Fibrifold
Introduction In mathematics, a  '''fibrifold''' is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by , 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 groups8o:24−:24o:24+:22−:22o:22+:21o:28o4−4o4+2−2o2+1o8o/44−/44o/44+/42−/42o/42+/41o/48−o8oo8+o 4− −4−o4oo4+o4++2−o2oo2+o+ 36 cubic groupsClassPoint groupHexoctahedral*432 (mm)Hextetrahedral*332 (3m)Gyroidal432 (432)Diploidal3*2 (m)Tetartoidal332 (23)bc lattice (I)8o:2 (Imm)4o:2 (I3m)8+o (I432)8−o (I)4oo (I23)nc lattice (P)4−:2 (Pmm)2o:2 (P3m)4−o (P432)4− (Pm)2o (P23)4+:2 (Pnm)4+ (P4232)4+o (Pn)fc lattice (F)2−:2 (Fmm)1o:2 (F3m)2−o (F432)2− (Fm)1o (F23)2+:2 (Fdm)2+ (F4132)2+o (Fd)Otherlatticegroups8o (Pmn)8oo (Pnn)4− − (Fmc)4++ (Fdc)4o (P3n)2oo (F3c)Achiralquartergroups8o/4 (Iad)4o/4 (I3d)4+/4 (I4132)2+/4 (P4332,P4132)2−/4 (Pa)4−/4 (Ia)1o/4 (P213)2o/4 (I213)280px180px240px8 primary hexoctahedral hextetrahedral lattices of the cubic space groupsThe fibrifold cubic subgroup structure shown is based on extending symmetry of the tetragonal disphenoid fundamental domain of space group 216, similar to the squareIrreducible group symbols (indexed 195−230) in Hermann–Mauguin notation, Fibrifold notation, geometric notation, and Coxeter notation:Class(Orbifold point group) Space groupsTetartoidal23(332)195196197198199   P23  F23  I23  P213  I213  2o1o4oo1o/42o/4   P..  F.. I..  P..1  I..1   (4,3+,4,2+)  34+ (4,3+,4,2+)     Diploidal3m(3*2)200201202203204205206   Pm  Pn  Fm  Fd  I  Pa  Ia  4−4+o2−2+o8−o2−/44−/4   P4  Pn4  F4  Fd4  I4  Pb4  Ib4  4,3+,44,3+,4+ 4,(31,1)+ 34+ 4,3+,4   Gyroidal432(432)207208209210211212213214  P432  P4232  F432  F4132  I432  P4332P4132  I4132 4−o4+2−o2+8+o2+/44+/4  P..  P2..  F..  F1.. I..  P3..  P1..  I1..  4,3,4+ 4,3,4++ 4,31,1+ 34+  4,3,4+    Hextetrahedral3m(*332)215216217218219220  P3m  F3m  I3m  P3n  F3c  I3d  2o:21o:24o:24o2oo4o/4  P33  F33  I33  Pn3n3n  Fc3c3a Id3d3d  (4,3,4,2+)  34  (4,3,4,2+) (4,3,4,2+)+ +(4,{3),4}+   Hexoctahedralmm(*432)221222223224225226227228229230 Pmm  Pnn  Pmn  Pnm  Fmm  Fmc  Fdm  Fdc  Imm  Iad4−:28oo8o 4+:22−:24−− 2+:24++ 8o:28o/4 P43  Pn4n3n  P4n3n  Pn43  F43  F4c3a  Fd4n3  Fd4c3a  I43  Ib4d3d4,3,4 4,3,4+ (4+,2+)34 4,31,1 4,(3,4)+ 34 +(4,{3),4}+  4,3,4
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