# 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.

## History

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:

• 2 sin(π/l) · sin(π/m) = cos(π/n)

## Regular skew apeirohedra of Euclidean 3-space

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.

1. Mucube: {4,6|4}: 6 squares on a vertex (related to cubic honeycomb, constructed by cubic cells, removing two opposite faces from each, and linking sets of six together around a faceless cube.)
2. Muoctahedron: {6,4|4}: 4 hexagons on a vertex (related to bitruncated cubic honeycomb, constructed by truncated octahedron with their square faces removed and linking hole pairs of holes together.)
3. Mutetrahedron: {6,6|3}: 6 hexagons on a vertex (related to quarter cubic honeycomb, constructed by truncated tetrahedron cells, removing triangle faces, and linking sets of four around a faceless 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)]].

14 Compact regular skew apeirohedra
Coxeter group
Symmetry
Apeirohedron
{p,q|l}
Image Face
{p}
Hole
{l}
Vertex
figure
Related
honeycomb

[[4,3,4]]
[[4,3,4]+]
{4,6|4}
Mucube

animation

t0,3{4,3,4}

{6,4|4}
Muoctahedron

animation

2t{4,3,4}

[[3]]
[[3]+]
{6,6|3}
Mutetrahedron

animation

q{4,3,4}

## Regular skew apeirohedra in hyperbolic 3-space

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.

These represent 14 compact and 17 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}. For the special case of linear graph groups r = 2, this represents the Coxeter group [p,q,p]. It generates regular skews {2q,4|p} and {4,2q|p}. All of these exist as a subset of faces of the convex uniform honeycombs in hyperbolic space.

The skew apeirohedron shares the same antiprism vertex figure with the honeycomb, but only the zig-zag edge faces of the vertex figure are realized, while the other faces make "holes".

14 Compact regular skew apeirohedra
Coxeter
group
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure

[3,5,3]
{10,4|3}
2t{3,5,3}
{4,10|3}
t0,3{3,5,3}

[5,3,5]
{6,4|5}
2t{5,3,5}
{4,6|5}
t0,3{5,3,5}

[(4,3,3,3)]
{8,6|3}
ct{(4,3,3,3)}
{6,8|3}
ct{(3,3,4,3)}

[(5,3,3,3)]
{10,6|3}
ct{(5,3,3,3)}
{6,10|3}
ct{(3,3,5,3)}

[(4,3,4,3)]
{8,8|3}
ct{(4,3,4,3)}
{6,6|4}
ct{(3,4,3,4)}

[(5,3,4,3)]
{8,10|3}
ct{(4,3,5,3)}
{10,8|3}
ct{(5,3,4,3)}

[(5,3,5,3)]
{10,10|3}
ct{(5,3,5,3)}
{6,6|5}
ct{(3,5,3,5)}

17 Paracompact regular skew apeirohedra
Coxeter
group
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure
Apeirohedron
{p,q|l}
Face
{p}
Hole
{l}
Honeycomb Vertex
figure

[4,4,4]
{8,4|4}
2t{4,4,4}
{4,8|4}
t0,3{4,4,4}

[3,6,3]
{12,4|3}
2t{3,6,3}
{4,12|3}
t0,3{3,6,3}

[6,3,6]
{6,4|6}
2t{6,3,6}
{4,6|6}
t0,3{6,3,6}

[(4,4,4,3)]
{8,6|4}
ct{(4,4,3,4)}
{6,8|4}
ct{(3,4,4,4)}

[(4,4,4,4)]
{8,8|4}
q{4,4,4}

[(6,3,3,3)]
{12,6|3}
ct{(6,3,3,3)}
{6,12|3}
ct{(3,3,6,3)}

[(6,3,4,3)]
{12,8|3}
ct{(6,3,4,3)}
{8,12|3}
ct{(4,3,6,3)}

[(6,3,5,3)]
{12,10|3}
ct{(6,3,5,3)}
{10,12|3}
ct{(5,3,6,3)}

[(6,3,6,3)]
{12,12|3}
ct{(6,3,6,3)}
{6,6|6}
ct{(3,6,3,6)}