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, reresented 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:
# {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.)
# {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.)
# {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|∞}.

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

The second set have a form: {l, m , q}

See also

* Skew polygon
* Infinite skew polyhedron

References

* Coxeter, "Regular Polytopes", Third edition, (1973), Dover edition, ISBN 0-486-61480-8
* "Kaleidoscopes: Selected Writings of H.S.M. Coxeter", editied 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.
* Coxeter, "The Beauty of Geometry: Twelve Essays", Dover Publications, 1999, ISBN 0486409198 (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.