In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.

Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters.

As point group 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.CoxeterOrb.AbstractstructureOrderI[5,3]+532A560Ih[5,3]*532A5×2120

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.

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

Visualizations The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters. Schoe.(Orb.)CoxeternotationElementsMirror diagramsOrthogonalStereographic projectionIh(*532) [5,3] Mirrorlines:15 I(532) [5,3]+Gyrationpoints:125203302

Group structure Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. 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 order-3 gyration points. There are 5 different orientations of pyritohedral symmetry. The I is of order 60. The group I is isomorphic to A5, 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 Th with 20 versions of D3 (10 axes, 2 per axis), and 6 versions of D5.

The Ih has order 120. It has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the center corresponding to element (identity,-1), where Z2 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 (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).

Isomorphism of I with A5 It is useful to describe explicitly what the isomorphism between I and A5 looks like. In the following table, permutations Pi and Qi act on 5 and 12 elements respectively, while the rotation matrices Mi are the elements of I. If Pk is the product of taking the permutation Pi and applying Pj to it, then for the same values of i, j and k, it is also true that Qk is the product of taking Qi and applying Qj, and also that premultiplying a vector by Mk is the same as premultiplying that vector by Mi and then premultiplying that result with Mj, that is Mk = Mj × Mi. Since the permutations Pi are all the 60 even permutations of 12345, the one-to-one correspondence is made explicit, therefore the isomorphism too.

Rotation matrixPermutation of 5on 1 2 3 4 5Permutation of 12on 1 2 3 4 5 6 7 8 9 10 11 12 = () = () = (3 4 5) = (1 11 8)(2 9 6)(3 5 12)(4 7 10) = (3 5 4) = (1 8 11)(2 6 9)(3 12 5)(4 10 7) = (2 3)(4 5) = (1 12)(2 8)(3 6)(4 9)(5 10)(7 11) = (2 3 4) = (1 2 3)(4 5 6)(7 9 8)(10 11 12) = (2 3 5) = (1 7 5)(2 4 11)(3 10 9)(6 8 12) = (2 4 3) = (1 3 2)(4 6 5)(7 8 9)(10 12 11) = (2 4 5) = (1 10 6)(2 7 12)(3 4 8)(5 11 9) = (2 4)(3 5) = (1 9)(2 5)(3 11)(4 12)(6 7)(8 10) = (2 5 3) = (1 5 7)(2 11 4)(3 9 10)(6 12 8) = (2 5 4) = (1 6 10)(2 12 7)(3 8 4)(5 9 11) = (2 5)(3 4) = (1 4)(2 10)(3 7)(5 8)(6 11)(9 12) = (1 2)(4 5) = (1 3)(2 4)(5 8)(6 7)(9 10)(11 12) = (1 2)(3 4) = (1 5)(2 7)(3 11)(4 9)(6 10)(8 12) = (1 2)(3 5) = (1 12)(2 10)(3 8)(4 6)(5 11)(7 9) = (1 2 3) = (1 11 6)(2 5 9)(3 7 12)(4 10 8) = (1 2 3 4 5) = (1 6 5 3 9)(4 12 7 8 11) = (1 2 3 5 4) = (1 4 8 6 2)(5 7 10 12 9) = (1 2 4 5 3) = (1 8 7 3 10)(2 12 5 6 11) = (1 2 4) = (1 7 4)(2 11 8)(3 5 10)(6 9 12) = (1 2 4 3 5) = (1 2 9 11 7)(3 6 12 10 4) = (1 2 5 4 3) = (2 3 4 7 5)(6 8 10 11 9) = (1 2 5) = (1 9 8)(2 6 3)(4 5 12)(7 11 10) = (1 2 5 3 4) = (1 10 5 4 11)(2 8 9 3 12) = (1 3 2) = (1 6 11)(2 9 5)(3 12 7)(4 8 10) = (1 3 4 5 2) = (2 5 7 4 3)(6 9 11 10 8) = (1 3 5 4 2) = (1 10 3 7 8)(2 11 6 5 12) = (1 3)(4 5) = (1 7)(2 10)(3 11)(4 5)(6 12)(8 9) = (1 3 4) = (1 9 10)(2 12 4)(3 6 8)(5 11 7) = (1 3 5) = (1 3 4)(2 8 7)(5 6 10)(9 12 11) = (1 3)(2 4) = (1 12)(2 6)(3 9)(4 11)(5 8)(7 10) = (1 3 2 4 5) = (1 4 10 11 5)(2 3 8 12 9) = (1 3 5 2 4) = (1 5 9 6 3)(4 7 11 12 8) = (1 3)(2 5) = (1 2)(3 5)(4 9)(6 7)(8 11)(10 12) = (1 3 2 5 4) = (1 11 2 7 9)(3 10 6 4 12) = (1 3 4 2 5) = (1 8 2 4 6)(5 10 9 7 12) = (1 4 5 3 2) = (1 2 6 8 4)(5 9 12 10 7) = (1 4 2) = (1 4 7)(2 8 11)(3 10 5)(6 12 9) = (1 4 3 5 2) = (1 11 4 5 10)(2 12 3 9 8) = (1 4 3) = (1 10 9)(2 4 12)(3 8 6)(5 7 11) = (1 4 5) = (1 5 2)(3 7 9)(4 11 6)(8 10 12) = (1 4)(3 5) = (1 6)(2 3)(4 9)(5 8)(7 12)(10 11) = (1 4 5 2 3) = (1 9 7 2 11)(3 12 4 6 10) = (1 4)(2 3) = (1 8)(2 10)(3 4)(5 12)(6 7)(9 11) = (1 4 2 3 5) = (2 7 3 5 4)(6 11 8 9 10) = (1 4 2 5 3) = (1 3 6 9 5)(4 8 12 11 7) = (1 4 3 2 5) = (1 7 10 8 3)(2 5 11 12 6) = (1 4)(2 5) = (1 12)(2 9)(3 11)(4 10)(5 6)(7 8) = (1 5 4 3 2) = (1 9 3 5 6)(4 11 8 7 12) = (1 5 2) = (1 8 9)(2 3 6)(4 12 5)(7 10 11) = (1 5 3 4 2) = (1 7 11 9 2)(3 4 10 12 6) = (1 5 3) = (1 4 3)(2 7 8)(5 10 6)(9 11 12) = (1 5 4) = (1 2 5)(3 9 7)(4 6 11)(8 12 10) = (1 5)(3 4) = (1 12)(2 11)(3 10)(4 8)(5 9)(6 7) = (1 5 4 2 3) = (1 5 11 10 4)(2 9 12 8 3) = (1 5)(2 3) = (1 10)(2 12)(3 11)(4 7)(5 8)(6 9) = (1 5 2 3 4) = (1 3 8 10 7)(2 6 12 11 5) = (1 5 2 4 3) = (1 6 4 2 8)(5 12 7 9 10) = (1 5 3 2 4) = (2 4 5 3 7)(6 10 9 8 11) = (1 5)(2 4) = (1 11)(2 10)(3 12)(4 9)(5 7)(6 8)

This non-abelian simple group is the only non-trivial normal subgroup of the symmetric group on five letters. Since the Galois group of the general quintic equation is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals. The proof of the Abel–Ruffini theorem uses this simple fact, and Felix Klein wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation. Dover edition, translated from A modern exposition is given in .

Commonly confused groups The following groups all have order 120, but are not isomorphic: S5, the symmetric group on 5 elements Ih, the full icosahedral group (subject of this article, also known as H3) 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 3-dimensional representation (as the icosahedral rotation group), but 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: the projective special linear group, see here for a proof; the projective general linear group; the special linear group.

Conjugacy classes The 120 symmetries fall into 10 conjugacy classes. + Conjugacy classesIAdditional classes of Ih identity, order 1 12 × rotation by ±72°, order 5, around the 6 axes through the face centers of the dodecahedron 12 × rotation by ±144°, order 5, around the 6 axes through the face centers of the dodecahedron 20 × rotation by ±120°, order 3, around the 10 axes through vertices of the dodecahedron 15 × rotation by 180°, order 2, around the 15 axes through midpoints of edges of the dodecahedron central inversion, order 2 12 × rotoreflection by ±36°, order 10, around the 6 axes through the face centers of the dodecahedron 12 × rotoreflection by ±108°, order 10, around the 6 axes through the face centers of the dodecahedron 20 × rotoreflection by ±60°, order 6, around the 10 axes through the vertices of the dodecahedron 15 × reflection, order 2, at 15 planes through edges of the dodecahedron

Subgroups of the full icosahedral symmetry group

Each line in the following table represents one class of conjugate (i.e., geometrically equivalent) subgroups. The column "Mult." (multiplicity) gives the number of different subgroups in the conjugacy class.

Explanation of colors: green = the groups that are generated by reflections, red = the chiral (orientation-preserving) groups, which contain only rotations.

The groups are described geometrically in terms of the dodecahedron.

The abbreviation "h.t.s.(edge)" means "halfturn swapping this edge with its opposite edge", and similarly for "face" and "vertex".

Schön.CoxeterOrb.H-MStructureCyc.OrderIndexMult.DescriptionIh[5,3]*5322/mA5×Z2 12011full groupD2h[2,2]*222mmmD4×D2=D23 8155fixing two opposite edges, possibly swapping themC5v *55 5mD10 10 126fixing a faceC3v *33 3mD6=S3 6 2010fixing a vertexC2v *22 2mmD4=D22 4 3015fixing an edgeCs [ ] * or mD2 2 6015reflection swapping two endpoints of an edgeTh[3+,4]3*2mA4×Z2 2455pyritohedral groupD5d [2+,10] 2*5 m2D20=Z2×D10 20 66fixing two opposite faces, possibly swapping themD3d [2+,6] 2*3 mD12=Z2×D612 1010fixing two opposite vertices, possibly swapping them D1d = C2h [2+,2] 2* 2/mD4=Z2×D24 3015halfturn around edge midpoint, plus central inversionS10 [2+,10+] 5× Z10=Z2×Z510 126rotations of a face, plus central inversionS6 [2+,6+] 3× Z6=Z2×Z36 2010rotations about a vertex, plus central inversionS2 [2+,2+] × Z22 601central inversionI[5,3]+532532A5 6021all rotationsT[3,3]+332332A4 12105rotations of a contained tetrahedronD5[2,5]+522522D1010126rotations around the center of a face, and h.t.s.(face)D3[2,3]+322322D6=S362010rotations around a vertex, and h.t.s.(vertex)D2[2,2]+222222D4=Z224305halfturn around edge midpoint, and h.t.s.(edge)C5+555Z55246rotations around a face centerC3+333Z3=A334010rotations around a vertexC2+222Z226015half-turn around edge midpointC1[ ]+111Z111201trivial group

Vertex stabilizers Stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate.