Infinite skew polyhedron

In geometry, infinite skew polyhedra are an extended definition of polyhedra, created by regular polygon faces, and nonplanar vertex figures.

Many are directly related to a convex uniform honeycomb, being the polygonal surface of a honeycomb with some of the cells removed. As "solids" they are called partial honeycombs and also sponges.

These polyhedra have also been called hyperbolic tessellations because they can be seen as related to hyperbolic space tessellations which also have negative angle defects. They are examples of the more general class of infinite polyhedra, or apeirohedra

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 {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)

Coxeter and Petrie found three of these that filled 3-space:

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)
** 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.
* J. R. Gott, "Pseudopolyhedrons", American Mathematical Monthly, Vol 74, p. 497-504, 1967.
* A. F. Wells, "Three-Dimensional Nets and Polyhedra", Wiley, 1977.

External links

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*GlossaryForHyperspace | anchor=Skew | title=Skew polytope
* [http://www.uwgb.edu/dutchs/symmetry/hyperbol.htm "Hyperbolic" Tessellations]
* [http://www.superliminal.com/geometry/infinite/infinite.htm Infinite Regular Polyhedra] [http://www.superliminal.com/geometry/ogeometry.htm]
* [http://Polyhedra.Doskey.com/PartialHoneycombs.html Infinite Repeating Polyhedra - Partial Honeycombs in 3-Space]
* [http://www.math.neu.edu/~schulte/symchapter.pdf 18 SYMMETRY OF POLYTOPES AND POLYHEDRA, Egon Schulte: 18.3 REGULAR SKEW POLYHEDRA]
* [http://www.3doro.de/infini-poly.htm Infinite Polyhedra, T.E. Dorozinski ]