In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface. Skew apeirohedra have also been called polyhedral sponges. Many are directly related to a

convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells. Twenty-eight such honey ...

, being the polygonal surface of a honeycomb with some of the

cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...

removed. Characteristically, an infinite skew polyhedron divides 3-dimensional space into two halves. If one half is thought of as ''solid'' the figure is sometimes called a partial honeycomb.

Regular skew apeirohedra

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to ''regular skew polyhedra'' (apeirohedra). Coxeter and Petrie found three of these that filled 3-space: There also exist chiral skew apeirohedra of types , , and . These skew apeirohedra are vertex-transitive, edge-transitive, and face-transitive, but not mirror symmetric . Beyond Euclidean 3-space, in 1967 C. W. L. Garner published a set of 31 regular skew polyhedra in hyperbolic 3-space.

Gott's regular pseudopolyhedrons

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 , , and and four new ones: , , , .The Symmetries of things, Pseudo-platonic polyhedra, p.340-344 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

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

, with by p-gonal faces, ''q'' around each vertex. However neither the term "pseudopolyhedron" nor Gott's definition of regularity have achieved wide usage. Crystallographer

A.F. Wells AF, af, Af, etc. may refer to: Arts and entertainment * A-F Records, an independent record label in Pittsburgh, Pennsylvania, US, founded by the band Anti-Flag *'' Almost Family'' episode titles tend to be "'' djective' AF" Businesses and organiz ...

in 1960's also published a list of skew apeirohedra. Melinda Green publishe
many more
in 1998.

Prismatic forms

There are two ''prismatic'' forms: # : 5 squares on a vertex (Two parallel square tilings connected by

cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

holes.) # : 8 triangles on a vertex (Two parallel triangle tilings connected by octahedral holes.)

Other forms

is also formed from parallel planes of ''triangular tilings'', with alternating octahedral holes going both ways. is composed of 3 coplanar

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

s 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'' and ''triangular tiling'' can be curved into approximating infinite cylinders in 3-space.

Theorems

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

Uniform skew apeirohedra

There are many other uniform ( vertex-transitive) skew apeirohedra. Wachmann, Burt and Kleinmann (1974) discovered many examples but it is not known whether their list is complete. A few are illustrated here. They can be named by their vertex configuration, although it is not a unique designation for skew forms. Others can be constructed as augmented chains of polyhedra:

References

* Coxeter, ''Regular Polytopes'', Third edition, (1973), Dover edition, * ''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,

** (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra", '' Scripta Mathematica'' 6 (1939) 240-244. * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) ''The Symmetries of Things'', (Chapter 23, Objects with prime symmetry, pseudo-platonic polyhedra, p340-344) *

* A. F. Wells, ''Three-Dimensional Nets and Polyhedra'', Wiley, 1977

*A. Wachmann, M. Burt and M. Kleinmann, ''Infinite polyhedra'', Technion, 1974. 2nd Edn. 2005. * E. Schulte, J.M. Will
On Coxeter's regular skew polyhedra
Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262

External links

* * *

18 SYMMETRY OF POLYTOPES AND POLYHEDRA, Egon Schulte: 18.3 REGULAR SKEW POLYHEDRA

{{Polyhedra Polyhedra