Icosahedral symmetry

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 T_h with 20 versions of D₃ (10 axes, 2 per axis), and 6 versions of D₅.

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 × Z₂, or A₅ × Z₂, with the inversion in the center corresponding to element (identity,-1), where Z₂ 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 S₅, 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 PSL₂(5), but I_h 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 P_i and Q_i act on 5 and 12 elements respectively, while the rotation matrices M_i are the elements of I. If P_k is the product of taking the permutation P_i and applying P_j to it, then for the same values of i, j and k, it is also true that Q_k is the product of taking Q_i and applying Q_j, and also that premultiplying a vector by M_k is the same as premultiplying that vector by M_i and then premultiplying that result with M_j, that is M_k = M_j × M_i. Since the permutations P_i 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
• I_h, 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

${\displaystyle 1\to A_5\to S_5\to Z_2\to 1}$

${\displaystyle I_h=A_5\times Z_2}$

${\displaystyle 1\to Z_2\to 2I\to A_5\to 1}$

In words,

• ${\displaystyle A_5}$ is a normal subgroup of ${\displaystyle S_5}$
• ${\displaystyle A_5}$ is a factor of ${\displaystyle I_h}$, which is a direct product
• ${\displaystyle A_5}$ is a quotient group of ${\displaystyle 2I}$

Note that ${\displaystyle A_5}$ has an exceptional irreducible 3-dimensional representation (as the icosahedral rotation group), but ${\displaystyle S_5}$ 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:

• ${\displaystyle A_5\cong \mathrm{PSL}(2,5),}$ the projective special linear group, see here for a proof;
• ${\displaystyle S_5\cong \mathrm{PGL}(2,5),}$ the projective general linear group;
• ${\displaystyle 2I\cong \mathrm{SL}(2,5),}$ the special linear group.

Conjugacy classes

The 120 symmetries fall into 10 conjugacy classes.

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".

Vertex stabilizers

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

• vertex stabilizers in I give cyclic groups C₃
• vertex stabilizers in I_h give dihedral groups D₃
• stabilizers of an opposite pair of vertices in I give dihedral groups D₃
• stabilizers of an opposite pair of vertices in I_h give ${\displaystyle D_3\times \pm 1}$

Edge stabilizers

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

• edges stabilizers in I give cyclic groups Z₂
• edges stabilizers in I_h give Klein four-groups ${\displaystyle Z_2\times Z_2}$
• stabilizers of a pair of edges in I give Klein four-groups ${\displaystyle Z_2\times Z_2}$; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
• stabilizers of a pair of edges in I_h give ${\displaystyle Z_2\times Z_2\times Z_2}$; there are 5 of these, given by reflections in 3 perpendicular axes.

Face stabilizers

Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the antiprism they generate.

• face stabilizers in I give cyclic groups C₅
• face stabilizers in I_h give dihedral groups D₅
• stabilizers of an opposite pair of faces in I give dihedral groups D₅
• stabilizers of an opposite pair of faces in I_h give ${\displaystyle D_5\times \pm 1}$

Polyhedron stabilizers

For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, ${\displaystyle I\stackrel{\sim }{\to }{A}_5<S_5}$.

• 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

The full icosahedral symmetry group [5,3] () of order 120 has generators represented by the reflection matrices R₀, R₁, R₂ below, with relations R₀² = R₁² = R₂² = (R₀×R₁)⁵ = (R₁×R₂)³ = (R₀×R₂)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 ${\displaystyle \varphi =\frac{\sqrt{5}+1}{2}}$ denotes the golden ratio.

Fundamental domain

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.

Polyhedra with icosahedral symmetry

Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.

Chiral polyhedra

Full icosahedral symmetry

Other objects with icosahedral symmetry

• Barth surfaces
• Virus structure, and Capsid
• In chemistry, the dodecaborate ion ([B₁₂H₁₂]²⁻) and the dodecahedrane molecule (C₂₀H₂₀)

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

Icosahedral nanoparticles

At small sizes, many elements form icosahedral nanoparticles, which are often lower in energy than single crystals.

Related geometries

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.

Klein's investigations continued with his discovery of order 7 and order 11 symmetries in (Klein 1878) 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.

Poincaré used the rotational symmetry group ${\displaystyle I\subset {\mathrm{S}\mathrm{O}}_3}$ to construct the Poincaré homology sphere as the quotient manifold ${\displaystyle X={\mathrm{S}\mathrm{O}}_3/I}$, an important example of a space whose homology is the same as that of the sphere ${\displaystyle S^3}$, but which is not homotopic to it.