Infinite skew polyhedron

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In geometry, an infinite skew polyhedron is an extension of the idea of a polyhedron, consisting of regular polygon faces with nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.

Infinite skew polyhedra have also been called polyhedral sponges, and also 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.

Many are directly related to a convex uniform honeycomb, being the polygonal surface of a honeycomb with some of the cells removed. Characteristically, an infinite skew polyhedron divides space into two halves. If one half is thought of as solid the figure is sometimes called a partial honeycomb.

Regular skew polyhedra[edit]

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, represented 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:

Regular skew polyhedra (partial)
Six-square skew polyhedron.png
{4,6|4}
mucube
Four-hexagon skew polyhedron.png
{6,4|4}
muoctahedron
Six-hexagon skew polyhedron.png
{6,6|3}
mutetrahedron

There also exist chiral skew polyhedra of types {4,6}, {6,4}, and {6,6}. These skew polyhedra are vertex-transitive, edge-transitive, and face-transitive, but not mirror symmetric (Schulte 2004).

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.

Gott's regular pseudopolyhedrons[edit]

Example: Parts of {3,7}, constructed from augmenting 8 octahedra around every icosahedron

J. Richard Gott in 1967 published a larger set of seven infinite skew polyhedra which he called regular pseudopolyhedrons, including the three from Coxeter as {4,6}, {6,4}, and {6,6} and four new ones: {5,5}, {4,5}, {3,8}, {3,10}.

Gott relaxed the definition of regularity to allow his new figures. Where Coxeter and Petrie had required that the vertices be symmetrical, Gott required only that they be congruent. Thus, Gott's new examples are not regular by Coxeter and Petrie's definition.

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

A.F. Wells also published a list of pseudopolyhedra in the 1960s, including different forms with the same symbol:

{p,q} Cells Space group Related H2
orbifold
notation
Cubic
space
group
Coxeter
notation
Fibrifold
notation
{4,5} Cubes Im3m [[4,3,4]] 8o:2 *4222
{4,5} Truncated octahedra
hexagonal prisms
I3 [[4,3+,4]] 80:2 2*42
{3,7} Icosahedra Fd3 [[3[4]]]+ 2o- 3222
{3,8} Octahedra Fd3m [[3[4]]] 2+:2 2*32
{3,8} Snub cubes Fm3m [4,(3,4)+] 2-- 32*
{3,9} Icosahedra I3 [[4,3+,4]] 80:2 22*2
{3,12} Octahedra Im3m [[4,3,4]] 8o:2 2*32

However neither the term "pseudopolyhedron" nor Gott's definition of regularity have achieved wide usage.

Prismatic forms[edit]

Five-square skew polyhedron.png
Prismatic form: {4,5}

There are two prismatic forms:

  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.)

Other forms[edit]

{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.

Gott also acknowledged that there are 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.

Theorems[edit]

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.

Semiregular infinite skew polyhedra[edit]

There are many other semiregular (vertex-transitive) infinite skew polyhedra. Wachmann, Burt and Kleinmann (1974) discovered many examples but it is not known whether their list is complete.

Three are illustrated here:

Skew polyhedron 4446a.png
A prismatic semiregular skew polyhedron with vertex configuration 4.4.4.6.
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Skew polyhedron 4848.png
A (partial) semiregular skew polyhedron with vertex configuration 4.8.4.8. Related to the omnitruncated cubic honeycomb.
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Skew polyhedron 34444.png
A (partial) semiregular skew polyhedron with vertex configuration 3.4.4.4.4. Related to the Runcitruncated cubic honeycomb.
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png

See also[edit]

References[edit]

  • 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 [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 0-486-40919-8 (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.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 23, Objects with prime symmetry, pseudo-platonic polyhedra, p340-344)
  • Schulte, Egon (2004), Chiral polyhedra in ordinary space. I, Discrete and Computational Geometry 32 (1): 55–99, doi:10.1007/s00454-004-0843-x, MR 2060817 .
  • A. F. Wells, Three-Dimensional Nets and Polyhedra, Wiley, 1977.
  • A. Wachmann, M. Burt and M. Kleinmann, Infinite polyhedra, Technion, 1974. 2nd Edn. 2005.

External links[edit]