# Regular skew polyhedron

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar faces or vertex figures.

These polyhedra have two forms: infinite polyhedra that span 3-space, and finite polyhedra that close into 4-space.

## History

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra.

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)

## Infinite regular skew polyhedra

There are 3 regular skew polyhedra, the first two being duals. John Conway named them mucube, muoctahedron, and mutetrahedron 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.)

Also solutions to the equation above are the Euclidean regular tilings {3,6}, {6,3}, {4,4}, represented as {3,6|6}, {6,3|6}, and {4,4|∞}.

Coxeter gives these regular skew polyhedra {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)]].[1]

Here are some partial representations, vertical projected views of their skew vertex figures, and partial corresponding uniform honeycombs.

Partial polyhedra

{4,6|4}
Mucube

{6,4|4}
Muoctahedron

{6,6|3}
Mutetrahedron
Vertex figures

{4,6}

{6,4}

{6,6}
Related convex uniform honeycombs

Runcinated cubic honeycomb

t0,3{4,3,4}

Bitruncated cubic

t1,2{4,3,4}

quarter cubic honeycomb

t0,1{3[4]}

## Finite regular skew polyhedra of 4-space

A4 Coxeter plane projections
{4, 6 | 3} {6, 4 | 3}
Runcinated 5-cell
(60 edges, 20 vertices)
Bitruncated 5-cell
(60 edges, 30 vertices)
F4 Coxeter plane projections
{4, 8 | 3} {8, 4 | 3}
Runcinated 24-cell
(576 edges, 144 vertices)
Bitruncated 24-cell
(576 edges, 288 vertices)
Some of the 4-dimensional regular skew polyhedra fits inside the uniform polychora as shown in these projections.

Coxeter also enumerated the a larger set of finite regular polyhedra in his paper "regular skew polyhedra in three and four dimensions, and their topological analogues".

Just like the infinite skew polyhedra represent manifold surfaces between the cells of the convex uniform honeycombs, the finite forms all represent manifold surfaces within the cells of the uniform polychora.

A first form, {l, m | n}, repeats the five convex Platonic solids, and one nonconvex Kepler-Poinsot solid:

{l, m | n} Faces Edges Vertices p Polyhedron Symmetry
order
{3,3| 3} = {3,3} 4 6 4 0 Tetrahedron 12
{3,4| 4} = {3,4} 8 12 6 0 Octahedron 24
{4,3| 4} = {4,3} 6 12 8 0 Cube 24
{3,5| 5} = {3,5} 20 30 12 0 Icosahedron 60
{5,3| 5} = {5,3} 12 30 20 0 Dodecahedron 60
{5,5| 3} = {5,5/2} 12 30 12 4 Great dodecahedron 60

The remaining solutions of the first form, {l, m | n} exist in 4-space. Polyhedra of the form {l, m | n} have a cyclic Coxeter group symmetry of [(l/2,n,m/2,n)], which reduces to the linear [n,l/2,n] when m is 4, and [n,m,/2,n] when l=4. {4,4|n} produces a double n-prism, or n-n duoprism, and specifically {4,4|4} fits inside of a {4}x{4} tesseract. {a,4|b} is represented by the {a} faces of the bitruncated {b,a/2,b} uniform polychoron, and {4,a|b} is represented by square faces of the runcinated {b,a/2,b}.

Even ordered solutions
{l, m | n} Faces Edges Vertices p Structure Symmetry Order Related uniform polychora
{4,4| 3} 9 18 9 1 D3xD3 [[3,2,3]] 18 3-3 duoprism
{4,4| 4} 16 32 16 1 D4xD4 [[4,2,4]] 32 4-4 duoprism or tesseract
{4,4| 5} 25 50 25 1 D5xD5 [[5,2,5]] 50 5-5 duoprism
{4,4| 6} 36 72 36 1 D6xD6 [[6,2,6]] 72 6-6 duoprism
{4,4| n} n2 2n2 n2 1 DnxDn [[n,2,n]] 2n2 n-n duoprism
{4,6| 3} 30 60 20 6 S5 [[3,3,3]] 120 Runcinated 5-cell
{6,4| 3} 20 60 30 6 S5 [[3,3,3]] 120 Bitruncated 5-cell
{4,8| 3} 288 576 144 73 [[3,4,3]] 1152 Runcinated 24-cell
{8,4| 3} 144 576 288 73 [[3,4,3]] 1152 Bitruncated 24-cell
pentagrammic solutions
{l, m | n} Faces Edges Vertices p Structure Symmetry Order Related uniform polychora
{4,5| 5} 90 180 72 10 A6 [[5/2,5,5/2]] 360 Runcinated grand stellated 120-cell
{5,4| 5} 72 180 90 10 A6 [[5/2,5,5/2]] 360 Bitruncated grand stellated 120-cell
{l, m | n} Faces Edges Vertices p Structure Order
{4,5| 4} 40 80 32 5 ? 160
{5,4| 4} 32 80 40 5 ? 160
{4,7| 3} 42 84 24 10 LF(2,7) 168
{7,4| 3} 24 84 42 10 LF(2,7) 168
{5,5| 4} 72 180 72 19 A6 360
{6,7| 3} 182 546 156 105 LF(2,13) 1092
{7,6| 3} 156 546 182 105 LF(2,13) 1092
{7,7| 3} 156 546 156 118 LF(2,13) 1092
{4,9| 3} 612 1224 272 171 LF(2,17) 2448
{9,4| 3} 272 1224 612 171 LF(2,17) 2448
{7,8| 3} 1536 5376 1344 1249 ? 10752
{8,7| 3} 1344 5376 1536 1249 ? 10752

A final set is based on Coxeter's further extended form {q1,m|q2,q3...} or with q2 unspecified: {l, m |, q}.

{l, m |, q} Faces Edges Vertices p Structure Order
{3,6|,q} 2q2 3q2 q2 1 ? 2q2
{3,2q|,3} 2q2 3q2 3q (q-1)*(q-2)/2 ? 2q2
{3,7|,4} 56 84 24 3 LF(2,7) 168
{3,8|,4} 112 168 42 8 PGL(2,7) 336
{4,6|,3} 84 168 56 15 PGL(2,7) 336
{3,7|,6} 364 546 156 14 LF(2,13) 1092
{3,7|,7} 364 546 156 14 LF(2,13) 1092
{3,8|,5} 720 1080 270 46 ? 2160
{3,10|,4} 720 1080 216 73 ? 2160
{4,6|,2} 12 24 8 3 S4×S2 48
{5,6|,2} 24 60 20 9 A5×S2 120
{3,11|,4} 2024 3036 552 231 LF(2,23) 6072
{3,7|,8} 3584 5376 1536 129 ? 10752
{3,9|,5} 12180 18270 4060 1016 LF(2,29)×A3 36540

## Notes

1. ^ Coxeter, Regular and Semi-Regular Polytopes II 2.34)

## References

• Peter McMullen, Four-Dimensional Regular Polyhedra, Discrete & Computational Geometry September 2007, Volume 38, Issue 2, pp 355-387
• Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra", Scripta Mathematica 6 (1939) 240-244.
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
• Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• Garner, C. W. L. Regular Skew Polyhedra in Hyperbolic Three-Space. Canad. J. Math. 19, 1179-1186, 1967.