Regular skew polyhedron#Finite regular skew polyhedra

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 {{math|{l,m{{!}}n}}} for these figures, with {{math|{l,m}}} implying the vertex figure, {{mvar|m}} {{mvar|l}}-gons around a vertex, and {{mvar|n}}-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.

The regular skew polyhedra, represented by {{math|{l,m{{!}}n}}}, follow this equation:

: $2\; \backslash cos\; \backslash frac\{\backslash pi\}\{l\}\; \backslash cos\; \backslash frac\{\backslash pi\}\{m\}\; =\; \backslash cos\; \backslash frac\{\backslash pi\}\{n\}$

A first set {{math|{l,m{{!}}n}}}, repeats the five convex Platonic solids, and one nonconvex Kepler–Poinsot solid:

:

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)]].Coxeter, Regular and Semi-Regular Polytopes II 2.34)

{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.

File:600-cell tet ring.png.]]

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File:5-cube_t0.svg, 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 {{pipe}}, q}. These can also be represented a regular finite map or {l, m}_2q, and group G^l,m,q.Coxeter and Moser, Generators and relations for discrete groups, Sec 8.6 Maps having specified Petrie polygons. p. 110

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.{{cite journal |last1=Deza |first1=Michael |last2=Shtogrin |first2=Mikhael |title=Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices |journal=Advanced Studies in Pure Mathematics |series=Arrangements – Tokyo 1998 |date=1998 |page=77 |doi=10.2969/aspm/02710073 |isbn=978-4-931469-77-8 |url=https://projecteuclid.org/euclid.aspm/1534788966 |accessdate=4 April 2020|doi-access=free }}

• Skew polygon
• Infinite skew polyhedron

{{reflist}}

• Peter McMullen, [https://link.springer.com/article/10.1007%2Fs00454-007-1342-7 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
• (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 [http://www.sciencedirect.com/science/article/pii/0012365X86900178 On Coxeter's regular skew polyhedra], Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262

Category:Polyhedra