Icosahedral symmetry


A regular icosahedron has 60 rotational 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 A5, and the full symmetry group is the product A5 × Z2. The latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation and Coxeter diagram.

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 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:
These correspond to the icosahedral groups being the 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.

Visualizations

Group structure

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, the compound of five octahedra, or either of the two compounds of five tetrahedra.
The group contains 5 versions of Th with 20 versions of D3, 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, 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. It acts on the compound of ten tetrahedra: I acts on the two chiral halves, 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.
I is also isomorphic to PSL2, but Ih is not isomorphic to SL2.

Commonly confused groups

The following groups all have order 120, but are not isomorphic:
They correspond to the following short exact sequences and product
In words,
Note that has an exceptional irreducible 3-dimensional representation, 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:

Subgroups of full icosahedral symmetry

All of these classes of subgroups 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

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.
For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism,.
The full icosahedral symmetry group of order 120 has generators represented by the reflection matrices R0, R1, R2 below, with relations R02 = R12 = R22 = 5 = 3 = 2 = Identity. The group + of order 60 is generated by any two of the rotations S0,1, S1,2, S0,2. A rotoreflection of order 10 is generated by V0,1,2, the product of all 3 reflections. Here denotes the golden ratio.

Fundamental domain

s for the icosahedral rotation group and the full icosahedral group are given by:

Icosahedral rotation group
I
Full icosahedral group
Ih
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

Chiral polyhedra

ClassSymbolsPicture
Archimedeansr
CatalanV3.3.3.3.5

Full icosahedral symmetry

Other objects with icosahedral symmetry

For the intermediate material phase called liquid crystals the existence of icosahedral symmetry was proposed by H. Kleinert and K. Maki
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.

Related geometries

Icosahedral symmetry is equivalently the projective special linear group PSL, and is the symmetry group of the modular curve X, and more generally PSL is the symmetry group of the modular curve X. The modular curve X 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 – 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 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 ; a modern exposition is given in.
Klein's investigations continued with his discovery of order 7 and order 11 symmetries in and and dessins d'enfants, the first yielding the Klein quartic, whose associated geometry has a tiling by 24 heptagons.
Similar geometries occur for PSL and more general groups for other modular curves.
More exotically, there are special connections between the groups PSL, PSL and PSL, which also admit geometric interpretations – PSL is the symmetries of the icosahedron, PSL of the Klein quartic, and PSL the buckyball surface. 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.