Regular skew polyhedron

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

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.

{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.

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 G^l,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

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]

• Skew polygon
• Infinite skew polyhedron

• 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
  • (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