# Regular skew apeirohedron

In geometry, a **regular skew apeirohedron** is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

## History

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to finite regular skew polyhedra in 4-dimensions, and infinite regular skew apeirohedra in 3-dimensions (described here).

Coxeter identified 3 forms, with planar faces and skew vertex figures, two are complements of each other. They are all named with a modified Schläfli symbol {*l*,*m*|*n*}, where there are *l*-gonal faces, *m* faces around each vertex, with *holes* identified as *n*-gonal missing faces.

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*)

## Regular skew apeirohedra of Euclidean 3-space

The three Euclidean solutions in 3-space are {4,6|4}, {6,4|4}, and {6,6|3}. John Conway named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron.[1]

**Mucube**: {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.)**Muoctahedron**: {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.)**Mutetrahedron**: {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.)

Coxeter gives these regular skew apeirohedra {2q,2r|p} with extended chiral symmetry [[(*p*,*q*,*p*,*r*)]^{+}] which he says is isomorphic to his abstract group (2*q*,2*r*|2,*p*). The related honeycomb has the extended symmetry [[(*p*,*q*,*p*,*r*)]].[2]

Coxeter group Symmetry |
Apeirohedron {p,q|l} | Image | Face {p} | Hole { l} | Vertex figure | Related honeycomb | |
---|---|---|---|---|---|---|---|

[[4,3,4]] [[4,3,4] ^{+}] |
{4,6|4} Mucube | animation | t _{0,3}{4,3,4} | ||||

{6,4|4} Muoctahedron | animation | 2t{4,3,4} | |||||

[[3 ^{[4]}]][[3 ^{[4]}]^{+}] | {6,6|3} Mutetrahedron | animation | q{4,3,4} |

## Regular skew apeirohedra in hyperbolic 3-space

In 1967, C. W. L. Garner identified 31 hyperbolic skew apeirohedra with regular skew polygon vertex figures, found in a similar search to the 3 above from Euclidean space.[3]

These represent 14 compact and 17 paracompact regular skew polyhedra in hyperbolic space, constructed from the symmetry of a subset of linear and cyclic Coxeter groups graphs of the form [[(*p*,*q*,*p*,*r*)]], These define *regular skew polyhedra* {2*q*,2*r*|*p*} and dual {2*r*,2*q*|*p*}. For the special case of linear graph groups *r* = 2, this represents the Coxeter group [*p*,*q*,*p*]. It generates regular skews {2*q*,*4*|*p*} and {4,2*q*|*p*}. All of these exist as a subset of faces of the convex uniform honeycombs in hyperbolic space.

The skew apeirohedron shares the same antiprism vertex figure with the honeycomb, but only the zig-zag edge faces of the vertex figure are realized, while the other faces make "holes".

Coxeter group |
Apeirohedron {p,q|l} | Face {p} | Hole {l} | Honeycomb | Vertex figure | Apeirohedron {p,q|l} | Face {p} | Hole {l} | Honeycomb | Vertex figure | |
---|---|---|---|---|---|---|---|---|---|---|---|

[3,5,3] | {10,4|3} | 2t{3,5,3} | {4,10|3} | t _{0,3}{3,5,3} | |||||||

[5,3,5] | {6,4|5} | 2t{5,3,5} | {4,6|5} | t _{0,3}{5,3,5} | |||||||

[(4,3,3,3)] | {8,6|3} | ct{(4,3,3,3)} | {6,8|3} | ct{(3,3,4,3)} | |||||||

[(5,3,3,3)] | {10,6|3} | ct{(5,3,3,3)} | {6,10|3} | ct{(3,3,5,3)} | |||||||

[(4,3,4,3)] | {8,8|3} | ct{(4,3,4,3)} | {6,6|4} | ct{(3,4,3,4)} | |||||||

[(5,3,4,3)] | {8,10|3} | ct{(4,3,5,3)} | {10,8|3} | ct{(5,3,4,3)} | |||||||

[(5,3,5,3)] | {10,10|3} | ct{(5,3,5,3)} | {6,6|5} | ct{(3,5,3,5)} |

Coxeter group |
Apeirohedron {p,q|l} | Face {p} | Hole {l} | Honeycomb | Vertex figure | Apeirohedron {p,q|l} | Face {p} | Hole {l} | Honeycomb | Vertex figure | |
---|---|---|---|---|---|---|---|---|---|---|---|

[4,4,4] | {8,4|4} | 2t{4,4,4} | {4,8|4} | t _{0,3}{4,4,4} | |||||||

[3,6,3] | {12,4|3} | 2t{3,6,3} | {4,12|3} | t _{0,3}{3,6,3} | |||||||

[6,3,6] | {6,4|6} | 2t{6,3,6} | {4,6|6} | t _{0,3}{6,3,6} | |||||||

[(4,4,4,3)] | {8,6|4} | ct{(4,4,3,4)} | {6,8|4} | ct{(3,4,4,4)} | |||||||

[(4,4,4,4)] | {8,8|4} | q{4,4,4} | |||||||||

[(6,3,3,3)] | {12,6|3} | ct{(6,3,3,3)} | {6,12|3} | ct{(3,3,6,3)} | |||||||

[(6,3,4,3)] | {12,8|3} | ct{(6,3,4,3)} | {8,12|3} | ct{(4,3,6,3)} | |||||||

[(6,3,5,3)] | {12,10|3} | ct{(6,3,5,3)} | {10,12|3} | ct{(5,3,6,3)} | |||||||

[(6,3,6,3)] | {12,12|3} | ct{(6,3,6,3)} | {6,6|6} | ct{(3,6,3,6)} |

## See also

## References

- The Symmetry of Things, 2008, Chapter 23
*Objects with Primary Symmetry*,*Infinite Platonic Polyhedra*, pp. 333–335 - Coxeter,
*Regular and Semi-Regular Polytopes II*2.34) - Garner, C. W. L.
*Regular Skew Polyhedra in Hyperbolic Three-Space.*Can. J. Math. 19, 1179–1186, 1967. Note: His paper says there are 32, but one is self-dual, leaving 31.

- Petrie–Coxeter Maps Revisited PDF, Isabel Hubard, Egon Schulte, Asia Ivic Weiss, 2005
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5, - Peter McMullen,
*Four-Dimensional Regular Polyhedra*, Discrete & Computational Geometry September 2007, Volume 38, Issue 2, pp 355–387 - 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- (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra",
*Scripta Mathematica*6 (1939) 240–244. - (Paper 22) H.S.M. Coxeter,
*Regular and Semi Regular Polytopes I*, [Math. Zeit. 46 (1940) 380–407, MR 2,10] - (Paper 23) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes II*, [Math. Zeit. 188 (1985) 559–591]

- (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra",
- 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, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)- Coxeter, H. S. M.
*Regular Skew Polyhedra in Three and Four Dimensions.*Proc. London Math. Soc. 43, 33–62, 1937.

- Coxeter, H. S. M.