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.
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Irreducible cubic space groups
The 35 irreducible space groups correspond to the cubic space group.
35 irreducible space groups| 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 |
| ClassPoint group | Hexoctahedral*432 (m3m) | Hextetrahedral*332 (43m) | Gyroidal432 (432) | Diploidal3*2 (m3) | Tetartoidal332 (23) |
|---|---|---|---|---|---|
| bc lattice (I) | 8o:2 (Im3m) | 4o:2 (I43m) | 8+o (I432) | 8−o (I3) | 4oo (I23) |
| nc lattice (P) | 4−:2 (Pm3m) | 2o:2 (P43m) | 4−o (P432) | 4− (Pm3) | 2o (P23) |
| 4+:2 (Pn3m) | 4+ (P4232) | 4+o (Pn3) | |||
| fc lattice (F) | 2−:2 (Fm3m) | 1o:2 (F43m) | 2−o (F432) | 2− (Fm3) | 1o (F23) |
| 2+:2 (Fd3m) | 2+ (F4132) | 2+o (Fd3) | |||
| Otherlatticegroups | 8o (Pm3n)8oo (Pn3n)4− − (Fm3c)4++ (Fd3c) | 4o (P43n)2oo (F43c) | |||
| Achiralquartergroups | 8o/4 (Ia3d) | 4o/4 (I43d) | 4+/4 (I4132)2+/4 (P4332,P4132) | 2−/4 (Pa3)4−/4 (Ia3) | 1o/4 (P213)2o/4 (I213) |
| 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:
| Class(Orbifold point group) | Space groups | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Tetartoidal23(332) | 195 | 196 | 197 | 198 | 199 | |||||
| P23 | F23 | I23 | P213 | I213 | ||||||
| 2o | 1o | 4oo | 1o/4 | 2o/4 | ||||||
| P3.3.2 | F3.3.2 | I3.3.2 | P3.3.21 | I3.3.21 | ||||||
| [(4,3+,4,2+)] | [3[4]]+ | [[(4,3+,4,2+)]] | ||||||||
| Diploidal43m(3*2) | 200 | 201 | 202 | 203 | 204 | 205 | 206 | |||
| Pm3 | Pn3 | Fm3 | Fd3 | I3 | Pa3 | Ia3 | ||||
| 4− | 4+o | 2− | 2+o | 8−o | 2−/4 | 4−/4 | ||||
| P43 | Pn43 | F43 | Fd43 | I43 | Pb43 | Ib43 | ||||
| [4,3+,4] | [[4,3+,4]+] | [4,(31,1)+] | [[3[4]]]+ | [[4,3+,4]] | ||||||
| Gyroidal432(432) | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | ||
| P432 | P4232 | F432 | F4132 | I432 | P4332 | P4132 | I4132 | |||
| 4−o | 4+ | 2−o | 2+ | 8+o | 2+/4 | 4+/4 | ||||
| P4.3.2 | P42.3.2 | F4.3.2 | F41.3.2 | I4.3.2 | P43.3.2 | P41.3.2 | I41.3.2 | |||
| [4,3,4]+ | [[4,3,4]+]+ | [4,31,1]+ | [[3[4]]]+ | [[4,3,4]]+ | ||||||
| Hextetrahedral43m(*332) | 215 | 216 | 217 | 218 | 219 | 220 | ||||
| P43m | F43m | I43m | P43n | F43c | I43d | |||||
| 2o:2 | 1o:2 | 4o:2 | 4o | 2oo | 4o/4 | |||||
| P33 | F33 | I33 | Pn3n3n | Fc3c3a | Id3d3d | |||||
| [(4,3,4,2+)] | [3[4]] | [[(4,3,4,2+)]] | [[(4,3,4,2+)]+] | [+(4,{3),4}+] | ||||||
| Hexoctahedralm3m(*432) | 221 | 222 | 223 | 224 | 225 | 226 | 227 | 228 | 229 | 230 |
| Pm3m | Pn3n | Pm3n | Pn3m | Fm3m | Fm3c | Fd3m | Fd3c | Im3m | Ia3d | |
| 4−:2 | 8oo | 8o | 4+:2 | 2−:2 | 4−− | 2+:2 | 4++ | 8o:2 | 8o/4 | |
| P43 | Pn4n3n | P4n3n | Pn43 | F43 | F4c3a | Fd4n3 | Fd4c3a | I43 | Ib4d3d | |
| [4,3,4] | [[4,3,4]+] | [(4+,2+)[3[4]]] | [4,31,1] | [4,(3,4)+] | [[3[4]]] | [[+(4,{3),4}+]] | [[4,3,4]] | |||
- 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