Infinite skew polyhedron

From Wikipedia, the free encyclopedia

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

Contents 1 Regular skew polyhedra 1.1 Prismatic regular skew polyhedra 2 Pseudopolyhedrons 3 Semiregular pseudopolyhedrons 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:

1. {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.)
2. {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.)
3. {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.

Partial polyhedra
{4,6|4}	{6,4|4}	{6,6|3}
Vertex figures
{4,6}	{6,4}	{6,6}
partial corresponding convex uniform honeycombs
Cubic honeycomb t0{4,3,4}	Bitruncated cubic t1,2{4,3,4}	quarter cubic honeycomb t0,1[P4]

[edit] Prismatic regular skew polyhedra

Prismatic form: {4,5}

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

1. {4,5}: 5 squares on a vertex (Two parallel square tilings connected by cubic holes.)
2. {3,8}: 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] Pseudopolyhedrons

J. Richard Gott in 1967 published a larger set of seven regular pseudopolyhedrons, including the three from Coxeter, the two coplanar ones {3,8}, and {4,5}, and two new ones: {3,10}, {5,5}.

Gott called the full set of regular polyhedra, regular tilings, and regular pseudopolyhedrons as regular generalized polyhedra, and representable by a {p,q} Schläfli symbol, with by p-gonal faces, q around each vertex.

{3,10} is also formed from parallel planes of triangular tilings, with alternating octahedral holes going both ways.

{5,5} is composed of 3 coplanar pentagons around a vertex and two perpendicular pentagons filling the gap.

He also acknowledged there's other periodic forms of the regular planar tessellations. Both the square tiling {4,4} and triangular tiling {3,6} can be curved into approximating infinite cylinders in 3-space.

He wrote some theorems:

1. For every regular polyhedron {p,q}: (p-2)*(q-2)<4. For Every regular tessellation: (p-2)*(q-2)=4. For every regular pseudopolyhedron: (p-2)*(q-2)>4.
2. The number of faces surrounding a given face is p*(q-2) in any regular generalized polyhedron.
3. Every regular pseudopolyhedron approximates a negatively curved surface.
4. The seven regular pseudopolyhedron are repeating structures.

A.F. Wells also published a list of pseudopolyhedra in the 1960's, including different forms with the same symbol: {4,5}, {3,7}, {3,8}, {3,10}, {3,12}.

[edit] Semiregular pseudopolyhedrons

There are many other semiregular (vertex-transitive) skew polyhedra. Some examples:

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

[edit] External links

• Eric W. Weisstein, Regular Skew Polyhedron at MathWorld.
• Eric W. Weisstein, Honeycombs and sponges at MathWorld.
• Olshevsky, George, Skew polytope at Glossary for Hyperspace.
• "Hyperbolic" Tessellations
• Infinite Regular Polyhedra [2]
• Infinite Repeating Polyhedra - Partial Honeycombs in 3-Space
• 18 SYMMETRY OF POLYTOPES AND POLYHEDRA, Egon Schulte: 18.3 REGULAR SKEW POLYHEDRA
• Infinite Polyhedra, T.E. Dorozinski