Fibrifold

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.

Irreducible cubic space groups

The 35/36 irreducible cubic space groups in fibrifold and international index and Hermann–Mauguin notation. 212 and 213 are enantiomorphous pairs giving the same fibrifold notation.

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
36 cubic groups
Class
Point group 		Hexoctahedral
*432 (m3m) 		Hextetrahedral
*332 (43m) 		Gyroidal
432 (432) 		Diploidal
3*2 (m3) 		Tetartoidal
332 (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)
Other
lattice
groups 		8o (Pm3n)
8oo (Pn3n)
4− − (Fm3c)
4++ (Fd3c) 		4o (P43n)
2oo (F43c)
Achiral
quarter
groups 		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
Tetartoidal
23
(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+)]]  
Diploidal
43m
(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]]  
Gyroidal
432
(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]]+  
Hextetrahedral
43m
(*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}+]  
Hexoctahedral
m3m
(*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]]

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


Stub icon

This geometry-related article is a stub. You can help Wikipedia by adding missing information.

Content Disclaimer

Informasi ini disarikan dari Wikipedia dan disajikan kembali untuk tujuan edukasi. Konten tersedia di bawah lisensi CC BY-SA 3.0. Kami tidak bertanggung jawab atas ketidakakuratan data yang bersumber dari kontribusi publik tersebut.

  1. The information displayed on this website is sourced in part or in whole from Wikipedia and has been adapted for the purpose of restating it. We strive to provide accurate and relevant information, however:
  2. There is no guarantee of absolute accuracy. Wikipedia is an open, collaborative project that can be edited by anyone, so information is subject to change.
  3. It is not intended to constitute professional advice. The content displayed is for informational and educational purposes only. For important decisions (e.g., medical, legal, or financial), please consult a professional.
  4. Content copyright. Wikipedia is licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA). This means that content may be reused with appropriate attribution and shared under a similar license.
  5. Responsible use. Any risk arising from the use of information from this website is entirely the responsibility of the user.