# Regular skew polyhedron

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In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.[1]

Infinite regular skew polyhedra that span 3-space or higher are called regular skew apeirohedra.

## 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:

${\displaystyle 2\cos {\frac {\pi }{l}}\cos {\frac {\pi }{m}}=\cos {\frac {\pi }{n}}}$

A first set {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

## Finite regular skew polyhedra

A4 Coxeter plane projections
{4, 6 | 3} {6, 4 | 3}
Runcinated 5-cell
(20 vertices, 60 edges)
Bitruncated 5-cell
(30 vertices, 60 edges)
F4 Coxeter plane projections
{4, 8 | 3} {8, 4 | 3}
Runcinated 24-cell
(144 vertices, 576 edges)
Bitruncated 24-cell
(288 vertices, 576 edges)
{3,8|,4} = {3,8}8 {4,6|,3} = {4,6}6
42 vertices, 168 edges 56 vertices, 168 edges
Some of the 4-dimensional regular skew polyhedra fits inside the uniform polychora as shown in the top 4 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 4-polytopes.

Polyhedra of the form {2p, 2q | r} are related to Coxeter group symmetry of [(p,r,q,r)], which reduces to the linear [r,p,r] when q is 2. Coxeter gives these symmetry as [[(p,r,q,r)]+] which he says is isomorphic to his abstract group (2p,2q|2,r). The related honeycomb has the extended symmetry [[(p,r,q,r)]].[2]

{2p,4|r} is represented by the {2p} faces of the bitruncated {r,p,r} uniform 4-polytope, and {4,2p|r} is represented by square faces of the runcinated {r,p,r}.

{4,4|n} produces a n-n duoprism, and specifically {4,4|4} fits inside of a {4}x{4} tesseract.

 The {4,4| n} solutions represent the square faces of the duoprisms, with the n-gonal faces as holes and represent a clifford torus, and an approximation of a duocylinder {4,4|6} has 36 square faces, seen in perspective projection as squares extracted from a 6,6 duoprism. {4,4|4} has 16 square faces and exists as a subset of faces in a tesseract.
A ring of 60 triangles make a regular skew polyhedron within a subset of faces of a 600-cell.
Finite polyhedra in 4 dimensions
{l, m | n} Faces Edges Vertices p Structure Symmetry Order Related uniform polychora
{4,4| 3} 9 18 9 1 D3xD3 [[3,2,3]+] 9 3-3 duoprism
{4,4| 4} 16 32 16 1 D4xD4 [[4,2,4]+] 16 4-4 duoprism or tesseract
{4,4| 5} 25 50 25 1 D5xD5 [[5,2,5]+] 25 5-5 duoprism
{4,4| 6} 36 72 36 1 D6xD6 [[6,2,6]+] 36 6-6 duoprism
{4,4| n} n2 2n2 n2 1 DnxDn [[n,2,n]+] n2 n-n duoprism
{4,6| 3} 30 60 20 6 S5 [[3,3,3]+] 60 Runcinated 5-cell
{6,4| 3} 20 60 30 6 S5 [[3,3,3]+] 60 Bitruncated 5-cell
{4,8| 3} 288 576 144 73 [[3,4,3]+] 576 Runcinated 24-cell
{8,4| 3} 144 576 288 73 [[3,4,3]+] 576 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
{4,5| 4} can be realized within the 32 vertices and 80 edges of a 5-cube, seen here in B5 Coxeter plane projection showing vertices and edges. The 80 square faces of the 5-cube become 40 square faces of the skew polyhedron and 40 square holes.
{l, m | n} Faces Edges Vertices p Structure Order Related uniform polytopes
{4,5| 4} 40 80 32 5 ? 160 5-cube vertices (±1,±1,±1,±1,±1) and edges
{5,4| 4} 32 80 40 5 ? 160 Rectified 5-orthoplex vertices (±1,±1,0,0,0)
{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}. These can also be represented a regular finite map or {l, m}2q, and group Gl,m,q.[3]

{l, m |, q} or {l, m}2q Faces Edges Vertices p Structure Order Related complex polyhedra
{3,6|,q} = {3,6}2q 2q2 3q2 q2 1 G3,6,2q 2q2
{3,2q|,3} = {3,2q}6 2q2 3q2 3q (q-1)*(q-2)/2 G3,6,2q 2q2
{3,7|,4} = {3,7}8 56 84 24 3 LF(2,7) 168
{3,8|,4} = {3,8}8 112 168 42 8 PGL(2,7) 336 (1 1 114)4,
{4,6|,3} = {4,6}6 84 168 56 15 PGL(2,7) 336 (14 14 11)(3),
{3,7|,6} = {3,7}12 364 546 156 14 LF(2,13) 1092
{3,7|,7} = {3,7}14 364 546 156 14 LF(2,13) 1092
{3,8|,5} = {3,8}10 720 1080 270 46 G3,8,10 2160 (1 1 114)5,
{3,10|,4} = {3,10}8 720 1080 216 73 G3,8,10 2160 (1 1 115)4,
{4,6|,2} = {4,6}4 12 24 8 3 S4×S2 48
{5,6|,2} = {5,6}4 24 60 20 9 A5×S2 120
{3,11|,4} = {3,11}8 2024 3036 552 231 LF(2,23) 6072
{3,7|,8} = {3,7}16 3584 5376 1536 129 G3,7,17 10752
{3,9|,5} = {3,9}10 12180 18270 4060 1016 LF(2,29)×A3 36540

## Higher dimensions

Regular skew polyhedra can also be constructed in dimensions higher than 4 as embeddings into regular polytopes or honeycombs. For example, the regular icosahedron can be embedded into the vertices of the 6-demicube; this was named the regular skew icosahedron by H. S. M. Coxeter. The dodecahedron can be similarly embedded into the 10-demicube.[4]

## Notes

1. ^ Abstract regular polytopes, p.7, p.17
2. ^ Coxeter, Regular and Semi-Regular Polytopes II 2.34)
3. ^ Coxeter and Moser, Generators and relations for discrete groups, Sec 8.6 Maps having specified Petrie polygons. p. 110
4. ^ Deza, Michael; Shtogrin, Mikhael (1998). "Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices". Advanced Studies in Pure Mathematics. Arrangements – Tokyo 1998: 77. doi:10.2969/aspm/02710073. ISBN 978-4-931469-77-8. Retrieved 4 April 2020.

## 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. Can. J. Math. 19, 1179-1186, 1967.
• E. Schulte, J.M. Wills On Coxeter's regular skew polyhedra, Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262