Regular skew apeirohedron

In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to finite regular skew polyhedra in 4-dimensions, and infinite regular skew apeirohedra in 3-dimensions (described here).

Coxeter identified 3 forms, with planar faces and skew vertex figures, two are complements of each other. They are all named with a modified Schläfli symbol {l,m|n}, where there are l-gonal faces, m faces around each vertex, with holes identified as n-gonal missing faces.

Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.

The regular skew polyhedra, represented by {l,m|n}, follow this equation:

The three Euclidean solutions in 3-space are {4,6|4}, {6,4|4}, and {6,6|3}. John Conway named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron.

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). The related honeycomb has the extended symmetry [[(p,q,p,r)]].

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In 1967, C. W. L. Garner identified 31 hyperbolic skew apeirohedra with regular skew polygon vertex figures, found in a similar search to the 3 above from Euclidean space.

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