# 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. 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:

2*sin(π/l)*sin(π/m)=cos(π/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 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.

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.
A ring of 60 triangles make a regular skew polyhedron within a subset of faces of a 600-cell.
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]+] 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
{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. ^ Abstract regular polytopes, p.7, p.17
2. ^ 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.
• E. Schulte, J.M. Wills On Coxeter's regular skew polyhedra, Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262