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 about each 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 about each 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 about each 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 (2q,2r|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 |
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![]() ![]() ![]() t0,3{4,3,4} |
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{6,4|4} Muoctahedron |
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![]() ![]() ![]() 2t{4,3,4} |
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![]() ![]() ![]() [[3[4]]] [[3[4]]+] |
{6,6|3} Mutetrahedron |
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![]() ![]() ![]() q{4,3,4} |
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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 {2q,2r|p} and dual {2r,2q|p}. For the special case of linear graph groups r = 2, this represents the Coxeter group [p,q,p]. It generates regular skews {2q,4|p} and {4,2q|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} | ![]() |
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![]() ![]() ![]() ![]() 2t{3,5,3} |
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{4,10|3} | ![]() |
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![]() ![]() ![]() ![]() t0,3{3,5,3} |
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![]() ![]() ![]() [5,3,5] |
{6,4|5} | ![]() |
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![]() ![]() ![]() 2t{5,3,5} |
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{4,6|5} | ![]() |
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![]() ![]() ![]() t0,3{5,3,5} |
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![]() ![]() ![]() ![]() [(4,3,3,3)] |
{8,6|3} | ![]() |
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![]() ![]() ![]() ![]() ct{(4,3,3,3)} |
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{6,8|3} | ![]() |
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![]() ![]() ![]() ![]() ct{(3,3,4,3)} |
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![]() ![]() ![]() ![]() [(5,3,3,3)] |
{10,6|3} | ![]() |
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![]() ![]() ![]() ![]() ct{(5,3,3,3)} |
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{6,10|3} | ![]() |
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![]() ![]() ![]() ![]() ct{(3,3,5,3)} |
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![]() ![]() ![]() ![]() ![]() [(4,3,4,3)] |
{8,8|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(4,3,4,3)} |
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{6,6|4} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(3,4,3,4)} |
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![]() ![]() ![]() ![]() ![]() [(5,3,4,3)] |
{8,10|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(4,3,5,3)} |
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{10,8|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(5,3,4,3)} |
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![]() ![]() ![]() ![]() ![]() [(5,3,5,3)] |
{10,10|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(5,3,5,3)} |
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{6,6|5} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(3,5,3,5)} |
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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} | ![]() |
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![]() ![]() ![]() ![]() 2t{4,4,4} |
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{4,8|4} | ![]() |
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![]() ![]() ![]() ![]() t0,3{4,4,4} |
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![]() ![]() ![]() ![]() [3,6,3] |
{12,4|3} | ![]() |
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![]() ![]() ![]() ![]() 2t{3,6,3} |
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{4,12|3} | ![]() |
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![]() ![]() ![]() ![]() t0,3{3,6,3} |
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![]() ![]() ![]() [6,3,6] |
{6,4|6} | ![]() |
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![]() ![]() ![]() 2t{6,3,6} |
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{4,6|6} | ![]() |
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![]() ![]() ![]() t0,3{6,3,6} |
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![]() ![]() ![]() ![]() [(4,4,4,3)] |
{8,6|4} | ![]() |
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![]() ![]() ![]() ![]() ct{(4,4,3,4)} |
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{6,8|4} | ![]() |
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![]() ![]() ![]() ![]() ct{(3,4,4,4)} |
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![]() ![]() ![]() ![]() ![]() [(4,4,4,4)] |
{8,8|4} | ![]() |
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![]() ![]() ![]() ![]() ![]() q{4,4,4} |
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![]() ![]() ![]() ![]() ![]() [(6,3,3,3)] |
{12,6|3} | ![]() |
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![]() ![]() ![]() ![]() ct{(6,3,3,3)} |
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{6,12|3} | ![]() |
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![]() ![]() ![]() ![]() ct{(3,3,6,3)} |
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![]() ![]() ![]() ![]() ![]() [(6,3,4,3)] |
{12,8|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(6,3,4,3)} |
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{8,12|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(4,3,6,3)} |
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![]() ![]() ![]() ![]() ![]() [(6,3,5,3)] |
{12,10|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(6,3,5,3)} |
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{10,12|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(5,3,6,3)} |
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![]() ![]() ![]() ![]() ![]() [(6,3,6,3)] |
{12,12|3} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(6,3,6,3)} |
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{6,6|6} | ![]() |
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![]() ![]() ![]() ![]() ![]() ct{(3,6,3,6)} |
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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. [1] 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 [2]
- (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]
- 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.
- Polyhedra