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

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)
	4^+:2 (Pn3m)		4^+ (P4232)	4^+o (Pn3)
fc lattice (F)	2^−:2 (Fm3m)	1^o:2 (F43m)	2^−o (F432)	2^− (Fm3)	1^o (F23)
	2^+:2 (Fd3m)		2^+ (F4132)	2^+o (Fd3)
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 (I4132) 2^+/4 (P4332, P4132)	2^−/4 (Pa3) 4^−/4 (Ia3)	1^o/4 (P213) 2^o/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
	2^o	1^o	4^oo	1^o/4	2^o/4
	P3.3.2	F3.3.2	I3.3.2	P3.3.21	I3.3.21
	[(4,3^+,4,2^+)]	[3^[4]]^+	[[(4,3<sup>+</sup>,4,2<sup>+</sup>)]]
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,(3^1,1)^+]	[[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]^+]^+	[4,3^1,1]^+	[[3^[4]]]^+	[[4,3,4]]^+
Hextetrahedral 43m (*332)	215	216	217	218	219	220
	P43m	F43m	I43m	P43n	F43c	I43d
	2^o:2	1^o:2	4^o:2	4^o	2^oo	4^o/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	8^oo	8^o	4^+:2	2^−:2	4^−−	2^+:2	4^++	8^o:2	8^o/4
	P43	Pn4n3n	P4n3n	Pn43	F43	F4c3a	Fd4n3	Fd4c3a	I43	Ib4d3d
	[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]]

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