Icosahedral symmetry

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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 Details
2 Conjugacy classes
3 Subgroups
4 Solids with full icosahedral symmetry
5 Chiral Archimedean and Catalan solids
6 Chiral nonconvex uniform polyhedra
7 See also

[edit] Details

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. 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. (The five objects being permuted by I in the case at hand are the five inscribed cubes in the dual 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. The group contains 10 versions of D_3d and 6 versions of D_5d (symmetries like antiprisms).

Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order
I	[3,5]+	532	60
I_h	[3,5]	*532	120

Presentations

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.

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.

The icosahedral group I_h with fundamental domain

[edit] Conjugacy classes

The conjugacy classes of I are:

identity
12 × rotation by 72°
12 × rotation by 144°
20 × rotation by 120°
15 × rotation by 180°

Those of I_h include also each with inversion:

inversion
12 × rotoreflection by 108°
12 × rotoreflection by 36°
20 × rotoreflection by 60°
15 × reflection

[edit] Subgroups

I contains 5 copies of T.

I_h contains 5 copies of T_h.

[edit] Solids with full icosahedral symmetry

(For details see below.)

Platonic solids - regular polyhedra (all faces of the same type)

{5,3}

{3,5}

Archimedean solids - polyhedra with more than one polygon face type.

3.10.10

4.6.10

5.6.6

3.4.5.4

3.5.3.5

Catalan solids - duals of the Archimedean solids.

V3.10.10

V4.6.10

V5.6.6

V3.4.5.4

V3.5.3.5

[edit] Platonic solids

Name	Picture	Faces	Edges	Vertices	Edges per face	Faces meeting at each vertex
dodecahedron	(Animation)	12	30	20	5	3
icosahedron	(Animation)	20	30	12	3	5

[edit] Achiral Archimedean solids

Name	picture	Faces		Edges	Vertices	Vertex configuration
icosidodecahedron (quasi-regular: vertex- and edge-uniform)	(Video)	32	20 triangles 12 pentagons	60	30	3,5,3,5
truncated dodecahedron	(Video)	32	20 triangles 12 decagons	90	60	3,10,10
truncated icosahedron or commonly football (soccer ball)	(Video)	32	12 pentagons 20 hexagons	90	60	5,6,6
rhombicosidodecahedron or small rhombicosidodecahedron	(Video)	62	20 triangles 30 squares 12 pentagons	120	60	3,4,5,4
truncated icosidodecahedron or great rhombicosidodecahedron	(Video)	62	30 squares 20 hexagons 12 decagons	180	120	4,6,10

[edit] Achiral Catalan solids

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon
rhombic triacontahedron (quasi-regular dual: face- and edge-uniform)	(Video)	icosidodecahedron	30	60	32	rhombus
triakis icosahedron	(Video)	truncated dodecahedron	60	90	32	isosceles triangle
pentakis dodecahedron	(Video)	truncated icosahedron	60	90	32	isosceles triangle
deltoidal hexecontahedron	(Video)	rhombicosidodecahedron	60	120	62	kite
disdyakis triacontahedron or hexakis icosahedron	(Video)	truncated icosidodecahedron	120	180	62	scalene triangle