Icosahedral symmetry

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A football (soccer ball) with full icosahedral symmetry. This commonplace object is not, however, a regular icosahedron; it is a spherical truncated icosahedron.

A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a total of 120 symmetries 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 (the alternating group on 5 letters), and the full symmetry group (including reflections) is the product A₅ × C₂ of A₅ with a cyclic group of order 2.

1 As point group
2 Group structure
3 Fundamental Domain
4 Solids with icosahedral symmetry
- 4.1 Full icosahedral symmetry
5 See also

[edit] As point group

The icosahedral rotation group I with fundamental domain

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.

Schönflies crystallographic notation	Coxeter notation	Orbifold notation	Order
I	[3,5]+	532	60
I_h	[3,5]	*532	120

Presentations corresponding to the above are:

I: $\langle s,t \mid s^2, t^3, (st)^5 \rangle$

I_h: $\langle s,t\mid s^3(st)^{-2}, t^5(st)^{-2}\rangle$

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

[edit] Group structure

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 × C₂, or A₅ × C₂, with the inversion in the center corresponding to element (identity,-1), where C₂ 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), and I_h is isomorphic to SL₂(5) (they are both central extensions by C₂).

[edit] 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 icosahderal group (subject of this article)
2I, the binary icosahedral group

They correspond to the following short exact sequences (which do not split) and product

$1\to A_5 \to S_5 \to C_2 \to 1$

$I_h = A_5 \times C_2$

$1\to C_2 \to 2I\to A_5 \to 1$

In words,

$A 5$ is a normal subgroup of $S 5$
$A 5$ is a factor of $I h$ , which is a direct product
$A 5$ is a quotient group of $2 I$

[edit] Conjugacy classes

The conjugacy classes of I are:

identity
12 × rotation by 72°, order 5
12 × rotation by 144°, order 5
20 × rotation by 120°, order 3
15 × rotation by 180°, order 2

Those of I_h include also each with inversion:

inversion
12 × rotoreflection by 108°, order 10
12 × rotoreflection by 36°, order 10
20 × rotoreflection by 60°, order 6
15 × reflection, order 2

[edit] Subgroups

All of these classes of subgroups are conjugate (i.e., all vertex stabilizers are conjugate), and admit geometric interpretations.

Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, since $- 1$ is central.

[edit] 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 $D_3 \times \pm 1$

[edit] 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 C₂
edges stabilizers in I_h give Klein four-groups $C_2 \times C_2$
stabilizers of a pair of edges in I give Klein four-groups $C_2 \times C_2$ ; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
stabilizers of a pair of edges in I_h give $C_2 \times C_2 \times C_2$ ; these are 5 of these, given by reflections in 3 perpendicular axes.

[edit] Face stabilizers

Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism 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 $D_5 \times \pm 1$

[edit] Polyhedron stabilizers

For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, $I \stackrel{\sim}\to A_5 < S_5$ .

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_h
stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in I are a copy of O
stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in I_h are a copy of O_h

[edit] Fundamental Domain

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

The icosahedral rotation group I with fundamental domain

The full icosahedral group I_h with fundamental domain

Fundamental domain in the disdyakis triacontahedron

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.