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 H3. It may be represented by Coxeter notation [5,3] and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 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.
Presentations corresponding to the above are:
I : ⟨ s , t ∣ s 2 , t 3 , ( s t ) 5 ⟩ {\displaystyle I:\langle s,t\mid s^{2},t^{3},(st)^{5}\rangle \ }
I h : ⟨ s , t ∣ s 3 ( s t ) − 2 , t 5 ( s t ) − 2 ⟩ . {\displaystyle I_{h}:\langle s,t\mid s^{3}(st)^{-2},t^{5}(st)^{-2}\rangle .\ }
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 H3. It may be represented by Coxeter notation [5,3] and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 on 5 letters.
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 icosahedral rotation group 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 full icosahedral group 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.
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. A modern exposition is given in Tóth (2002).
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
1 → A 5 → S 5 → Z 2 → 1 {\displaystyle 1\to A_{5}\to S_{5}\to Z_{2}\to 1}
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