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 in red. 212 and 213 are enantiomorphous pairs giving the same fibrifold notation. Subgroup indices 1,2,4,8,16 are partitioned from top to bottom, with /4 groups (in blue) with their indices times 4.

The 35 irreducible space groups correspond to the cubic space group.

35 irreducible space groups
8o:24−:24o:24+:22−:22o:22+:21o:2
8o4−4o4+2−2o2+1o
8o/44−/44o/44+/42−/42o/42+/41o/4
8−o8oo8+o4− −4−o4oo4+o4++2−o2oo2+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)		 195196197198199 
P23F23I23P213I213 
2o1o4oo1o/42o/4 
P3.3.2F3.3.2I3.3.2P3.3.21I3.3.21 
[(4,3+,4,2+)][3[4]]+[[(4,3+,4,2+)]] 
Diploidal
43m
(3*2)		 200201202203204205206 
Pm3Pn3Fm3Fd3I3Pa3Ia3 
4−4+o2−2+o8−o2−/44−/4 
P43Pn43F43Fd43I43Pb43Ib43 
[4,3+,4][[4,3+,4]+][4,(31,1)+][[3[4]]]+[[4,3+,4]] 
Gyroidal
432
(432)		 207208209210211212213214 
P432P4232F432F4132I432P4332P4132I4132 
4−o4+2−o2+8+o2+/44+/4 
P4.3.2P42.3.2F4.3.2F41.3.2I4.3.2P43.3.2P41.3.2I41.3.2 
[4,3,4]+[[4,3,4]+]+[4,31,1]+[[3[4]]]+[[4,3,4]]+ 
Hextetrahedral
43m
(*332)		 215216217218219220 
P43mF43mI43mP43nF43cI43d 
2o:21o:24o:24o2oo4o/4 
P33F33I33Pn3n3nFc3c3aId3d3d 
[(4,3,4,2+)][3[4]][[(4,3,4,2+)]][[(4,3,4,2+)]+][+(4,{3),4}+] 
Hexoctahedral
m3m
(*432)		 221222223224225226227228229230
Pm3mPn3nPm3nPn3mFm3mFm3cFd3mFd3cIm3mIa3d
4−:28oo8o4+:22−:24−−2+:24++8o:28o/4
P43Pn4n3nP4n3nPn43F43F4c3aFd4n3Fd4c3aI43Ib4d3d
[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, doi:10.1063/1.2426416
  • Huson, Daniel H. (2008), The Fibrifold Notation and Classification for 3D Space Groups (PDF)
  • Conway, John H.; Burgiel, Heidi; Goodman-Strass, 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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