ICOSAHEDRAL SYMMETRY

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 ''A''₅ (the alternating group on 5 letters), and the full symmetry group (including reflections) is the product ''A''₅ × ''C''₂ of ''A''₅ with a cyclic group of order 2.

Contents

Details

Conjugacy classes

Subgroups

Solids with full icosahedral symmetry

Platonic solids

Achiral Archimedean solids

Achiral Catalan solids

Kepler-Poinsot solids

Achiral nonconvex uniform polyhedra

Chiral Archimedean and Catalan solids

Chiral nonconvex uniform polyhedra

See also

Details

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 ''A''₅, 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'' × ''C''₂, or ''A''₅ × ''C''₂, with the inversion in the center corresponding to element (identity,-1), where ''C''₂ 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
''Ih''	[3,5]	★ 532	120

Presentations
''I'': $langle\; s,t\; mid\; s^2,\; t^3,\; (st)^5$
angle
''I_h'': $langle\; s,tmid\; s^3(st)^\{-2\},\; t^5(st)^\{-2\}$
angle
Note that other presentations are possible.

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

Subgroups

''I'' contains 5 copies of ''T''.
''I_h'' contains 5 copies of ''T_h''.

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

Platonic solids

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

Achiral Archimedean solids

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

Achiral Catalan solids

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

Kepler-Poinsot solids

Achiral nonconvex uniform polyhedra

Chiral Archimedean and Catalan solids

Archimedean solids:

Name	picture	Faces		Edges	Vertices	Vertex configuration
snub dodecahedron or snub icosidodecahedron (2 chiral forms)	() ()	92	80 triangles 12 pentagons	150	60	3,3,3,3,5

Catalan solids:

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon
pentagonal hexecontahedron	()()	snub dodecahedron	60	150	92	irregular pentagon

Stellated Archimedean solids:

★ the snub dodecadodecahedron

★ the great inverted snub icosidodecahedron or great vertisnub icosidodecahedron

★ the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron

Chiral nonconvex uniform polyhedra

See also

★ tetrahedral symmetry

★ octahedral symmetry

★ binary icosahedral group