Fibrifold

{{inline |date=May 2024}} 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.svg|320px|thumb|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. {| 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<BR>Point group !Hexoctahedral<BR>*432 (m{{overline|3}}m) !Hextetrahedral<BR>*332 ({{overline|4}}3m) !Gyroidal<BR>432 (432) !Diploidal<BR>3*2 (m{{overline|3}}) !Tetartoidal<BR>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<BR>lattice<BR>groups |8^o (Pm{{overline|3}}n)<BR>8^oo (Pn{{overline|3}}n)<BR>4^{− −} (Fm{{overline|3}}c)<BR>4⁺⁺ (Fd{{overline|3}}c) |4^o (P{{overline|4}}3n)<BR>2^oo (F{{overline|4}}3c) | | | |- align=center valign=top !Achiral<BR>quarter<BR>groups |8^o/4 (Ia{{overline|3}}d) |4^o/4 (I{{overline|4}}3d) |4⁺/4 (I4₁32)<BR>2⁺/4 (P4₃32,<BR>P4₁32) |2⁻/4 (Pa{{overline|3}})<BR>4⁻/4 (Ia{{overline|3}}) |1^o/4 (P2₁3)<BR>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=wikitable |- !Class<br>(Orbifold point group) !colspan=10| Space groups |-align=center !rowspan=5|Tetartoidal<BR>23<br>(332) !195||196||197||198||199 || colspan=5|&nbsp; |- BGCOLOR="#ffe0e0" align=center | P23 || F23 || I23 || P2₁3 || I2₁3 || colspan=5|&nbsp; |- BGCOLOR="#e0e0ff" align=center |2^o||1^o||4^oo||1^o/4||2^o/4 || colspan=5|&nbsp; |- 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|&nbsp; |- BGCOLOR="#e0ffe0" align=center | [(4,3⁺,4,2⁺)] || [3^[4]]⁺|| (4,3⁺,4,2⁺) || || || colspan=5|&nbsp; |- align=center !rowspan=5|Diploidal<BR>{{overline|4}}3m<br>(3*2) !200||201||202||203||204||205||206 ||colspan=3|&nbsp; |- 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|&nbsp; |- BGCOLOR="#e0e0ff" align=center |4⁻||4^+o||2⁻||2^+o||8^−o||2⁻/4||4⁻/4 ||colspan=3|&nbsp; |- 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|&nbsp; |- BGCOLOR="#e0ffe0" align=center |[4,3⁺,4]||<nowiki>[[</nowiki>4,3⁺,4]⁺] ||[4,(3^1,1)⁺] ||<nowiki>[[</nowiki>3^[4]<nowiki>]]</nowiki>⁺ ||<nowiki></nowiki>4,3⁺,4<nowiki></nowiki>|| || ||colspan=3|&nbsp;

|- align=center !rowspan=5|Gyroidal<BR>432<br>(432) !207||208||209||210||211||212||213||214||colspan=2|&nbsp;

|- BGCOLOR="#ffe0e0" align=center | P432 || P4₂32 || F432 || F4₁32 || I432 || P4₃32||P4₁32 || I4₁32||colspan=2|&nbsp; |- BGCOLOR="#e0e0ff" align=center ||4^−o||4⁺||2^−o||2⁺||8^+o||colspan=2|2⁺/4||4⁺/4||colspan=2|&nbsp;

|- 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|&nbsp; |- BGCOLOR="#e0ffe0" align=center |[4,3,4]⁺ ||<nowiki>[[</nowiki>4,3,4]⁺]⁺ ||[4,3^1,1]⁺ ||<nowiki>[[</nowiki>3^[4]<nowiki>]]</nowiki>⁺ ||<nowiki></nowiki>4,3,4<nowiki></nowiki>⁺ || colspan=2| || ||colspan=2|&nbsp; |- align=center !rowspan=5|Hextetrahedral<BR>{{overline|4}}3m<br>(*332) !215||216||217||218||219||220|| colspan=4|&nbsp; |- 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|&nbsp; |- BGCOLOR="#e0e0ff" align=center ||2^o:2||1^o:2||4^o:2||4^o||2^oo||4^o/4|| colspan=4|&nbsp;

|- BGCOLOR="#ffffd0" align=center || P33 || F33 || I33 || P_n3_n3_n || F_c3_c3_a|| I_d3_d3_d || colspan=4|&nbsp; |- BGCOLOR="#e0ffe0" align=center |[(4,3,4,2⁺)] || [3^[4]] || <nowiki></nowiki>(4,3,4,2⁺)<nowiki></nowiki> ||<nowiki>[[</nowiki>(4,3,4,2⁺)]⁺] ||[⁺(4,{3),4}⁺] || || colspan=4|&nbsp;

|- align=center !rowspan=5|Hexoctahedral<BR>m{{overline|3}}m<br>(*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]|| ||<nowiki>[[</nowiki>4,3,4]⁺] ||[(4⁺,2⁺)[3^[4]<nowiki>]]</nowiki> ||[4,3^1,1] ||[4,(3,4)⁺] ||<nowiki>[[</nowiki>3^[4]<nowiki>]]</nowiki> ||<nowiki></nowiki>⁺(4,{3),4}⁺<nowiki></nowiki> || <nowiki></nowiki>4,3,4<nowiki></nowiki> || |}

==References== {{refbegin}} *{{Citation | 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}} *{{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 }} *{{citation |first1=John H. |last1=Conway |first2=Heidi |last2=Burgiel |first3=Chaim |last3=Goodman-Strauss |title=The Symmetries of Things |publisher=Taylor & Francis |year=2008 |isbn=978-1-56881-220-5 |zbl=1173.00001}} *{{citation |first=H.S.M. |last=Coxeter |author-link=H. S. M. 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 }} {{refend}}

Category:Symmetry Category:Finite groups Category:Discrete groups

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