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Fibrifold
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In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by John Horton Conway, Olaf Delgado Friedrichs, and Daniel H. Huson et al. (2001), who introduced a system of notation for 3-dimensional fibrifolds and used this to assign names to the 219 affine space group types. 184 of these are considered reducible, and 35 irreducible.
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (May 2024) |
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Irreducible cubic space groups
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The 35 irreducible space groups correspond to the cubic space group.
8o:2 | 4−:2 | 4o:2 | 4+:2 | 2−:2 | 2o:2 | 2+:2 | 1o:2 | |||
8o | 4− | 4o | 4+ | 2− | 2o | 2+ | 1o | |||
8o/4 | 4−/4 | 4o/4 | 4+/4 | 2−/4 | 2o/4 | 2+/4 | 1o/4 | |||
8−o | 8oo | 8+o | 4− − | 4−o | 4oo | 4+o | 4++ | 2−o | 2oo | 2+o |
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8 primary hexoctahedral hextetrahedral lattices of the cubic space groups | The fibrifold cubic subgroup structure shown is based on extending symmetry of the tetragonal disphenoid fundamental domain of space group 216, similar to the square |
Irreducible group symbols (indexed 195−230) in Hermann–Mauguin notation, Fibrifold notation, geometric notation, and Coxeter notation:
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References
- Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie, 42 (2): 475–507, ISSN 0138-4821, MR 1865535
- Hestenes, David; Holt, Jeremy W. (February 2007), "The Crystallographic Space Groups in Geometric Algebra" (PDF), Journal of Mathematical Physics, 48 (2): 023514, Bibcode:2007JMP....48b3514H, doi:10.1063/1.2426416
- Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), The Symmetries of Things, Taylor & Francis, ISBN 978-1-56881-220-5, Zbl 1173.00001
- Coxeter, H.S.M. (1995), "Regular and Semi Regular Polytopes III", in Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; et al. (eds.), Kaleidoscopes: Selected Writings of H.S.M. Coxeter, Wiley, pp. 313–358, ISBN 978-0-471-01003-6, Zbl 0976.01023
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