12 pure apeirohedra in 3D Euclidean space
The compact skew apeirohedron {4,6 | 5}