# 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 irreducible space groups correspond to the cubic space group.
35 irreducible space groups :
| 8$^o$:2 | 4$^−$:2 | 4$^o$:2 | 4$^+$:2 | 2$^−$:2 | 2$^o$:2 | 2$^+$:2 | 1$^o$:2 |
| 8$^o$ | 4$^−$ | 4$^o$ | 4$^+$ | 2$^−$ | 2$^o$ | 2$^+$ | 1$^o$ |
| 8$^o$/4 | 4$^−$/4 | 4$^o$/4 | 4$^+$/4 | 2$^−$/4 | 2$^o$/4 | 2$^+$/4 | 1$^o$/4 |
| 8$^{−o}$ | 8$^{oo}$ | 8$^{+o}$ | 4$^{−−}$ | 4$^{−o}$ | 4$^{oo}$ | 4$^{+o}$ | 4$^{++}$ | 2$^{−o}$ | 2$^{oo}$ | 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) | 8$^o$:2 (Im3m) | 4$^o$:2 (I43m) | 8$^{+o}$ (I432) | 8$^{−o}$ (I3) | 4$^{oo}$ (I23) |
| nc lattice (P) | 4$^−$:2 (Pm3m) | 2$^o$:2 (P43m) | 4$^{−o}$ (P432) | 4$^−$ (Pm3) | 2$^o$ (P23) |
| nc lattice (P) | 4$^+$:2 (Pn3m) | 2$^o$:2 (P43m) | 4$^+$ (P4$_2$32) | 4$^{+o}$ (Pn3) | 2$^o$ (P23) |
| fc lattice (F) | 2$^−$:2 (Fm3m) | 1$^o$:2 (F43m) | 2$^{−o}$ (F432) | 2$^−$ (Fm3) | 1$^o$ (F23) |
| fc lattice (F) | 2$^+$:2 (Fd3m) | 1$^o$:2 (F43m) | 2$^+$ (F4$_1$32) | 2$^{+o}$ (Fd3) | 1$^o$ (F23) |
| Other lattice groups | 8$^o$ (Pm3n) 8$^{oo}$ (Pn3n) 4$^{−−}$ (Fm3c) 4$^{++}$ (Fd3c) | 4$^o$ (P43n) 2$^{oo}$ (F43c) | | | |
| Achiral quarter groups | 8$^o$/4 (Ia3d) | 4$^o$/4 (I43d) | 4$^+$/4 (I4$_1$32) 2$^+$/4 (P4$_3$32, P4$_1$32) | 2$^−$/4 (Pa3) 4$^−$/4 (Ia3) | 1$^o$/4 (P2$_1$3) 2$^o$/4 (I2$_1$3) |
| 8 primary hexoctahedral hextetrahedral lattices of the cubic space groups | The fibrifold cubic subgroup structure shown is based on extending symmetry of the tetragonal disphenoid fundamental domain of space group 216, similar to the square | The fibrifold cubic subgroup structure shown is based on extending symmetry of the tetragonal disphenoid fundamental domain of space group 216, similar to the square |
Irreducible group symbols (indexed 195−230) in Hermann–Mauguin notation, Fibrifold notation, geometric notation, and Coxeter notation:
| Class (Orbifold point group) | Space groups | Space groups | Space groups | Space groups | Space groups | Space groups | Space groups | Space groups | Space groups | Space groups |
| Tetartoidal 23 (332) | 195 | 196 | 197 | 198 | 199 | | | | | |
| Tetartoidal 23 (332) | P23 | F23 | I23 | P2$_1$3 | I2$_1$3 | | | | | |
| Tetartoidal 23 (332) | 2$^o$ | 1$^o$ | 4$^{oo}$ | 1$^o$/4 | 2$^o$/4 | | | | | |
| Tetartoidal 23 (332) | P3.3.2 | F3.3.2 | I3.3.2 | P3.3.2$_1$ | I3.3.2$_1$ | | | | | |
| Tetartoidal 23 (332) | [(4,3$^+$,4,2$^+$)] | [3$^{[4]}$]$^+$ | [[(4,3$^+$,4,2$^+$)]] | | | | | | | |
| Diploidal 43m (3*2) | 200 | 201 | 202 | 203 | 204 | 205 | 206 | | | |
| Diploidal 43m (3*2) | Pm3 | Pn3 | Fm3 | Fd3 | I3 | Pa3 | Ia3 | | | |
| Diploidal 43m (3*2) | 4$^−$ | 4$^{+o}$ | 2$^−$ | 2$^{+o}$ | 8$^{−o}$ | 2$^−$/4 | 4$^−$/4 | | | |
| Diploidal 43m (3*2) | P43 | P$_n$43 | F43 | F$_d$43 | I43 | P$_b$43 | I$_b$43 | | | |
| Diploidal 43m (3*2) | [4,3$^+$,4] | [[4,3$^+$,4]$^+$] | [4,(3$^{1,1}$)$^+$] | [[3$^{[4]}$]]$^+$ | [[4,3$^+$,4]] | | | | | |
| Gyroidal 432 (432) | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | | |
| Gyroidal 432 (432) | P432 | P4$_2$32 | F432 | F4$_1$32 | I432 | P4$_3$32 | P4$_1$32 | I4$_1$32 | | |
| Gyroidal 432 (432) | 4$^{−o}$ | 4$^+$ | 2$^{−o}$ | 2$^+$ | 8$^{+o}$ | 2$^+$/4 | 2$^+$/4 | 4$^+$/4 | | |
| Gyroidal 432 (432) | P4.3.2 | P4$_2$.3.2 | F4.3.2 | F4$_1$.3.2 | I4.3.2 | P4$_3$.3.2 | P4$_1$.3.2 | I4$_1$.3.2 | | |
| Gyroidal 432 (432) | [4,3,4]$^+$ | [[4,3,4]$^+$]$^+$ | [4,3$^{1,1}$]$^+$ | [[3$^{[4]}$]]$^+$ | [[4,3,4]]$^+$ | | | | | |
| Hextetrahedral 43m (*332) | 215 | 216 | 217 | 218 | 219 | 220 | | | | |
| Hextetrahedral 43m (*332) | P43m | F43m | I43m | P43n | F43c | I43d | | | | |
| Hextetrahedral 43m (*332) | 2$^o$:2 | 1$^o$:2 | 4$^o$:2 | 4$^o$ | 2$^{oo}$ | 4$^o$/4 | | | | |
| Hextetrahedral 43m (*332) | P33 | F33 | I33 | P$_n$3$_n$3$_n$ | F$_c$3$_c$3$_a$ | I$_d$3$_d$3$_d$ | | | | |
| Hextetrahedral 43m (*332) | [(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 |
| Hexoctahedral m3m (*432) | Pm3m | Pn3n | Pm3n | Pn3m | Fm3m | Fm3c | Fd3m | Fd3c | Im3m | Ia3d |
| Hexoctahedral m3m (*432) | 4$^−$:2 | 8$^{oo}$ | 8$^o$ | 4$^+$:2 | 2$^−$:2 | 4$^{−−}$ | 2$^+$:2 | 4$^{++}$ | 8$^o$:2 | 8$^o$/4 |
| Hexoctahedral m3m (*432) | P43 | P$_n$4$_n$3$_n$ | P4$_n$3$_n$ | P$_n$43 | F43 | F4$_c$3$_a$ | F$_d$4$_n$3 | F$_d$4$_c$3$_a$ | I43 | I$_b$4$_d$3$_d$ |
| Hexoctahedral m3m (*432) | [4,3,4] | | [[4,3,4]$^+$] | [(4$^+$,2$^+$)[3$^{[4]}$]] | [4,3$^{1,1}$] | [4,(3,4)$^+$] | [[3$^{[4]}$]] | [[$^+$(4,{3),4}$^+$]] | [[4,3,4]] | |