Icosahedral symmetry
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Involutional symmetry C_{s}, (*) [ ] = 
Cyclic symmetry C_{nv}, (*nn) [n] = 
Dihedral symmetry D_{nh}, (*n22) [n,2] =  
Polyhedral group, [n,3], (*n32)  

Tetrahedral symmetry T_{d}, (*332) [3,3] = 
Octahedral symmetry O_{h}, (*432) [4,3] = 
Icosahedral symmetry I_{h}, (*532) [5,3] = 
A regular icosahedron has 60 rotational (or orientationpreserving) 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 full symmetry group (including reflections) is known as the Coxeter group H_{3}, and is also represented by Coxeter notation [5,3] and Coxeter diagram . The set of orientationpreserving symmetries forms a subgroup that is isomorphic to the group A_{5} (the alternating group on 5 letters).
As point group[edit]
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 discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups.
Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups.
Schö.  Coxeter  Orb.  Abstract structure 
Order  

I  [5,3]^{+}  532  A_{5}  60  
I_{h}  [5,3]  *532  A_{5}×2  120 
Presentations corresponding to the above are:
These correspond to the icosahedral groups (rotational and full) being the (2,3,5) triangle groups.
The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus.^{[1]}
Note that other presentations are possible, for instance as an alternating group (for I).
Visualizations[edit]
Schoe. (Orb.) 
Coxeter notation 
Elements  Mirror diagrams  

Orthogonal  Stereographic projection  
I_{h} (*532) 
[5,3] 
Mirror lines: 15 

I (532) 
[5,3]^{+} 
Gyration points: 12_{5} 20_{3} 30_{2} 
Group structure[edit]
The edges of a spherical compound of five octahedra represent the 15 mirror planes as colored great circles. Each octahedron can represent 3 orthogonal mirror planes by its edges.  
The pyritohedral symmetry is an index 5 subgroup of icosahedral symmetry, with 3 orthogonal green reflection lines and 8 red order3 gyration points. As an index 5 subgroup there are 5 other orientations of pyritohedral symmetry. 
The 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 compounds of five tetrahedra (which are enantiomorphs, and inscribe in the dodecahedron).
The group contains 5 versions of T_{h} with 20 versions of D_{3} (10 axes, 2 per axis), and 6 versions of D_{5}.
The full icosahedral group I_{h} has order 120. It has I as normal subgroup of index 2. The group I_{h} is isomorphic to I × Z_{2}, or A_{5} × Z_{2}, with the inversion in the center corresponding to element (identity,1), where Z_{2} is written multiplicatively.
I_{h} 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 (compounds of five tetrahedra), and −1 interchanges the two halves. Notably, it does not act as S_{5}, and these groups are not isomorphic; see below for details.
The group contains 10 versions of D_{3d} and 6 versions of D_{5d} (symmetries like antiprisms).
I is also isomorphic to PSL_{2}(5), but I_{h} is not isomorphic to SL_{2}(5).
Commonly confused groups[edit]
The following groups all have order 120, but are not isomorphic:
 S_{5}, the symmetric group on 5 elements
 I_{h}, the full icosahedral group (subject of this article, also known as H_{3})
 2I, the binary icosahedral group
They correspond to the following short exact sequences (the latter of which does not split) and product
In words,
 is a normal subgroup of
 is a factor of , which is a direct product
 is a quotient group of
Note that has an exceptional irreducible 3dimensional representation (as the icosahedral rotation group), but does not have an irreducible 3dimensional 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:
 the projective special linear group, see here for a proof;
 the projective general linear group;
 the special linear group.
Conjugacy classes[edit]
I  I_{h} 



Subgroups of full icosahedral symmetry[edit]
Schön.  Coxeter  Orb.  HM  Structure  Cyc.  Order  Index  

I_{h}  [5,3]  *532  532/m  A_{5}×Z_{2}  120  1  
D_{2h}  [2,2]  *222  mmm  Dih_{2}×Dih_{1}=Dih_{1}^{3}  8  15  
C_{5v}  [5]  *55  5m  Dih_{5}  10  12  
C_{3v}  [3]  *33  3m  Dih_{3}=S_{3}  6  20  
C_{2v}  [2]  *22  2mm  Dih_{2}=Dih_{1}^{2}  4  30  
C_{s}  [ ]  *  2 or m  Dih_{1}  2  60  
T_{h}  [3^{+},4]  3*2  m3  A_{4}×Z_{2}  24  5  
D_{5d}  [2^{+},10]  2*5  10m2  Dih_{10}=Z_{2}×Dih_{5}  20  6  
D_{3d}  [2^{+},6]  2*3  3m  Dih_{6}=Z_{2}×Dih_{3}  12  10  
D_{1d} = C_{2h}  [2^{+},2]  2*  2/m  Dih_{2}=Z_{2}×Dih_{1}  4  30  
S_{10}  [2^{+},10^{+}]  5×  5  Z_{10}=Z_{2}×Z_{5}  10  12  
S_{6}  [2^{+},6^{+}]  3×  3  Z_{6}=Z_{2}×Z_{3}  6  20  
S_{2}  [2^{+},2^{+}]  ×  1  Z_{2}  2  60  
I  [5,3]^{+}  532  532  A_{5}  60  2  
T  [3,3]^{+}  332  332  A_{4}  12  10  
D_{5}  [2,5]^{+}  522  522  Dih_{5}  10  12  
D_{3}  [2,3]^{+}  322  322  Dih_{3}=S_{3}  6  20  
D_{2}  [2,2]^{+}  222  222  Dih_{2}=Z_{2}^{2}  4  30  
C_{5}  [5]^{+}  55  5  Z_{5}  5  24  
C_{3}  [3]^{+}  33  3  Z_{3}=A_{3}  3  40  
C_{2}  [2]^{+}  22  2  Z_{2}  2  60  
C_{1}  [ ]^{+}  11  1  Z_{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 stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, since is central.
Vertex stabilizers[edit]
Stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate.
 vertex stabilizers in I give cyclic groups C_{3}
 vertex stabilizers in I_{h} give 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 I_{h} give
Edge stabilizers[edit]
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 I_{h} give Klein fourgroups
 stabilizers of a pair of edges in I give Klein fourgroups ; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
 stabilizers of a pair of edges in I_{h} give ; there are 5 of these, given by reflections in 3 perpendicular axes.
Face stabilizers[edit]
Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the antiprism they generate.
 face stabilizers in I give cyclic groups C_{5}
 face stabilizers in I_{h} 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 I_{h} give
Polyhedron stabilizers[edit]
For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, .
 stabilizers of the inscribed tetrahedra in I are a copy of T
 stabilizers of the inscribed tetrahedra in I_{h} are a copy of T
 stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedra) in I are a copy of T
 stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedra) in I_{h} are a copy of T_{h}
Coxeter group generators[edit]
The full icosahedral symmetry group [5,3] () of order 120 has generators represented by the reflection matrices R_{0}, R_{1}, R_{2} below, with relations R_{0}^{2} = R_{1}^{2} = R_{2}^{2} = (R_{0}×R_{1})^{5} = (R_{1}×R_{2})^{3} = (R_{0}×R_{2})^{2} = Identity. The group [5,3]^{+} () of order 60 is generated by any two of the rotations S_{0,1}, S_{1,2}, S_{0,2}. A rotoreflection of order 10 is generated by V_{0,1,2}, the product of all 3 reflections. Here denotes the golden ratio.
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  5  3  2  10 
Matrix  
(1,0,0)_{n}  _{n}  (0,1,0)_{n}  (φ,1,0)_{axis}  (1,1,1)_{axis}  (1,0,0)_{axis} 
Fundamental domain[edit]
Fundamental domains for the icosahedral rotation group and the full icosahedral group are given by:
Icosahedral rotation group I 
Full icosahedral group I_{h} 
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.
Polyhedra with icosahedral symmetry[edit]
Chiral polyhedra[edit]
Class  Symbols  Picture 

Archimedean  sr{5,3} 

Catalan  V3.3.3.3.5 
Full icosahedral symmetry[edit]
Platonic solid  Kepler–Poinsot polyhedra  Archimedean solids  

{5,3} 
{5/2,5} 
{5/2,3} 
t{5,3} 
t{3,5} 
r{3,5} 
rr{3,5} 
tr{3,5} 
Platonic solid  Kepler–Poinsot polyhedra  Catalan solids  
{3,5} = 
{5,5/2} = 
{3,5/2} = 
V3.10.10 
V5.6.6 
V3.5.3.5 
V3.4.5.4 
V4.6.10 
Other objects with icosahedral symmetry[edit]
 Barth surfaces
 Virus structure, and Capsid
 In chemistry, the dodecaborate ion ([B_{12}H_{12}]^{2−}) and the dodecahedrane molecule (C_{20}H_{20})
Liquid crystals with icosahedral symmetry[edit]
For the intermediate material phase called liquid crystals the existence of icosahedral symmetry was proposed by H. Kleinert and K. Maki^{[2]} and its structure was first analyzed in detail in that paper. See the review article 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[edit]
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 groups 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 (Klein 1888); a modern exposition is given in (Tóth 2002, Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, p. 66).
Klein's investigations continued with his discovery of order 7 and order 11 symmetries in (Klein & 1878/79b) and (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 "trinity" in the sense of Vladimir Arnold, which gives a framework for the various relationships; see trinities for details.
There is a close relationship to other Platonic solids.
See also[edit]
References[edit]
 ^ Sir William Rowan Hamilton (1856), "Memorandum respecting a new System of Roots of Unity" (PDF), Philosophical Magazine, 12: 446
 ^ Kleinert, H. & Maki, K. (1981). "Lattice Textures in Cholesteric Liquid Crystals" (PDF). Fortschritte der Physik. 29 (5): 219–259. doi:10.1002/prop.19810290503.
 Klein, F. (1878). "Ueber die Transformation siebenter Ordnung der elliptischen Functionen" [On the orderseven transformation of elliptic functions]. Mathematische Annalen. 14 (3): 428–471. doi:10.1007/BF01677143. Translated in Levy, Silvio, ed. (1999). The Eightfold Way. Cambridge University Press. ISBN 9780521660662. MR 1722410.CS1 maint: ref=harv (link)
 Klein, F. (1879), "Ueber die Transformation elfter Ordnung der elliptischen Functionen (On the eleventh order transformation of elliptic functions)", Mathematische Annalen, 15 (3–4): 533–555, doi:10.1007/BF02086276, collected as pp. 140–165 in Oeuvres, Tome 3
 Klein, Felix (1888), Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Trübner & Co., ISBN 0486495280trans. George Gavin Morrice
 Tóth, Gábor (2002), Finite Möbius groups, minimal immersions of spheres, and moduli
 Peter R. Cromwell, Polyhedra (1997), p. 296
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups