fibrifold

In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by {{harvs | txt| last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Delgado Friedrichs | first2=Olaf | last3=Huson | first3=Daniel H. | last4=Thurston | first4=William P. | author4-link=William Thurston | title=On three-dimensional space groups | url=http://www.emis.de/journals/BAG/vol.42/no.2/17.html | mr=1865535 | year=2001 | journal=Beiträge zur Algebra und Geometrie | issn=0138-4821 | volume=42 | issue=2 | pages=475–507}}, 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

File:35 cubic fibrifold groups.svgs 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.

class=wikitable \|+ 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

class=wikitable \|+ 36 cubic groups
Class Point group !Hexoctahedral 432 (m{{overline\|3}}m) !Hextetrahedral 332 ({{overline\|4}}3m) !Gyroidal 432 (432) !Diploidal 3*2 (m{{overline\|3}}) !Tetartoidal 332 (23)
align=center !bc lattice (I) \|8^o:2 (Im{{overline\|3}}m) \|4^o:2 (I{{overline\|4}}3m) \|8^+o (I432) \|8^−o (I{{overline\|3}}) \|4^oo (I23)
align=center !rowspan=2\|nc lattice (P) \|4⁻:2 (Pm{{overline\|3}}m) \|rowspan=2\|2^o:2 (P{{overline\|4}}3m) \|4^−o (P432) \|4⁻ (Pm{{overline\|3}}) \|rowspan=2\|2^o (P23)
align=center \|4⁺:2 (Pn{{overline\|3}}m) \|4⁺ (P4₂32) \|4^+o (Pn{{overline\|3}})
align=center !rowspan=2\|fc lattice (F) \|2⁻:2 (Fm{{overline\|3}}m) \|rowspan=2\|1^o:2 (F{{overline\|4}}3m) \|2^−o (F432) \|2⁻ (Fm{{overline\|3}}) \|rowspan=2\|1^o (F23)
align=center \|2⁺:2 (Fd{{overline\|3}}m) \|2⁺ (F4₁32) \|2^+o (Fd{{overline\|3}})
align=center valign=top !Other lattice groups \|8^o (Pm{{overline\|3}}n) 8^oo (Pn{{overline\|3}}n) 4^{− −} (Fm{{overline\|3}}c) 4⁺⁺ (Fd{{overline\|3}}c) \|4^o (P{{overline\|4}}3n) 2^oo (F{{overline\|4}}3c) \| \| \|
align=center valign=top !Achiral quarter groups \|8^o/4 (Ia{{overline\|3}}d) \|4^o/4 (I{{overline\|4}}3d) \|4⁺/4 (I4₁32) 2⁺/4 (P4₃32, P4₁32) \|2⁻/4 (Pa{{overline\|3}}) 4⁻/4 (Ia{{overline\|3}}) \|1^o/4 (P2₁3) 2^o/4 (I2₁3)

class=wikitable

|+ 36 cubic groups

Class
Point group

!Hexoctahedral
*432 (m{{overline|3}}m)

!Hextetrahedral
*332 ({{overline|4}}3m)

!Gyroidal
432 (432)

!Diploidal
3*2 (m{{overline|3}})

!Tetartoidal
332 (23)

align=center

!bc lattice (I)

|8^o:2 (Im{{overline|3}}m)

|4^o:2 (I{{overline|4}}3m)

|8^+o (I432)

|8^−o (I{{overline|3}})

|4^oo (I23)

align=center

!rowspan=2|nc lattice (P)

|4⁻:2 (Pm{{overline|3}}m)

|rowspan=2|2^o:2 (P{{overline|4}}3m)

|4^−o (P432)

|4⁻ (Pm{{overline|3}})

|rowspan=2|2^o (P23)

align=center

|4⁺:2 (Pn{{overline|3}}m)

|4⁺ (P4₂32)

|4^+o (Pn{{overline|3}})

align=center

!rowspan=2|fc lattice (F)

|2⁻:2 (Fm{{overline|3}}m)

|rowspan=2|1^o:2 (F{{overline|4}}3m)

|2^−o (F432)

|2⁻ (Fm{{overline|3}})

|rowspan=2|1^o (F23)

align=center

|2⁺:2 (Fd{{overline|3}}m)

|2⁺ (F4₁32)

|2^+o (Fd{{overline|3}})

align=center valign=top

!Other
lattice
groups

|8^o (Pm{{overline|3}}n)
8^oo (Pn{{overline|3}}n)
4^{− −} (Fm{{overline|3}}c)
4⁺⁺ (Fd{{overline|3}}c)

|4^o (P{{overline|4}}3n)
2^oo (F{{overline|4}}3c)

align=center valign=top

!Achiral
quarter
groups

|8^o/4 (Ia{{overline|3}}d)

|4^o/4 (I{{overline|4}}3d)

|4⁺/4 (I4₁32)
2⁺/4 (P4₃32,
P4₁32)

|2⁻/4 (Pa{{overline|3}})
4⁻/4 (Ia{{overline|3}})

|1^o/4 (P2₁3)
2^o/4 (I2₁3)

class=wikitable width=580

|280px

|180px

|240px

valign=top

|8 primary hexoctahedral hextetrahedral lattices of the cubic space groups

|colspan=2|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) !colspan=10\| Space groups
class=wikitable
align=center !rowspan=5\|Tetartoidal 23 (332) !195	196	197	198	199	colspan=5\|
BGCOLOR="#ffe0e0" align=center \| P23	F23	I23	P2₁3	I2₁3	colspan=5\|
BGCOLOR="#e0e0ff" align=center \|2^o	1^o	4^oo	1^o/4	2^o/4	colspan=5\|
BGCOLOR="#ffffd0" align=center \| P{{overline\|3}}.{{overline\|3}}.{{overline\|2}}	F{{overline\|3}}.{{overline\|3}}.{{overline\|2}}	I{{overline\|3}}.{{overline\|3}}.{{overline\|2}}	P{{overline\|3}}.{{overline\|3}}.{{overline\|2}}₁	I{{overline\|3}}.{{overline\|3}}.{{overline\|2}}₁	colspan=5\|
BGCOLOR="#e0ffe0" align=center \| [(4,3⁺,4,2⁺)]	[3^[4]]⁺	(4,3⁺,4,2⁺)			colspan=5\|
align=center !rowspan=5\|Diploidal {{overline\|4}}3m (3*2) !200	201	202	203	204	205	206	colspan=3\|
BGCOLOR="#ffe0e0" align=center \| Pm{{overline\|3}}	Pn{{overline\|3}}	Fm{{overline\|3}}	Fd{{overline\|3}}	I{{overline\|3}}	Pa{{overline\|3}}	Ia{{overline\|3}}	colspan=3\|
BGCOLOR="#e0e0ff" align=center \|4⁻	4^+o	2⁻	2^+o	8^−o	2⁻/4	4⁻/4	colspan=3\|
BGCOLOR="#ffffd0" align=center \| P4{{overline\|3}}	P_n4{{overline\|3}}	F4{{overline\|3}}	F_d4{{overline\|3}}	I4{{overline\|3}}	P_b4{{overline\|3}}	I_b4{{overline\|3}}	colspan=3\|
BGCOLOR="#e0ffe0" align=center \|[4,3⁺,4]	4,3⁺,4]⁺]	[4,(3^1,1)⁺]	[[3^[4]⁺	4,3⁺,4			colspan=3\|
align=center !rowspan=5\|Gyroidal 432 (432) !207	208	209	210	211	212	213	214	colspan=2\|
BGCOLOR="#ffe0e0" align=center \| P432	P4₂32	F432	F4₁32	I432	P4₃32	P4₁32	I4₁32	colspan=2\|
BGCOLOR="#e0e0ff" align=center	4^−o	4⁺	2^−o	2⁺	8^+o	colspan=2\|2⁺/4	4⁺/4	colspan=2\|
BGCOLOR="#ffffd0" align=center \| P{{overline\|4}}.{{overline\|3}}.{{overline\|2}}	P{{overline\|4}}₂.{{overline\|3}}.{{overline\|2}}	F{{overline\|4}}.{{overline\|3}}.{{overline\|2}}	F{{overline\|4}}₁.{{overline\|3}}.{{overline\|2}}	I{{overline\|4}}.{{overline\|3}}.{{overline\|2}}	P{{overline\|4}}₃.{{overline\|3}}.{{overline\|2}}	P{{overline\|4}}₁.{{overline\|3}}.{{overline\|2}}	I{{overline\|4}}₁.{{overline\|3}}.{{overline\|2}}	colspan=2\|
BGCOLOR="#e0ffe0" align=center \|[4,3,4]⁺	4,3,4]⁺]⁺	[4,3^1,1]⁺	[[3^[4]⁺	4,3,4⁺	colspan=2\|		colspan=2\|
align=center !rowspan=5\|Hextetrahedral {{overline\|4}}3m (*332) !215	216	217	218	219	220	colspan=4\|
BGCOLOR="#ffe0e0" align=center \| P{{overline\|4}}3m	F{{overline\|4}}3m	I{{overline\|4}}3m	P{{overline\|4}}3n	F{{overline\|4}}3c	I{{overline\|4}}3d	colspan=4\|
BGCOLOR="#e0e0ff" align=center	2^o:2	1^o:2	4^o:2	4^o	2^oo	4^o/4	colspan=4\|
BGCOLOR="#ffffd0" align=center	P33	F33	I33	P_n3_n3_n	F_c3_c3_a	I_d3_d3_d	colspan=4\|
BGCOLOR="#e0ffe0" align=center \|[(4,3,4,2⁺)]	[3^[4]]	(4,3,4,2⁺)	[[(4,3,4,2⁺)]⁺]	[⁺(4,{3),4}⁺]		colspan=4\|
align=center !rowspan=5\|Hexoctahedral m{{overline\|3}}m (*432) !221	222	223	224	225	226	227	228	229	230
BGCOLOR="#ffe0e0" align=center \| Pm{{overline\|3}}m	Pn{{overline\|3}}n	Pm{{overline\|3}}n	Pn{{overline\|3}}m	Fm{{overline\|3}}m	Fm{{overline\|3}}c	Fd{{overline\|3}}m	Fd{{overline\|3}}c	Im{{overline\|3}}m	Ia{{overline\|3}}d
BGCOLOR="#e0e0ff" align=center	4⁻:2	8^oo	8^o	4⁺:2	2⁻:2	4⁻⁻	2⁺:2	4⁺⁺	8^o:2	8^o/4
BGCOLOR="#ffffd0" align=center	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
BGCOLOR="#e0ffe0" align=center	[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

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{{citation |first1=David |last1=Hestenes |first2=Jeremy W. |last2=Holt |title=The Crystallographic Space Groups in Geometric Algebra |journal=Journal of Mathematical Physics |volume=48 |issue=2 |pages=023514 |date=February 2007 |doi=10.1063/1.2426416 |bibcode=2007JMP....48b3514H |url=https://davidhestenes.net/geocalc/pdf/CrystalGA.pdf }}
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{{citation |first=H.S.M. |last=Coxeter |chapter=Regular and Semi Regular Polytopes III |chapter-url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA313 |editor-first=F. Arthur |editor-last=Sherk |editor-first2=Peter |editor2-last=McMullen |editor3-first=Anthony C. |editor3-last=Thompson |editor4-first=Asia Ivić |display-editors=3 |editor4-last=Weiss |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |publisher=Wiley |year=1995 |isbn=978-0-471-01003-6 |pages=[https://archive.org/details/kaleidoscopessel0000coxe/page/313 313–358] |zbl=0976.01023 |url=https://archive.org/details/kaleidoscopessel0000coxe/page/313 }}

Category:Symmetry

Category:Finite groups

Category:Discrete groups