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2012-02-29T08:56:38Z

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[[File:Icosahedral reflection domains.png|thumb|Icosahedral symmetry fundamental domains]]
[[File:Soccer ball.svg|thumb|right|A [[Soccer ball]], a common example of a [[Spherical polyhedron|spherical]] [[truncated icosahedron]], has full icosahedral symmetry.]]
A regular [[icosahedron]] has 60 rotational (or orientation-preserving) symmetries, and a [[symmetry order]] of 120 including transformations that combine a reflection and a rotation. A [[regular dodecahedron]] has the same set of symmetries, since it is the dual of the icosahedron.

The set of orientation-preserving symmetries forms a group referred to as ''A''5 (the [[alternating group]] on 5 letters), and the full symmetry group (including reflections) is the product ''A''5 × ''Z''2. The latter group is also known as the [[Coxeter group]] ''H''3, and is also represented by [[Coxeter notation]], [5,3] and [[Coxeter diagram]] {{CDD|node|5|node|3|node}}.

== As point group ==

[[File:Sphere symmetry group i.png|thumb|The icosahedral rotation group ''I'' with [[fundamental domain]]]]

Apart from the two infinite series of prismatic and antiprismatic symmetry, '''rotational icosahedral symmetry''' or '''chiral icosahedral symmetry''' of chiral objects and '''full icosahedral symmetry''' or '''achiral icosahedral symmetry''' are the [[Point groups in three dimensions|discrete point symmetries]] (or equivalently, [[List of spherical symmetry groups|symmetries on the sphere]]) with the largest [[symmetry group]]s.

Icosahedral symmetry is ''not'' compatible with [[translational symmetry]], so there are no associated [[Crystal system#Overview of point groups by crystal system|crystallographic point groups]] or [[space group]]s.

{| class=wikitable
![[Arthur Moritz Schönflies|Schönflies]] crystallographic notation
![[Coxeter notation|Coxeter notation]]
![[Orbifold notation|Orbifold notation]]
![[Symmetry order|Order]]
|- align=center
|''I''
|[3,5]+
|532
|60
|- align=center
|''Ih''
|[3,5]
|*532
|120
|}

[[Presentation of a group|Presentations]] corresponding to the above are:

:<math>I: \langle s,t \mid s^2, t^3, (st)^5 \rangle\ </math>
:<math>I_h: \langle s,t\mid s^3(st)^{-2}, t^5(st)^{-2}\rangle.\ </math>
These correspond to the icosahedral groups (rotational and full) being the (2,3,5) [[triangle group]]s.

The first presentation was given by [[William Rowan Hamilton]] in 1856, in his paper on [[icosian calculus]].<ref>{{citation
|title=Memorandum respecting a new System of Roots of Unity
|author=Sir William Rowan Hamilton
|author-link=William Rowan Hamilton
|url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf
|journal=[[Philosophical Magazine]]
|volume=12
|year=1856
|pages=446
}}</ref>

Note that other presentations are possible, for instance as an [[alternating group]] (for ''I'').

== Group structure ==
The '''{{visible anchor|icosahedral rotation group}}''' '''''I''''' is of order 60. The group ''I'' is [[isomorphic]] to ''A''5, the [[alternating group]] of even permutations of five objects. This isomorphism can be realized by ''I'' acting on various compounds, notably the [[compound of five cubes]] (which inscribe in the [[dodecahedron]]), the [[compound of five octahedra]], or either of the two [[compound of five tetrahedra|compounds of five tetrahedra]] (which are [[enantiomorphs]], and inscribe in the dodecahedron).

The group contains 5 versions of ''T''h with 20 versions of ''D3'' (10 axes, 2 per axis), and 6 versions of ''D5''.

The '''{{visible anchor|full icosahedral group}}''' '''''Ih''''' has order 120. It has ''I'' as [[normal subgroup]] of [[index of a subgroup|index]] 2. The group ''Ih'' is isomorphic to ''I'' × ''Z''2, or ''A''5 × ''Z''2, with the [[Inversion in a point|inversion in the center]] corresponding to element (identity,-1), where ''Z''2 is written multiplicatively.

''Ih'' acts on the [[compound of five cubes]] and the [[compound of five octahedra]], but -1 acts as the identity (as cubes and octahedra are centrally symmetric). It acts on the [[compound of ten tetrahedra]]: ''I'' acts on the two chiral halves ([[compound of five tetrahedra|compounds of five tetrahedra]]), and -1 interchanges the two halves.
Notably, it does ''not'' act as S5, and these groups are not isomorphic; see below for details.

The group contains 10 versions of ''D3d'' and 6 versions of ''D5d'' (symmetries like antiprisms).

''I'' is also isomorphic to PSL2(5), but ''Ih'' is not isomorphic to SL2(5).

=== Commonly confused groups ===
The following groups all have order 120, but are not isomorphic:
* ''S''5, the [[symmetric group]] on 5 elements
* ''Ih'', the full icosahedral group (subject of this article, also known as ''H''3)
* 2''I'', the [[binary icosahedral group]]
They correspond to the following [[short exact sequence]]s (which do not split) and product
:<math>1\to A_5 \to S_5 \to Z_2 \to 1</math>
:<math>I_h = A_5 \times Z_2</math>
:<math>1\to Z_2 \to 2I\to A_5 \to 1</math>
In words,
* <math>A_5</math> is a ''[[normal subgroup]]'' of <math>S_5</math>
* <math>A_5</math> is a ''factor'' of <math>I_h</math>, which is a ''[[direct product of groups|direct product]]''
* <math>A_5</math> is a ''[[quotient group]]'' of <math>2I</math>
Note that <math>A_5</math> has an [[exceptional object|exceptional]] irreducible 3-dimensional [[linear representation|representation]] (as the icosahedral rotation group), but <math>S_5</math> does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group.

These can also be related to linear groups over the [[finite field]] with five elements, which exhibit the subgroups and covering groups directly; none of these are the full icosahedral group:
* <math>A_5 \cong \operatorname{PSL}(2,5),</math> the [[projective special linear group]], see [[Projective linear group#Action_on_p_points|here]] for a proof;
* <math>S_5 \cong \operatorname{PGL}(2,5),</math> the [[projective general linear group]];
* <math>2I \cong \operatorname{SL}(2,5),</math> the [[special linear group]].

=== Conjugacy classes ===
The [[conjugacy class]]es of ''I'' are:
*identity
*12 × rotation by 72°, order 5
*12 × rotation by 144°, order 5
*20 × rotation by 120°, order 3
*15 × rotation by 180°, order 2

Those of ''Ih'' include also each with inversion:
*inversion
*12 × rotoreflection by 108°, order 10
*12 × rotoreflection by 36°, order 10
*20 × rotoreflection by 60°, order 6
*15 × reflection, order 2

=== Subgroups of chiral icosahedral symmetry ===
{| class=wikitable
![[Schoenflies notation|Schoenflies notation]]
![[Coxeter notation|Coxeter notation]]
![[Orbifold notation|Orbifold notation]]
![[Hermann–Mauguin notation|Hermann Mauguin notation]]
![[Symmetry number|Order]]
![[Index of a subgroup|Index]]
!Subgroup relations
|-align=center
|I||[5,3]+||532||532||60||1
|rowspan=9|[[File:Chiral_icosahedral_subgroup_tree.png|240px]]
|- align=center
|T||[3,3]+||332||332||12||5
|- align=center
|D5||[5,2]+||522||522||10||6
|- align=center
|D3||[3,2]+||322||322||6||10
|- align=center
|C5||[5]+||55||5||5||12
|- align=center
|D2||[2,2]+||222||222||4||15
|- align=center
|C3||[3]+||33||3||3||20
|- align=center
|C2||[2]+||22||2||2||30
|- align=center
|C1||[ ]+||11||1||1||60

|}

=== Subgroups of full icosahedral symmetry ===

{| class=wikitable
![[Schoenflies notation|Schoenflies notation]]
![[Coxeter notation|Coxeter notation]]
![[Orbifold notation|Orbifold notation]]
![[Hermann–Mauguin notation|Hermann Mauguin notation]]
![[Symmetry number|Order]]
![[Index of a subgroup|Index]]
!Subgroup relations
|-align=center
|Ih||[5,3]||*532||{{overline|53}}2/m||120||1
|rowspan=22|[[File:Icosahedral_subgroup_tree.png|640px]]
|-align=center
|I||[5,3]+||532||532||60||2
|-align=center
|D5d ||[2+,10] ||2*5 ||{{overline|10}}m2 ||20 ||6
|- align=center
|Th||[4,3+]||3*2||m{{overline|3}}||12||10
|- align=center
|T||[3,3]+||332||332||12||10
|-align=center
|D3d ||[2+,6] ||2*3 ||{{overline|3}}m ||12 ||10

|- align=center
|D5||[5,2]+||522||522||10||12
|-align=center
|C5v ||[5] ||*55 ||5m ||10 ||12
|-align=center
|S10 ||[2+,10+] ||5× ||{{overline|5}} ||10 ||12
|- align=center
|D2h||[2,2]||*222||mmm||8||15
|- align=center
|D3||[3,2]+||322||322||6||20
|-align=center
|C3v ||[3] ||*33 ||3m ||6 ||20
|-align=center
|S6 ||[2+,6+] ||3× ||{{overline|3}} ||6 ||20
|- align=center
|C5||[5]+||55||5||5||24
|- align=center
|D2||[2,2]+||222||222||4||30
|-align=center
|C2h ||[2,2+] ||2* ||2/m ||4 ||30
|-align=center
|C2v ||[2] ||*22 ||mm2 ||4 ||30
|- align=center
|C3||[3]+||33||3||3||40
|-align=center
|Cs ||[ ] ||* ||{{overline|2}} or m ||2 ||60
|- align=center
|C2||[2]+||22||2||2||60
|-align=center
|S2 ||[2+,2+] ||× ||{{overline|1}} ||2 ||60
|- align=center
|C1||[ ]+||11||1||1||120

|}

All of these classes of subgroups are conjugate (i.e., all vertex stabilizers are conjugate), and admit geometric interpretations.

Note that the [[Group action#Orbits and stabilizers|stabilizer]] of a vertex/edge/face/polyhedron and its opposite are equal, since <math>-1</math> is central.

==== Vertex stabilizers ====
Stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate.
* vertex stabilizers in ''I'' give [[cyclic group]]s ''C''3
* vertex stabilizers in ''Ih'' give [[Dihedral symmetry in three dimensions|dihedral groups]] ''D''3
* stabilizers of an opposite pair of vertices in ''I'' give dihedral groups ''D''3
* stabilizers of an opposite pair of vertices in ''Ih'' give <math>D_3 \times \pm 1</math>

==== Edge stabilizers ====
Stabilizers of an opposite pair of edges can be interpreted as stabilizers of the rectangle they generate.
* edges stabilizers in ''I'' give cyclic groups ''Z''2
* edges stabilizers in ''Ih'' give [[Klein four-group]]s <math>Z_2 \times Z_2</math>
* stabilizers of a pair of edges in ''I'' give [[Klein four-group]]s <math>Z_2 \times Z_2</math>; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
* stabilizers of a pair of edges in ''Ih'' give <math>Z_2 \times Z_2 \times Z_2</math>; there are 5 of these, given by reflections in 3 perpendicular axes.

==== Face stabilizers ====
Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the [[anti-prism]] they generate.
* face stabilizers in ''I'' give cyclic groups ''C''5
* face stabilizers in ''Ih'' give dihedral groups ''D''5
* stabilizers of an opposite pair of faces in ''I'' give dihedral groups ''D''5
* stabilizers of an opposite pair of faces in ''Ih'' give <math>D_5 \times \pm 1</math>

==== Polyhedron stabilizers ====
For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, <math>I \stackrel{\sim}\to A_5 < S_5</math>.
* stabilizers of the inscribed tetrahedra in ''I'' are a copy of ''T''
* stabilizers of the inscribed tetrahedra in ''Ih'' are a copy of ''Th''
* stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in ''I'' are a copy of ''O''
* stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in ''Ih'' are a copy of ''Oh''

== Fundamental domain ==
[[Fundamental domain]]s for the icosahedral rotation group and the full icosahedral group are given by:
{| class=wikitable width=450
|- align=center
|[[File:Sphere symmetry group i.png|150px]] Icosahedral rotation group ''I''
|[[File:Sphere symmetry group ih.png|150px]] Full icosahedral group ''I''h
|[[File:Disdyakistriacontahedron.jpg|150px]] Faces of [[disdyakis triacontahedron]] are the fundamental domain
|}

In the [[disdyakis triacontahedron]] one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.

== Solids with icosahedral symmetry ==
{{details|Solids with icosahedral symmetry}}

===Chiral solids===

{| class=wikitable
!Class
! Name
! picture
! Faces
! Edges
! Vertices
|-align=center
![[Archimedean solid]]
! [[snub dodecahedron]]
| [[image:snubdodecahedronccw.jpg|50px]]|| 92|| 150|| 60
|-align=center
![[Catalan solid]]
![[pentagonal hexecontahedron]]
|[[image:pentagonalhexecontahedronccw.jpg|50px]]||60||50||92
|}

=== Full icosahedral symmetry ===
{| class=wikitable
!colspan=5|[[Platonic solid]]s
|- align=center
|[[File:Dodecahedron.jpg|50px]] {5,3}
|[[File:Icosahedron.jpg|50px]] {3,5}
|| || ||
|-
!colspan=5|[[Archimedean solid]]s
|- align=center
|[[File:truncateddodecahedron.jpg|50px]] 3.10.10
|[[File:truncatedicosidodecahedron.jpg|50px]] 4.6.10
|[[File:truncatedicosahedron.jpg|50px]] 5.6.6
|[[File:rhombicosidodecahedron.jpg|50px]] 3.4.5.4
|[[File:icosidodecahedron.jpg|50px]] 3.5.3.5
|-
!colspan=5|[[Catalan solid]]s
|- align=center
|[[File:triakisicosahedron.jpg|50px]] V3.10.10
|[[File:disdyakistriacontahedron.jpg|50px]] V4.6.10
|[[File:pentakisdodecahedron.jpg|50px]] V5.6.6
|[[File:deltoidalhexecontahedron.jpg|50px]] V3.4.5.4
|[[File:rhombictriacontahedron.jpg|50px]] V3.5.3.5
|}

== Other objects with icosahedral symmetry ==
{{Expand section|date=January 2010}}
* [[Barth surface]]s
* [[Virus structure]], and [[Capsid]]

== Liquid crystals with icosahedral symmetry ==

For the intermediate material phase called [[liquid qrystals]] the existence of icosahedral symmetry was proposed by [[Hagen Kleinert|H. Kleinert]] and K. Maki<ref>
{{cite journal
| title = Lattice Textures in Cholesteric Liquid Crystals
| author = [[Hagen Kleinert|Kleinert, H.]] and Maki, K.
| journal = Fortschritte der Physik
| volume = 29
| issue = 5
| pages = 219–259
| year = 1981
| doi = 10.1002/prop.19810290503
| url = http://www.physik.fu-berlin.de/~kleinert/75/75.pdf}}
</ref>
and its structure was first analyzed in detail in that paper. See the review article [http://chemgroups.northwestern.edu/seideman/Publications/The%20liquid-crystalline%20blue%20phases.pdf here].
In aluminum, the icosahedral structure was discovered experimentally three years after this
by [[Dan Shechtman]], which earned him the Nobel Prize in 2011.

== Related geometries ==
Icosahedral symmetry is equivalently the [[projective special linear group]] PSL(2,5), and is the symmetry group of the [[modular curve]] X(5), and more generally PSL(2,''p'') is the symmetry group of the modular curve X(''p''). The modular curve X(5) is geometrically a dodecahedron with a cusp at the center of each polygonal face, which demonstrates the symmetry group.

This geometry, and associated symmetry group, was studied by [[Felix Klein]] as the [[monodromy group]]s of a Belyi surface – a Riemann surface with a holomorphic map to the Riemann sphere, ramified only at 0, 1, and infinity (a [[Belyi function]]) – the cusps are the points lying over infinity, while the vertices and the centers of each edge lie over 0 and 1; the degree of the covering (number of sheets) equals 5.

This arose from his efforts to give a geometric setting for why icosahedral symmetry arose in the solution of the [[quintic equation]], with the theory given in the famous {{Harv|Klein|1888}}; a modern exposition is given in {{Harv|Tóth|2002|loc=Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, [http://books.google.com/books?id=i76mmyvDHYUC&pg=PA66 p. 66]}}.

Klein's investigations continued with his discovery of order 7 and order 11 symmetries in {{Harv|Klein|1878/79b}} and {{Harv|Klein|1879}} (and associated coverings of degree 7 and 11) and [[dessins d'enfants]], the first yielding the [[Klein quartic]], whose associated geometry has a tiling by 24 heptagons (with a cusp at the center of each).

Similar geometries occur for PSL(2,''n'') and more general groups for other modular curves.

More exotically, there are special connections between the groups PSL(2,5) (order 60), [[PSL(2,7)]] (order 168) and PSL(2,11) (order 660), which also admit geometric interpretations – PSL(2,5) is the symmetries of the icosahedron (genus 0), PSL(2,7) of the [[Klein quartic]] (genus 3), and PSL(2,11) the [[buckyball surface]] (genus 70). These groups form a "[[ADE classification#Trinities|trinity]]" in the sense of [[Vladimir Arnold]], which gives a framework for the various relationships; see ''[[ADE classification#Trinities|trinities]]'' for details.

There is a close relationship to other [[Platonic solids]].

== See also ==
*[[tetrahedral symmetry]]
*[[octahedral symmetry]]
*[[binary icosahedral group]]
*[[Icosian calculus]]

== References ==
{{reflist}}
{{refbegin}}
* {{cite doi|10.1007/BF01677143}}
* {{cite doi|10.1007/BF02086276}}
* {{citation | first = Felix | last = Klein | title = Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree | publisher = Trübner & Co. | year = 1888 | isbn = 0-486-49528-0 | postscript = trans. George Gavin Morrice }}
* {{citation | title = Finite Möbius groups, minimal immersions of spheres, and moduli
| first = Gábor | last = Tóth | year = 2002 }}
* Peter R. Cromwell, ''Polyhedra'' (1997), p.296
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5
* '''Kaleidoscopes: Selected Writings of [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', Manuscript, (2011) Chapter 11: Finite symmetry groups
{{refend}}
==External links==
* {{mathworld | urlname = IcosahedralGroup | title = Icosahedral group }}

[[Category:Finite groups]]
[[Category:Rotational symmetry]]

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en>M4gnum0n: no references