Ingeometry, askew apeirohedron is an infiniteskew polyhedron consisting of nonplanarfaces or nonplanarvertex figures, allowing the figure to extend indefinitely without folding round to form aclosed surface.

Skew apeirohedra have also been calledpolyhedral sponges.

Many are directly related to aconvex uniform honeycomb, being thepolygonal surface of ahoneycomb with some of thecells removed. Characteristically, an infinite skew polyhedron divides 3-dimensional space into two halves. If one half is thought of assolid the figure is sometimes called apartial honeycomb.

According toCoxeter, in 1926John Flinders Petrie generalized the concept ofregular skew polygons (nonplanar polygons) toregular skew polyhedra (apeirohedra).^[1]

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

Regular skew apeirohedra
{4,6|4} mucube	{6,4|4} muoctahedron	{6,6|3} mutetrahedron

There also existchiral skew apeirohedra of types {4,6}, {6,4}, and {6,6}. These skew apeirohedra arevertex-transitive,edge-transitive, andface-transitive, but notmirror symmetric (Schulte 2004).

Beyond Euclidean 3-space, in 1967 C. W. L. Garner published a set of 31 regular skew polyhedra in hyperbolic 3-space.^[2]

J. Richard Gott in 1967 published a larger set of seven infinite skew polyhedra which he calledregular 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}.^[3][4]

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 ofregular polyhedra,regular tilings, andregular pseudopolyhedra asregular generalized polyhedra, representable by a {p,q}Schläfli symbol, with by p-gonal faces,q around each vertex. However neither the term "pseudopolyhedron" nor Gott's definition of regularity have achieved wide usage.

CrystallographerA.F. Wells in 1960's also published a list of skew apeirohedra.Melinda Green publishedmany more in 1998.

{p,q}	Cells around a vertex	Vertex faces	Larger pattern	Space group			Related H2 orbifold notation
				Cubic space group	Coxeter notation	Fibrifold notation
{4,5}	3cubes			Im3m	[[4,3,4]]	8°:2	*4222
{4,5}	1truncated octahedron 2hexagonal prisms			I3	[[4,3+,4]]	8°:2	2*42
{3,7}	1octahedron 1icosahedron			Fd3	[[3[4]]]+	2°−	3222
{3,8}	2snub cubes			Fm3m	[4,(3,4)+]	2−−	32*
{3,9}	1tetrahedron 3octahedra			Fd3m	[[3[4]]]	2+:2	2*32
{3,9}	1icosahedron 2octahedra			I3	[[4,3+,4]]	8°:2	22*2
{3,12}	5 octahedra			Im3m	[[4,3,4]]	8°:2	2*32

Prismatic form: {4,5}

There are twoprismatic forms:

1. {4,5}: 5 squares on a vertex (Two parallelsquare tilings connected bycubic holes.)
2. {3,8}: 8 triangles on a vertex (Two paralleltriangle tilings connected byoctahedral holes.)

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

{5,5} is composed of 3 coplanarpentagons 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 thesquare tiling {4,4} andtriangular 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.

There are many otheruniform (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 theirvertex configuration, although it is not a unique designation for skew forms.

Uniform skew apeirohedra related to uniform honeycombs

4.4.6.6		6.6.8.8
Related tocantitruncated cubic honeycomb,		Related toruncicantic cubic honeycomb,
4.4.4.6	4.8.4.8	3.3.3.3.3.3.3
Related to theomnitruncated cubic honeycomb:
4.4.4.6	4.4.4.8	3.4.4.4.4
		Related to theruncitruncated cubic honeycomb.

Prismatic uniform skew apeirohedra

4.4.4.4.4	4.4.4.6
Related to	Related to

Others can be constructed as augmented chains of polyhedra:

Uniform Boerdijk–Coxeter helix	Stacks of cubes

• Petrie polygon
• Regular skew polyhedron

• 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[2]
  • (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,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.[3]Archived 2016-03-04 at theWayback Machine
• A. F. Wells,Three-Dimensional Nets and Polyhedra, Wiley, 1977.[4]
• A. Wachmann, M. Burt and M. Kleinmann,Infinite polyhedra, Technion, 1974. 2nd Edn. 2005.
• E. Schulte, J.M. WillsOn Coxeter's regular skew polyhedra, Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262

• Weisstein, Eric W."Regular Skew Polyhedron".MathWorld.
• Weisstein, Eric W."Honeycombs and sponges".MathWorld.
• Olshevsky, George."Skew polytope".Glossary for Hyperspace. Archived fromthe original on 4 February 2007.
• "Hyperbolic" Tessellations
• Infinite Regular Polyhedra[5]Archived 2016-03-12 at theWayback Machine
• 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

• Polyhedra

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