Infinite skew polyhedron

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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

1 Regular skew polyhedra
2 Prismatic regular skew polyhedra
3 Semiregular skew polyhedra
4 References
5 External links

[edit] 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)

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.

{4,6\|4}	{6,4\|4}	{6,6\|3}
{4,6}	{6,4}	{6,6}
Cubic honeycomb t₀{4,3,4}	Bitruncated cubic t_1,2{4,3,4}	quarter cubic honeycomb

[edit] Prismatic regular skew polyhedra

There are also two regular prismatic forms, disqualified by Coxeter (among others) from being called regular because they have adjacent coplanar faces.

5 squares on a vertex (Two parallel square tilings connected by cubic holes.)
8 triangles on a vertex (Two parallel triangle tilings connected by octahedral holes.)

Beyond Euclidean 3-space, C. W. L. Garner determined a set of 32 regular skew polyhedra in hyperbolic 3-space, derived from the 4 regular hyperbolic honeycombs.

[edit] Semiregular skew polyhedra

There are many other semiregular (vertex-transitive) skew polyhedra, discovered by A.F. Wells and J.R. Gott (he called them pseudopolyhedrons) in the 1960's.

A prismatic semiregular skew polyhedron with vertex configuration 4.4.4.6.

A (partial) semiregular skew polyhedron with vertex configuration 4.8.4.8. Related to the omnitruncated cubic honeycomb.

A (partial) semiregular skew polyhedron with vertex configuration 3.4.4.4.4. Related to the Runcitruncated cubic honeycomb.

[edit] 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 [1]
- (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.