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 full symmetry group (including reflections) is known as the Coxeter group H3, and is also represented by Coxeter notation [5,3] and Coxeter diagram CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png. The set of orientation-preserving symmetries forms a subgroup that is isomorphic to the group A5 (the alternating group on 5 letters).

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.

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).

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).

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.

They correspond to the following short exact sequences (the latter of which does not split) and product

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:

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".

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

Stabilizers of an opposite pair of edges can be interpreted as stabilizers of the rectangle they generate.

Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate.

Fundamental domains for the icosahedral rotation group and the full icosahedral group are given by:

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.

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 . In aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011.

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, ).

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.