Fibrifold Explained

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 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 (mm)	Hextetrahedral *332 (3m)	Gyroidal 432 (432)	Diploidal 3*2 (m)	Tetartoidal 332 (23)
bc lattice (I)	8^o:2 (Imm)	4^o:2 (I3m)	8^+o (I432)	8^−o (I)	4^oo (I23)
nc lattice (P)	4⁻:2 (Pmm)	2^o:2 (P3m)	4^−o (P432)	4⁻ (Pm)	2^o (P23)
nc lattice (P)	4⁺:2 (Pnm)	2^o:2 (P3m)	4⁺ (P4₂32)	4^+o (Pn)	2^o (P23)
fc lattice (F)	2⁻:2 (Fmm)	1^o:2 (F3m)	2^−o (F432)	2⁻ (Fm)	1^o (F23)
fc lattice (F)	2⁺:2 (Fdm)	1^o:2 (F3m)	2⁺ (F4₁32)	2^+o (Fd)	1^o (F23)
Other lattice groups	8^o (Pmn) 8^oo (Pnn) 4^{− −} (Fmc) 4⁺⁺ (Fdc)	4^o (P3n) 2^oo (F3c)
Achiral quarter groups	8^o/4 (Iad)	4^o/4 (I3d)	4⁺/4 (I4₁32) 2⁺/4 (P4₃32, P4₁32)	2⁻/4 (Pa) 4⁻/4 (Ia)	1^o/4 (P2₁3) 2^o/4 (I2₁3)

Irreducible group symbols (indexed 195−230) in Hermann–Mauguin notation, Fibrifold notation, geometric notation, and Coxeter notation:

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Fibrifold".

Class (Orbifold point group)	Space groups
Tetartoidal 23 (332)	195	196	197	198	199
	P23	F23	I23	P2₁3	I2₁3
	2^o	1^o	4^oo	1^o/4	2^o/4
	P..	F..	I..	P..₁	I..₁
	[(4,3<sup>+</sup>,4,2<sup>+</sup>)]	[3<sup>[4]]⁺
Diploidal 3m (3*2)	200	201	202	203	204	205	206
	Pm	Pn	Fm	Fd	I	Pa	Ia
	4⁻	4^+o	2⁻	2^+o	8^−o	2⁻/4	4⁻/4
	P4	P_n4	F4	F_d4	I4	P_b4	I_b4
	[4,3<sup>+</sup>,4]	[[4,3⁺,4]⁺]	[4,(3<sup>1,1</sup>)<sup>+</sup>]	[[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
	P..	P₂..	F..	F₁..	I..	P₃..	P₁..	I₁..
	[4,3,4]⁺	[[4,3,4]⁺]⁺	[4,3<sup>1,1</sup>]⁺	[[3^[4]]]⁺	[[4,3,4]]⁺
Hextetrahedral 3m (*332)	215	216	217	218	219	220
	P3m	F3m	I3m	P3n	F3c	I3d
	2^o:2	1^o:2	4^o:2	4^o	2^oo	4^o/4
	P33	F33	I33	P_n3_n3_n	F_c3_c3_a	I_d3_d3_d
	[(4,3,4,2<sup>+</sup>)]	[3<sup>[4]]	[[(4,3,4,2⁺)]]	[[(4,3,4,2⁺)]⁺]	[<sup>+</sup>(4,{3),4}<sup>+</sup>]
Hexoctahedral mm (*432)	221	222	223	224	225	226	227	228	229	230
	Pmm	Pnn	Pmn	Pnm	Fmm	Fmc	Fdm	Fdc	Imm	Iad
	4⁻:2	8^oo	8^o	4⁺:2	2⁻:2	4⁻⁻	2⁺:2	4⁺⁺	8^o:2	8^o/4
	P43	P_n4_n3_n	P4_n3_n	P_n43	F43	F4_c3_a	F_d4_n3	F_d4_c3_a	I43	I_b4_d3_d
	[4,3,4]		[[4,3,4]⁺]	[(4<sup>+</sup>,2<sup>+</sup>)[3<sup>[4]]]	[4,3<sup>1,1</sup>]	[4,(3,4)<sup>+</sup>]	[[3^[4]]]	[[⁺(4,⁺]]	[[4,3,4]]