Skew apeirohedron

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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, being the polygonal surface of a honeycomb with some of the cells 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[edit]

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra (apeirohedra).[1]

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

Regular skew apeirohedra

There also exist chiral skew apeirohedra of types {4,6}, {6,4}, and {6,6}. These skew apeirohedra are vertex-transitive, edge-transitive, and face-transitive, but not mirror 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]

Gott's regular pseudopolyhedrons[edit]

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

Crystallographer A.F. Wells in 1960's also published a list of skew apeirohedra.

{p,q} Cells
around a vertex
Space group Related H2
{4,5} 3 cubes Pseudo-platonic cubic polyhedron vertex.png Pseudo-platonic cubic polyhedron.png Im3m [[4,3,4]] 8°:2 *4222
{4,5} 1 truncated octahedron
2 hexagonal prisms
Pseudo-platonic hexagonal prism truncated octahedral polyhedron vertex.png I3 [[4,3+,4]] 8°:2 2*42
{3,7} 1 octahedron
1 icosahedron
Pseudo-platonic octa-icosahedral vertex.png Pseudo-platonic octa-icosahedral.png Fd3 [[3[4]]]+ 3222
{3,8} 1 tetrahedron
3 octahedra
Pseudo-platonic tetra-octahedral polyhedron vertex.png Pseudo-platonic tetra-octahedral polyhedron2.png Fd3m [[3[4]]] 2+:2 2*32
{3,8} 2 snub cubes Pseudo-platonic snub cubic polyhedron vertex.png Uniform apeirohedron snub cube 33333333.png Fm3m [4,(3,4)+] 2−− 32*
{3,9} 1 icosahedron
2 octahedra
Pseudo-platonic pyritohedral polyhedron vertex.png I3 [[4,3+,4]] 8°:2 22*2
{3,12} 5 octahedra Pseudo-platonic octahedral polyhedron vertex.png Sk12x3.gif Im3m [[4,3,4]] 8°:2 2*32

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.


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.

Uniform skew apeirohedra[edit]

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.

Uniform skew apeirohedra related to uniform honeycombs
Cantitruncated cubic honeycomb apeirohedron 4466.png Omnitruncated cubic honeycomb apeirohedron 4466.png Runcicantic cubic honeycomb apeirohedron 6688.png
Related to cantitruncated cubic honeycomb, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png Related to runcicantic cubic honeycomb, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Omnitruncated cubic honeycomb apeirohedron 4446.png Skew polyhedron 4848.png Icosahedron octahedron infinite skew pseudoregular polyhedron.png
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
Apeirohedron truncated octahedra and hexagonal prism 4446.png Octagonal prism apeirohedron 4448.png Skew polyhedron 34444.png
Related to the runcitruncated cubic honeycomb.
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Prismatic uniform skew apeirohedra
Pseudoregular apeirohedron prismatic 44444.png
Related to CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Skew polyhedron 4446a.png
Related to CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png

Others can be constructed as augmented chains of polyhedra:

Coxeter helix 3 colors.png
Coxeter helix 3 colors cw.png
Cube stack diagonal-face helix apeirogon.png
Boerdijk–Coxeter helix
Stacks of cubes

See also[edit]


  1. ^ Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
  2. ^ Garner, C. W. L. Regular Skew Polyhedra in Hyperbolic Three-Space. Can. J. Math. 19, 1179-1186, 1967. [1]
  3. ^ J. R. Gott, Pseudopolyhedrons, American Mathematical Monthly, Vol 74, p. 497-504, 1967.
  4. ^ The Symmetries of things, Pseudo-platonic polyhedra, p.340-344
  • 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]
  • 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. Wills On Coxeter's regular skew polyhedra, Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262

External links[edit]