Icosahedral symmetry

Selected point groups in three dimensions
Polyhedral group, [n,3], (*n32)
Involutional symmetry C_s, (*) [ ] =	Cyclic symmetry C_nv, (*nn) [n] =	Dihedral symmetry D_nh, (*n22) [n,2] =
Tetrahedral symmetry T_d, (*332) [3,3] =	Octahedral symmetry O_h, (*432) [4,3] =	Icosahedral symmetry I_h, (*532) [5,3] =

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.

Icosahedral symmetry fundamental domains

A soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry.

Rotations and reflections form the symmetry group of a great icosahedron.

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 full symmetry group is the Coxeter group of type $H 3$ . It may be represented by Coxeter notation $[5,3]$ and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group $A 5$ on 5 letters.

Description edit

Icosahedral symmetry is a mathematical property of objects indicating that an object has the same symmetries as a regular icosahedron.

As point group edit

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.

Schö.	Coxeter		Orb.	Abstract structure	Order
I	[5,3]⁺		532	A₅	60
I_h	[5,3]		*532	A₅×2	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 .\

These correspond to the icosahedral groups (rotational and full) being the (2,3,5) triangle groups.

The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus.^[1]

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

Visualizations edit

The full symmetry group is the Coxeter group of type $H 3$ . It may be represented by Coxeter notation $[5,3]$ and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group $A 5$ on 5 letters.

Schoe. (Orb.)	Coxeter notation	Elements	Mirror diagrams
Schoe. (Orb.)	Coxeter notation	Elements	Orthogonal	Stereographic projection
I_h (*532)	[5,3]	Mirror lines: 15
I (532)	[5,3]⁺	Gyration points: 12₅ 20₃ 30₂

Group structure edit

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 edges of a spherical compound of five octahedra represent the 15 mirror planes as colored great circles. Each octahedron can represent 3 orthogonal mirror planes by its edges.

The pyritohedral symmetry is an index 5 subgroup of icosahedral symmetry, with 3 orthogonal green reflection lines and 8 red order-3 gyration points. There are 5 different orientations of pyritohedral symmetry.

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 edit

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.

Rotation matrix	Permutation of 5 on 1 2 3 4 5	Permutation of 12 on 1 2 3 4 5 6 7 8 9 10 11 12
$M_{1}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}$	$P_{1}$ = ()	$Q_{1}$ = ()
$M_{2}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{2}$ = (3 4 5)	$Q_{2}$ = (1 11 8)(2 9 6)(3 5 12)(4 7 10)
$M_{3}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{3}$ = (3 5 4)	$Q_{3}$ = (1 8 11)(2 6 9)(3 12 5)(4 10 7)
$M_{4}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{4}$ = (2 3)(4 5)	$Q_{4}$ = (1 12)(2 8)(3 6)(4 9)(5 10)(7 11)
$M_{5}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{5}$ = (2 3 4)	$Q_{5}$ = (1 2 3)(4 5 6)(7 9 8)(10 11 12)
$M_{6}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{6}$ = (2 3 5)	$Q_{6}$ = (1 7 5)(2 4 11)(3 10 9)(6 8 12)
$M_{7}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{7}$ = (2 4 3)	$Q_{7}$ = (1 3 2)(4 6 5)(7 8 9)(10 12 11)
$M_{8}={\begin{bmatrix}0&-1&0\\0&0&1\\-1&0&0\end{bmatrix}}$	$P_{8}$ = (2 4 5)	$Q_{8}$ = (1 10 6)(2 7 12)(3 4 8)(5 11 9)
$M_{9}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{9}$ = (2 4)(3 5)	$Q_{9}$ = (1 9)(2 5)(3 11)(4 12)(6 7)(8 10)
$M_{10}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{10}$ = (2 5 3)	$Q_{10}$ = (1 5 7)(2 11 4)(3 9 10)(6 12 8)
$M_{11}={\begin{bmatrix}0&0&-1\\-1&0&0\\0&1&0\end{bmatrix}}$	$P_{11}$ = (2 5 4)	$Q_{11}$ = (1 6 10)(2 12 7)(3 8 4)(5 9 11)
$M_{12}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{12}$ = (2 5)(3 4)	$Q_{12}$ = (1 4)(2 10)(3 7)(5 8)(6 11)(9 12)
$M_{13}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}$	$P_{13}$ = (1 2)(4 5)	$Q_{13}$ = (1 3)(2 4)(5 8)(6 7)(9 10)(11 12)
$M_{14}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{14}$ = (1 2)(3 4)	$Q_{14}$ = (1 5)(2 7)(3 11)(4 9)(6 10)(8 12)
$M_{15}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{15}$ = (1 2)(3 5)	$Q_{15}$ = (1 12)(2 10)(3 8)(4 6)(5 11)(7 9)
$M_{16}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{16}$ = (1 2 3)	$Q_{16}$ = (1 11 6)(2 5 9)(3 7 12)(4 10 8)
$M_{17}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{17}$ = (1 2 3 4 5)	$Q_{17}$ = (1 6 5 3 9)(4 12 7 8 11)
$M_{18}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{18}$ = (1 2 3 5 4)	$Q_{18}$ = (1 4 8 6 2)(5 7 10 12 9)
$M_{19}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{19}$ = (1 2 4 5 3)	$Q_{19}$ = (1 8 7 3 10)(2 12 5 6 11)
$M_{20}={\begin{bmatrix}0&0&1\\-1&0&0\\0&-1&0\end{bmatrix}}$	$P_{20}$ = (1 2 4)	$Q_{20}$ = (1 7 4)(2 11 8)(3 5 10)(6 9 12)
$M_{21}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{21}$ = (1 2 4 3 5)	$Q_{21}$ = (1 2 9 11 7)(3 6 12 10 4)
$M_{22}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{22}$ = (1 2 5 4 3)	$Q_{22}$ = (2 3 4 7 5)(6 8 10 11 9)
$M_{23}={\begin{bmatrix}0&1&0\\0&0&-1\\-1&0&0\end{bmatrix}}$	$P_{23}$ = (1 2 5)	$Q_{23}$ = (1 9 8)(2 6 3)(4 5 12)(7 11 10)
$M_{24}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{24}$ = (1 2 5 3 4)	$Q_{24}$ = (1 10 5 4 11)(2 8 9 3 12)
$M_{25}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{25}$ = (1 3 2)	$Q_{25}$ = (1 6 11)(2 9 5)(3 12 7)(4 8 10)
$M_{26}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{26}$ = (1 3 4 5 2)	$Q_{26}$ = (2 5 7 4 3)(6 9 11 10 8)
$M_{27}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{27}$ = (1 3 5 4 2)	$Q_{27}$ = (1 10 3 7 8)(2 11 6 5 12)
$M_{28}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{28}$ = (1 3)(4 5)	$Q_{28}$ = (1 7)(2 10)(3 11)(4 5)(6 12)(8 9)
$M_{29}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{29}$ = (1 3 4)	$Q_{29}$ = (1 9 10)(2 12 4)(3 6 8)(5 11 7)
$M_{30}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{30}$ = (1 3 5)	$Q_{30}$ = (1 3 4)(2 8 7)(5 6 10)(9 12 11)
$M_{31}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{31}$ = (1 3)(2 4)	$Q_{31}$ = (1 12)(2 6)(3 9)(4 11)(5 8)(7 10)
$M_{32}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{32}$ = (1 3 2 4 5)	$Q_{32}$ = (1 4 10 11 5)(2 3 8 12 9)
$M_{33}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{33}$ = (1 3 5 2 4)	$Q_{33}$ = (1 5 9 6 3)(4 7 11 12 8)
$M_{34}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{34}$ = (1 3)(2 5)	$Q_{34}$ = (1 2)(3 5)(4 9)(6 7)(8 11)(10 12)
$M_{35}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{35}$ = (1 3 2 5 4)	$Q_{35}$ = (1 11 2 7 9)(3 10 6 4 12)
$M_{36}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{36}$ = (1 3 4 2 5)	$Q_{36}$ = (1 8 2 4 6)(5 10 9 7 12)
$M_{37}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{37}$ = (1 4 5 3 2)	$Q_{37}$ = (1 2 6 8 4)(5 9 12 10 7)
$M_{38}={\begin{bmatrix}0&-1&0\\0&0&-1\\1&0&0\end{bmatrix}}$	$P_{38}$ = (1 4 2)	$Q_{38}$ = (1 4 7)(2 8 11)(3 10 5)(6 12 9)
$M_{39}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{39}$ = (1 4 3 5 2)	$Q_{39}$ = (1 11 4 5 10)(2 12 3 9 8)
$M_{40}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{40}$ = (1 4 3)	$Q_{40}$ = (1 10 9)(2 4 12)(3 8 6)(5 7 11)
$M_{41}={\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}$	$P_{41}$ = (1 4 5)	$Q_{41}$ = (1 5 2)(3 7 9)(4 11 6)(8 10 12)
$M_{42}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{42}$ = (1 4)(3 5)	$Q_{42}$ = (1 6)(2 3)(4 9)(5 8)(7 12)(10 11)
$M_{43}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}$	$P_{43}$ = (1 4 5 2 3)	$Q_{43}$ = (1 9 7 2 11)(3 12 4 6 10)
$M_{44}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{44}$ = (1 4)(2 3)	$Q_{44}$ = (1 8)(2 10)(3 4)(5 12)(6 7)(9 11)
$M_{45}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{45}$ = (1 4 2 3 5)	$Q_{45}$ = (2 7 3 5 4)(6 11 8 9 10)
$M_{46}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{46}$ = (1 4 2 5 3)	$Q_{46}$ = (1 3 6 9 5)(4 8 12 11 7)
$M_{47}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{47}$ = (1 4 3 2 5)	$Q_{47}$ = (1 7 10 8 3)(2 5 11 12 6)
$M_{48}={\begin{bmatrix}-1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}$	$P_{48}$ = (1 4)(2 5)	$Q_{48}$ = (1 12)(2 9)(3 11)(4 10)(5 6)(7 8)
$M_{49}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}$	$P_{49}$ = (1 5 4 3 2)	$Q_{49}$ = (1 9 3 5 6)(4 11 8 7 12)
$M_{50}={\begin{bmatrix}0&0&-1\\1&0&0\\0&-1&0\end{bmatrix}}$	$P_{50}$ = (1 5 2)	$Q_{50}$ = (1 8 9)(2 3 6)(4 12 5)(7 10 11)
$M_{51}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{51}$ = (1 5 3 4 2)	$Q_{51}$ = (1 7 11 9 2)(3 4 10 12 6)
$M_{52}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{52}$ = (1 5 3)	$Q_{52}$ = (1 4 3)(2 7 8)(5 10 6)(9 11 12)
$M_{53}={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}$	$P_{53}$ = (1 5 4)	$Q_{53}$ = (1 2 5)(3 9 7)(4 6 11)(8 12 10)
$M_{54}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{54}$ = (1 5)(3 4)	$Q_{54}$ = (1 12)(2 11)(3 10)(4 8)(5 9)(6 7)
$M_{55}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}$	$P_{55}$ = (1 5 4 2 3)	$Q_{55}$ = (1 5 11 10 4)(2 9 12 8 3)
$M_{56}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}$	$P_{56}$ = (1 5)(2 3)	$Q_{56}$ = (1 10)(2 12)(3 11)(4 7)(5 8)(6 9)
$M_{57}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{57}$ = (1 5 2 3 4)	$Q_{57}$ = (1 3 8 10 7)(2 6 12 11 5)
$M_{58}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{58}$ = (1 5 2 4 3)	$Q_{58}$ = (1 6 4 2 8)(5 12 7 9 10)
$M_{59}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$	$P_{59}$ = (1 5 3 2 4)	$Q_{59}$ = (2 4 5 3 7)(6 10 9 8 11)
$M_{60}={\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}$	$P_{60}$ = (1 5)(2 4)	$Q_{60}$ = (1 11)(2 10)(3 12)(4 9)(5 7)(6 8)

Commonly confused groups edit

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

1\to A_{5}\to S_{5}\to Z_{2}\to 1

I_{h}=A_{5}\times Z_{2}

1\to Z_{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 $2I$

Note that $A_{5}$ has an exceptional irreducible 3-dimensional representation (as the icosahedral rotation group), but $S_{5}$ 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:

$A_{5}\cong \operatorname {PSL} (2,5),$ the projective special linear group, see here for a proof;
$S_{5}\cong \operatorname {PGL} (2,5),$ the projective general linear group;
$2I\cong \operatorname {SL} (2,5),$ the special linear group.

Conjugacy classes edit

The 120 symmetries fall into 10 conjugacy classes.

conjugacy classes
I	additional classes of I_h
identity, order 1 12 × rotation by ±72°, order 5, around the 6 axes through the face centers of the dodecahedron 12 × rotation by ±144°, order 5, around the 6 axes through the face centers of the dodecahedron 20 × rotation by ±120°, order 3, around the 10 axes through vertices of the dodecahedron 15 × rotation by 180°, order 2, around the 15 axes through midpoints of edges of the dodecahedron	central inversion, order 2 12 × rotoreflection by ±36°, order 10, around the 6 axes through the face centers of the dodecahedron 12 × rotoreflection by ±108°, order 10, around the 6 axes through the face centers of the dodecahedron 20 × rotoreflection by ±60°, order 6, around the 10 axes through the vertices of the dodecahedron 15 × reflection, order 2, at 15 planes through edges of the dodecahedron

Subgroups of the full icosahedral symmetry group edit

Subgroup relations

Chiral subgroup relations

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".

Schön.	Coxeter	Orb.	H-M	Structure	Order	Index	Mult.	Description
I_h	[5,3]	*532	532/m	A₅×Z₂	120	1	1	full group
D_2h	[2,2]	*222	mmm	D₄×D₂=D₂³	8	15	5	fixing two opposite edges, possibly swapping them
C_5v	[5]	*55	5m	D₁₀	10	12	6	fixing a face
C_3v	[3]	*33	3m	D₆=S₃	6	20	10	fixing a vertex
C_2v	[2]	*22	2mm	D₄=D₂²	4	30	15	fixing an edge
C_s	[ ]	*	2 or m	D₂	2	60	15	reflection swapping two endpoints of an edge
T_h	[3⁺,4]	3*2	m3	A₄×Z₂	24	5	5	pyritohedral group
D_5d	[2⁺,10]	2*5	10m2	D₂₀=Z₂×D₁₀	20	6	6	fixing two opposite faces, possibly swapping them
D_3d	[2⁺,6]	2*3	3m	D₁₂=Z₂×D₆	12	10	10	fixing two opposite vertices, possibly swapping them
D_1d = C_2h	[2⁺,2]	2*	2/m	D₄=Z₂×D₂	4	30	15	halfturn around edge midpoint, plus central inversion
S₁₀	[2⁺,10⁺]	5×	5	Z₁₀=Z₂×Z₅	10	12	6	rotations of a face, plus central inversion
S₆	[2⁺,6⁺]	3×	3	Z₆=Z₂×Z₃	6	20	10	rotations about a vertex, plus central inversion
S₂	[2⁺,2⁺]	×	1	Z₂	2	60	1	central inversion
I	[5,3]⁺	532	532	A₅	60	2	1	all rotations
T	[3,3]⁺	332	332	A₄	12	10	5	rotations of a contained tetrahedron
D₅	[2,5]⁺	522	522	D₁₀	10	12	6	rotations around the center of a face, and h.t.s.(face)
D₃	[2,3]⁺	322	322	D₆=S₃	6	20	10	rotations around a vertex, and h.t.s.(vertex)
D₂	[2,2]⁺	222	222	D₄=Z₂²	4	30	15	halfturn around edge midpoint, and h.t.s.(edge)
C₅	[5]⁺	55	5	Z₅	5	24	6	rotations around a face center
C₃	[3]⁺	33	3	Z₃=A₃	3	40	10	rotations around a vertex
C₂	[2]⁺	22	2	Z₂	2	60	15	half-turn around edge midpoint
C₁	[ ]⁺	11	1	Z₁	1	120	1	trivial group

Vertex stabilizers edit

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 $D_{3}\times \pm 1$

Edge stabilizers edit

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 $Z_{2}\times Z_{2}$
stabilizers of a pair of edges in I give Klein four-groups $Z_{2}\times Z_{2}$ ; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
stabilizers of a pair of edges in I_h give $Z_{2}\times Z_{2}\times Z_{2}$ ; there are 5 of these, given by reflections in 3 perpendicular axes.

Face stabilizers edit

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 $D_{5}\times \pm 1$

Polyhedron stabilizers edit

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
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 edit

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 $\phi ={\tfrac {{\sqrt {5}}+1}{2}}$ denotes the golden ratio.

[5,3],
	Reflections			Rotations			Rotoreflection
Name	R₀	R₁	R₂	S_0,1	S_1,2	S_0,2	V_0,1,2
Group
Order	2	2	2	5	3	2	10
Matrix	$\left[{\begin{smallmatrix}-1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}-1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {-\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$
	(1,0,0)_n	$({\begin{smallmatrix}{\frac {\phi }{2}},{\frac {1}{2}},{\frac {\phi -1}{2}}\end{smallmatrix}})$ _n	(0,1,0)_n	$(0,-1,\phi )$ _axis	$(1-\phi ,0,\phi )$ _axis	$(0,0,1)$ _axis

Fundamental domain edit

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

Icosahedral rotation group I	Full icosahedral group I_h	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 edit

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

Chiral polyhedra edit

Class	Symbols	Picture
Archimedean	sr{5,3}
Catalan	V3.3.3.3.5

Full icosahedral symmetry edit

Platonic solid	Kepler–Poinsot polyhedra		Archimedean solids
{5,3}	{5/2,5}	{5/2,3}	t{5,3}	t{3,5}	r{3,5}	rr{3,5}	tr{3,5} April 11, 2024 Latest articles January 01, 1970 Burbur, West Azerbaijan January 01, 1970 Burgoyne's Cove January 01, 1970 Burgoyne baronets January 01, 1970 Burgo (food) January 01, 1970 Burgo people January 01, 1970 Burgoon Formation January 01, 1970 Burgauberg-Neudauberg January 01, 1970 Burgas region January 01, 1970 Burgenland – Saalekreis January 01, 1970 Burger King sliders Most Read January 01, 1970 Park Soo-gil January 01, 1970 Park Square Historic District (Franklinville, New York) January 01, 1970 Park Street Press January 01, 1970 Park Street Under January 01, 1970 Park Won-jae icosahedral, symmetry, this, article, includes, list, general, references, lacks, sufficient, corresponding, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, 2020, learn, when, remove, this, template, message, sel. This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations May 2020 Learn how and when to remove this template message Selected point groups in three dimensions Involutional symmetryCs Cyclic symmetryCnv nn n Dihedral symmetryDnh n22 n 2 Polyhedral group n 3 n32 Tetrahedral symmetryTd 332 3 3 Octahedral symmetryOh 432 4 3 Icosahedral symmetryIh 532 5 3 In mathematics and especially in geometry an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron the dual of the icosahedron and the rhombic triacontahedron Icosahedral symmetry fundamental domainsA soccer ball a common example of a spherical truncated icosahedron has full icosahedral symmetry Rotations and reflections form the symmetry group of a great icosahedron 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 full symmetry group is the Coxeter group of type H3 It may be represented by Coxeter notation 5 3 and Coxeter diagram The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 on 5 letters Contents 1 Description 2 As point group 2 1 Visualizations 3 Group structure 3 1 Isomorphism of I with A5 3 2 Commonly confused groups 3 3 Conjugacy classes 3 4 Subgroups of the full icosahedral symmetry group 3 4 1 Vertex stabilizers 3 4 2 Edge stabilizers 3 4 3 Face stabilizers 3 4 4 Polyhedron stabilizers 3 4 5 Coxeter group generators 4 Fundamental domain 5 Polyhedra with icosahedral symmetry 5 1 Chiral polyhedra 5 2 Full icosahedral symmetry 6 Other objects with icosahedral symmetry 6 1 Liquid crystals with icosahedral symmetry 7 Related geometries 8 See also 9 References 10 External linksDescription editIcosahedral symmetry is a mathematical property of objects indicating that an object has the same symmetries as a regular icosahedron As point group editApart 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 Scho Coxeter Orb Abstractstructure OrderI 5 3 nbsp nbsp nbsp nbsp nbsp 532 A5 60Ih 5 3 nbsp nbsp nbsp nbsp nbsp 532 A5 2 120Presentations corresponding to the above are I s t s2 t3 st 5 displaystyle I langle s t mid s 2 t 3 st 5 rangle nbsp Ih s t s3 st 2 t5 st 2 displaystyle I h langle s t mid s 3 st 2 t 5 st 2 rangle nbsp These correspond to the icosahedral groups rotational and full being the 2 3 5 triangle groups The first presentation was given by William Rowan Hamilton in 1856 in his paper on icosian calculus 1 Note that other presentations are possible for instance as an alternating group for I Visualizations edit The full symmetry group is the Coxeter group of type H3 It may be represented by Coxeter notation 5 3 and Coxeter diagram nbsp nbsp nbsp nbsp nbsp The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 on 5 letters Schoe Orb Coxeternotation Elements Mirror diagramsOrthogonal Stereographic projectionIh 532 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 3 Mirrorlines 15 nbsp nbsp nbsp nbsp nbsp I 532 nbsp nbsp nbsp nbsp nbsp nbsp 5 3 Gyrationpoints 125 nbsp 203 nbsp 302 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Group structure editEvery 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 nbsp nbsp The edges of a spherical compound of five octahedra represent the 15 mirror planes as colored great circles Each octahedron can represent 3 orthogonal mirror planes by its edges nbsp nbsp The pyritohedral symmetry is an index 5 subgroup of icosahedral symmetry with 3 orthogonal green reflection lines and 8 red order 3 gyration points There are 5 different orientations of pyritohedral symmetry 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 Th with 20 versions of D3 10 axes 2 per axis and 6 versions of D5 The full icosahedral group 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 identity 1 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 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 D3d and 6 versions of D5d symmetries like antiprisms I is also isomorphic to PSL2 5 but Ih is not isomorphic to SL2 5 Isomorphism of I with A5 edit It is useful to describe explicitly what the isomorphism between I and A5 looks like In the following table permutations Pi and Qi act on 5 and 12 elements respectively while the rotation matrices Mi are the elements of I If Pk is the product of taking the permutation Pi and applying Pj to it then for the same values of i j and k it is also true that Qk is the product of taking Qi and applying Qj and also that premultiplying a vector by Mk is the same as premultiplying that vector by Mi and then premultiplying that result with Mj that is Mk Mj Mi Since the permutations Pi are all the 60 even permutations of 12345 the one to one correspondence is made explicit therefore the isomorphism too Rotation matrix Permutation of 5on 1 2 3 4 5 Permutation of 12on 1 2 3 4 5 6 7 8 9 10 11 12M1 100010001 displaystyle M 1 begin bmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end bmatrix nbsp P1 displaystyle P 1 nbsp Q1 displaystyle Q 1 nbsp M2 1212ϕϕ2 12ϕϕ2 12 ϕ2 12 12ϕ displaystyle M 2 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P2 displaystyle P 2 nbsp 3 4 5 Q2 displaystyle Q 2 nbsp 1 11 8 2 9 6 3 5 12 4 7 10 M3 12 12ϕ ϕ212ϕϕ2 12ϕ2 12 12ϕ displaystyle M 3 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P3 displaystyle P 3 nbsp 3 5 4 Q3 displaystyle Q 3 nbsp 1 8 11 2 6 9 3 12 5 4 10 7 M4 1212ϕ ϕ212ϕ ϕ2 12 ϕ2 1212ϕ displaystyle M 4 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P4 displaystyle P 4 nbsp 2 3 4 5 Q4 displaystyle Q 4 nbsp 1 12 2 8 3 6 4 9 5 10 7 11 M5 ϕ21212ϕ12 12ϕ ϕ2 12ϕϕ2 12 displaystyle M 5 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P5 displaystyle P 5 nbsp 2 3 4 Q5 displaystyle Q 5 nbsp 1 2 3 4 5 6 7 9 8 10 11 12 M6 12ϕ ϕ212ϕ2 12 12ϕ1212ϕϕ2 displaystyle M 6 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P6 displaystyle P 6 nbsp 2 3 5 Q6 displaystyle Q 6 nbsp 1 7 5 2 4 11 3 10 9 6 8 12 M7 ϕ212 12ϕ12 12ϕϕ212ϕ ϕ2 12 displaystyle M 7 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P7 displaystyle P 7 nbsp 2 4 3 Q7 displaystyle Q 7 nbsp 1 3 2 4 6 5 7 8 9 10 12 11 M8 0 10001 100 displaystyle M 8 begin bmatrix 0 amp 1 amp 0 0 amp 0 amp 1 1 amp 0 amp 0 end bmatrix nbsp P8 displaystyle P 8 nbsp 2 4 5 Q8 displaystyle Q 8 nbsp 1 10 6 2 7 12 3 4 8 5 11 9 M9 ϕ21212ϕ1212ϕϕ212ϕϕ2 12 displaystyle M 9 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P9 displaystyle P 9 nbsp 2 4 3 5 Q9 displaystyle Q 9 nbsp 1 9 2 5 3 11 4 12 6 7 8 10 M10 12ϕϕ212 ϕ2 1212ϕ12 12ϕϕ2 displaystyle M 10 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P10 displaystyle P 10 nbsp 2 5 3 Q10 displaystyle Q 10 nbsp 1 5 7 2 11 4 3 9 10 6 12 8 M11 00 1 100010 displaystyle M 11 begin bmatrix 0 amp 0 amp 1 1 amp 0 amp 0 0 amp 1 amp 0 end bmatrix nbsp P11 displaystyle P 11 nbsp 2 5 4 Q11 displaystyle Q 11 nbsp 1 6 10 2 12 7 3 8 4 5 9 11 M12 12ϕ ϕ212 ϕ2 12 12ϕ12 12ϕ ϕ2 displaystyle M 12 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P12 displaystyle P 12 nbsp 2 5 3 4 Q12 displaystyle Q 12 nbsp 1 4 2 10 3 7 5 8 6 11 9 12 M13 1000 1000 1 displaystyle M 13 begin bmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end bmatrix nbsp P13 displaystyle P 13 nbsp 1 2 4 5 Q13 displaystyle Q 13 nbsp 1 3 2 4 5 8 6 7 9 10 11 12 M14 1212ϕϕ212ϕ ϕ212ϕ21212ϕ displaystyle M 14 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P14 displaystyle P 14 nbsp 1 2 3 4 Q14 displaystyle Q 14 nbsp 1 5 2 7 3 11 4 9 6 10 8 12 M15 12 12ϕ ϕ2 12ϕ ϕ212 ϕ21212ϕ displaystyle M 15 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P15 displaystyle P 15 nbsp 1 2 3 5 Q15 displaystyle Q 15 nbsp 1 12 2 10 3 8 4 6 5 11 7 9 M16 12 12ϕϕ212ϕϕ212 ϕ212 12ϕ displaystyle M 16 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P16 displaystyle P 16 nbsp 1 2 3 Q16 displaystyle Q 16 nbsp 1 11 6 2 5 9 3 7 12 4 10 8 M17 12ϕϕ2 12ϕ21212ϕ12 12ϕ ϕ2 displaystyle M 17 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P17 displaystyle P 17 nbsp 1 2 3 4 5 Q17 displaystyle Q 17 nbsp 1 6 5 3 9 4 12 7 8 11 M18 ϕ2 12 12ϕ1212ϕϕ2 12ϕ ϕ212 displaystyle M 18 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P18 displaystyle P 18 nbsp 1 2 3 5 4 Q18 displaystyle Q 18 nbsp 1 4 8 6 2 5 7 10 12 9 M19 12ϕ ϕ2 12 ϕ212 12ϕ1212ϕ ϕ2 displaystyle M 19 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P19 displaystyle P 19 nbsp 1 2 4 5 3 Q19 displaystyle Q 19 nbsp 1 8 7 3 10 2 12 5 6 11 M20 001 1000 10 displaystyle M 20 begin bmatrix 0 amp 0 amp 1 1 amp 0 amp 0 0 amp 1 amp 0 end bmatrix nbsp P20 displaystyle P 20 nbsp 1 2 4 Q20 displaystyle Q 20 nbsp 1 7 4 2 11 8 3 5 10 6 9 12 M21 12ϕϕ2 12 ϕ21212ϕ1212ϕϕ2 displaystyle M 21 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P21 displaystyle P 21 nbsp 1 2 4 3 5 Q21 displaystyle Q 21 nbsp 1 2 9 11 7 3 6 12 10 4 M22 ϕ2 1212ϕ1212ϕ ϕ212ϕϕ212 displaystyle M 22 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P22 displaystyle P 22 nbsp 1 2 5 4 3 Q22 displaystyle Q 22 nbsp 2 3 4 7 5 6 8 10 11 9 M23 01000 1 100 displaystyle M 23 begin bmatrix 0 amp 1 amp 0 0 amp 0 amp 1 1 amp 0 amp 0 end bmatrix nbsp P23 displaystyle P 23 nbsp 1 2 5 Q23 displaystyle Q 23 nbsp 1 9 8 2 6 3 4 5 12 7 11 10 M24 ϕ2 12 12ϕ12 12ϕ ϕ212ϕ ϕ212 displaystyle M 24 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P24 displaystyle P 24 nbsp 1 2 5 3 4 Q24 displaystyle Q 24 nbsp 1 10 5 4 11 2 8 9 3 12 M25 1212ϕ ϕ2 12ϕϕ212ϕ212 12ϕ displaystyle M 25 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P25 displaystyle P 25 nbsp 1 3 2 Q25 displaystyle Q 25 nbsp 1 6 11 2 9 5 3 12 7 4 8 10 M26 ϕ21212ϕ 1212ϕϕ212ϕ ϕ212 displaystyle M 26 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P26 displaystyle P 26 nbsp 1 3 4 5 2 Q26 displaystyle Q 26 nbsp 2 5 7 4 3 6 9 11 10 8 M27 12ϕ ϕ212 ϕ21212ϕ 12 12ϕ ϕ2 displaystyle M 27 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P27 displaystyle P 27 nbsp 1 3 5 4 2 Q27 displaystyle Q 27 nbsp 1 10 3 7 8 2 11 6 5 12 M28 12 12ϕϕ2 12ϕ ϕ2 12ϕ2 1212ϕ displaystyle M 28 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P28 displaystyle P 28 nbsp 1 3 4 5 Q28 displaystyle Q 28 nbsp 1 7 2 10 3 11 4 5 6 12 8 9 M29 12ϕϕ2 12 ϕ2 12 12ϕ 1212ϕϕ2 displaystyle M 29 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P29 displaystyle P 29 nbsp 1 3 4 Q29 displaystyle Q 29 nbsp 1 9 10 2 12 4 3 6 8 5 11 7 M30 ϕ2 12 12ϕ 12 12ϕ ϕ212ϕϕ2 12 displaystyle M 30 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P30 displaystyle P 30 nbsp 1 3 5 Q30 displaystyle Q 30 nbsp 1 3 4 2 8 7 5 6 10 9 12 11 M31 ϕ212 12ϕ1212ϕ ϕ2 12ϕ ϕ2 12 displaystyle M 31 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P31 displaystyle P 31 nbsp 1 3 2 4 Q31 displaystyle Q 31 nbsp 1 12 2 6 3 9 4 11 5 8 7 10 M32 12ϕ ϕ2 12ϕ212 12ϕ12 12ϕϕ2 displaystyle M 32 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P32 displaystyle P 32 nbsp 1 3 2 4 5 Q32 displaystyle Q 32 nbsp 1 4 10 11 5 2 3 8 12 9 M33 1212ϕϕ212ϕϕ2 12 ϕ21212ϕ displaystyle M 33 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P33 displaystyle P 33 nbsp 1 3 5 2 4 Q33 displaystyle Q 33 nbsp 1 5 9 6 3 4 7 11 12 8 M34 12ϕϕ212ϕ2 1212ϕ1212ϕ ϕ2 displaystyle M 34 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P34 displaystyle P 34 nbsp 1 3 2 5 Q34 displaystyle Q 34 nbsp 1 2 3 5 4 9 6 7 8 11 10 12 M35 ϕ2 1212ϕ12 12ϕϕ2 12ϕϕ212 displaystyle M 35 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P35 displaystyle P 35 nbsp 1 3 2 5 4 Q35 displaystyle Q 35 nbsp 1 11 2 7 9 3 10 6 4 12 M36 12 12ϕ ϕ212ϕ ϕ212 ϕ2 12 12ϕ displaystyle M 36 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P36 displaystyle P 36 nbsp 1 3 4 2 5 Q36 displaystyle Q 36 nbsp 1 8 2 4 6 5 10 9 7 12 M37 ϕ212 12ϕ 1212ϕ ϕ2 12ϕϕ212 displaystyle M 37 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P37 displaystyle P 37 nbsp 1 4 5 3 2 Q37 displaystyle Q 37 nbsp 1 2 6 8 4 5 9 12 10 7 M38 0 1000 1100 displaystyle M 38 begin bmatrix 0 amp 1 amp 0 0 amp 0 amp 1 1 amp 0 amp 0 end bmatrix nbsp P38 displaystyle P 38 nbsp 1 4 2 Q38 displaystyle Q 38 nbsp 1 4 7 2 8 11 3 10 5 6 12 9 M39 ϕ21212ϕ 12 12ϕ ϕ2 12ϕ ϕ212 displaystyle M 39 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P39 displaystyle P 39 nbsp 1 4 3 5 2 Q39 displaystyle Q 39 nbsp 1 11 4 5 10 2 12 3 9 8 M40 12ϕ ϕ2 12ϕ2 1212ϕ 12 12ϕϕ2 displaystyle M 40 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P40 displaystyle P 40 nbsp 1 4 3 Q40 displaystyle Q 40 nbsp 1 10 9 2 4 12 3 8 6 5 7 11 M41 001100010 displaystyle M 41 begin bmatrix 0 amp 0 amp 1 1 amp 0 amp 0 0 amp 1 amp 0 end bmatrix nbsp P41 displaystyle P 41 nbsp 1 4 5 Q41 displaystyle Q 41 nbsp 1 5 2 3 7 9 4 11 6 8 10 12 M42 12ϕϕ2 12ϕ2 12 12ϕ 12 12ϕ ϕ2 displaystyle M 42 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P42 displaystyle P 42 nbsp 1 4 3 5 Q42 displaystyle Q 42 nbsp 1 6 2 3 4 9 5 8 7 12 10 11 M43 ϕ212 12ϕ 12 12ϕϕ212ϕϕ212 displaystyle M 43 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P43 displaystyle P 43 nbsp 1 4 5 2 3 Q43 displaystyle Q 43 nbsp 1 9 7 2 11 3 12 4 6 10 M44 12ϕ ϕ2 12 ϕ2 1212ϕ 1212ϕ ϕ2 displaystyle M 44 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P44 displaystyle P 44 nbsp 1 4 2 3 Q44 displaystyle Q 44 nbsp 1 8 2 10 3 4 5 12 6 7 9 11 M45 1212ϕϕ2 12ϕ ϕ212ϕ2 12 12ϕ displaystyle M 45 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P45 displaystyle P 45 nbsp 1 4 2 3 5 Q45 displaystyle Q 45 nbsp 2 7 3 5 4 6 11 8 9 10 M46 1212ϕ ϕ212ϕϕ212ϕ2 1212ϕ displaystyle M 46 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P46 displaystyle P 46 nbsp 1 4 2 5 3 Q46 displaystyle Q 46 nbsp 1 3 6 9 5 4 8 12 11 7 M47 12 12ϕϕ2 12ϕϕ212 ϕ2 1212ϕ displaystyle M 47 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P47 displaystyle P 47 nbsp 1 4 3 2 5 Q47 displaystyle Q 47 nbsp 1 7 10 8 3 2 5 11 12 6 M48 10001000 1 displaystyle M 48 begin bmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end bmatrix nbsp P48 displaystyle P 48 nbsp 1 4 2 5 Q48 displaystyle Q 48 nbsp 1 12 2 9 3 11 4 10 5 6 7 8 M49 12ϕϕ212ϕ212 12ϕ 1212ϕ ϕ2 displaystyle M 49 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P49 displaystyle P 49 nbsp 1 5 4 3 2 Q49 displaystyle Q 49 nbsp 1 9 3 5 6 4 11 8 7 12 M50 00 11000 10 displaystyle M 50 begin bmatrix 0 amp 0 amp 1 1 amp 0 amp 0 0 amp 1 amp 0 end bmatrix nbsp P50 displaystyle P 50 nbsp 1 5 2 Q50 displaystyle Q 50 nbsp 1 8 9 2 3 6 4 12 5 7 10 11 M51 12ϕ ϕ212ϕ21212ϕ 1212ϕϕ2 displaystyle M 51 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P51 displaystyle P 51 nbsp 1 5 3 4 2 Q51 displaystyle Q 51 nbsp 1 7 11 9 2 3 4 10 12 6 M52 ϕ2 1212ϕ 12 12ϕϕ2 12ϕ ϕ2 12 displaystyle M 52 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P52 displaystyle P 52 nbsp 1 5 3 Q52 displaystyle Q 52 nbsp 1 4 3 2 7 8 5 10 6 9 11 12 M53 010001100 displaystyle M 53 begin bmatrix 0 amp 1 amp 0 0 amp 0 amp 1 1 amp 0 amp 0 end bmatrix nbsp P53 displaystyle P 53 nbsp 1 5 4 Q53 displaystyle Q 53 nbsp 1 2 5 3 9 7 4 6 11 8 12 10 M54 ϕ2 12 12ϕ 1212ϕϕ2 12ϕϕ2 12 displaystyle M 54 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P54 displaystyle P 54 nbsp 1 5 3 4 Q54 displaystyle Q 54 nbsp 1 12 2 11 3 10 4 8 5 9 6 7 M55 12ϕϕ212 ϕ212 12ϕ 12 12ϕϕ2 displaystyle M 55 begin bmatrix frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 end bmatrix nbsp P55 displaystyle P 55 nbsp 1 5 4 2 3 Q55 displaystyle Q 55 nbsp 1 5 11 10 4 2 9 12 8 3 M56 ϕ2 1212ϕ 1212ϕ ϕ212ϕ ϕ2 12 displaystyle M 56 begin bmatrix frac phi 2 amp frac 1 2 amp frac 1 2 phi frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 end bmatrix nbsp P56 displaystyle P 56 nbsp 1 5 2 3 Q56 displaystyle Q 56 nbsp 1 10 2 12 3 11 4 7 5 8 6 9 M57 12 12ϕ ϕ2 12ϕϕ2 12ϕ21212ϕ displaystyle M 57 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P57 displaystyle P 57 nbsp 1 5 2 3 4 Q57 displaystyle Q 57 nbsp 1 3 8 10 7 2 6 12 11 5 M58 1212ϕ ϕ2 12ϕ ϕ2 12 ϕ212 12ϕ displaystyle M 58 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P58 displaystyle P 58 nbsp 1 5 2 4 3 Q58 displaystyle Q 58 nbsp 1 6 4 2 8 5 12 7 9 10 M59 12 12ϕϕ212ϕ ϕ2 12ϕ212 12ϕ displaystyle M 59 begin bmatrix frac 1 2 amp frac 1 2 phi amp frac phi 2 frac 1 2 phi amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 2 phi end bmatrix nbsp P59 displaystyle P 59 nbsp 1 5 3 2 4 Q59 displaystyle Q 59 nbsp 2 4 5 3 7 6 10 9 8 11 M60 1000 10001 displaystyle M 60 begin bmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end bmatrix nbsp P60 displaystyle P 60 nbsp 1 5 2 4 Q60 displaystyle Q 60 nbsp 1 11 2 10 3 12 4 9 5 7 6 8 Commonly confused groups edit The following groups all have order 120 but are not isomorphic S5 the symmetric group on 5 elements Ih the full icosahedral group subject of this article also known as H3 2I the binary icosahedral groupThey correspond to the following short exact sequences the latter of which does not split and product 1 A5 S5 Z2 1 displaystyle 1 to A 5 to S 5 to Z 2 to 1 nbsp Ih A5 Z2 displaystyle I h A 5 times Z 2 nbsp 1 Z2 2I A5 1 displaystyle 1 to Z 2 to 2I to A 5 to 1 nbsp In words A5 displaystyle A 5 nbsp is a normal subgroup of S5 displaystyle S 5 nbsp A5 displaystyle A 5 nbsp is a factor of Ih displaystyle I h nbsp which is a direct product A5 displaystyle A 5 nbsp is a quotient group of 2I displaystyle 2I nbsp Note that A5 displaystyle A 5 nbsp has an exceptional irreducible 3 dimensional representation as the icosahedral rotation group but S5 displaystyle S 5 nbsp 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 A5 PSL 2 5 displaystyle A 5 cong operatorname PSL 2 5 nbsp the projective special linear group see here for a proof S5 PGL 2 5 displaystyle S 5 cong operatorname PGL 2 5 nbsp the projective general linear group 2I SL 2 5 displaystyle 2I cong operatorname SL 2 5 nbsp the special linear group Conjugacy classes edit The 120 symmetries fall into 10 conjugacy classes conjugacy classes I additional classes of Ihidentity order 1 12 rotation by 72 order 5 around the 6 axes through the face centers of the dodecahedron 12 rotation by 144 order 5 around the 6 axes through the face centers of the dodecahedron 20 rotation by 120 order 3 around the 10 axes through vertices of the dodecahedron 15 rotation by 180 order 2 around the 15 axes through midpoints of edges of the dodecahedron central inversion order 2 12 rotoreflection by 36 order 10 around the 6 axes through the face centers of the dodecahedron 12 rotoreflection by 108 order 10 around the 6 axes through the face centers of the dodecahedron 20 rotoreflection by 60 order 6 around the 10 axes through the vertices of the dodecahedron 15 reflection order 2 at 15 planes through edges of the dodecahedronSubgroups of the full icosahedral symmetry group edit nbsp Subgroup relations nbsp Chiral subgroup relationsEach 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 Schon Coxeter Orb H M Structure Cyc Order Index Mult DescriptionIh 5 3 nbsp nbsp nbsp nbsp nbsp 532 53 2 m A5 Z2 120 1 1 full groupD2h 2 2 nbsp nbsp nbsp nbsp nbsp 222 mmm D4 D2 D23 nbsp 8 15 5 fixing two opposite edges possibly swapping themC5v 5 nbsp nbsp nbsp 55 5m D10 nbsp 10 12 6 fixing a faceC3v 3 nbsp nbsp nbsp 33 3m D6 S3 nbsp 6 20 10 fixing a vertexC2v 2 nbsp nbsp nbsp 22 2mm D4 D22 nbsp 4 30 15 fixing an edgeCs nbsp 2 or m D2 nbsp 2 60 15 reflection swapping two endpoints of an edgeTh 3 4 nbsp nbsp nbsp nbsp nbsp 3 2 m3 A4 Z2 nbsp 24 5 5 pyritohedral groupD5d 2 10 nbsp nbsp nbsp nbsp nbsp 2 5 10 m2 D20 Z2 D10 nbsp 20 6 6 fixing two opposite faces possibly swapping themD3d 2 6 nbsp nbsp nbsp nbsp nbsp 2 3 3 m D12 Z2 D6 nbsp 12 10 10 fixing two opposite vertices possibly swapping themD1d C2h 2 2 nbsp nbsp nbsp nbsp nbsp 2 2 m D4 Z2 D2 nbsp 4 30 15 halfturn around edge midpoint plus central inversionS10 2 10 nbsp nbsp nbsp nbsp nbsp 5 5 Z10 Z2 Z5 nbsp 10 12 6 rotations of a face plus central inversionS6 2 6 nbsp nbsp nbsp nbsp nbsp 3 3 Z6 Z2 Z3 nbsp 6 20 10 rotations about a vertex plus central inversionS2 2 2 nbsp nbsp nbsp nbsp nbsp 1 Z2 nbsp 2 60 1 central inversionI 5 3 nbsp nbsp nbsp nbsp nbsp 532 532 A5 60 2 1 all rotationsT 3 3 nbsp nbsp nbsp nbsp nbsp 332 332 A4 nbsp 12 10 5 rotations of a contained tetrahedronD5 2 5 nbsp nbsp nbsp nbsp nbsp 522 522 D10 nbsp 10 12 6 rotations around the center of a face and h t s face D3 2 3 nbsp nbsp nbsp nbsp nbsp 322 322 D6 S3 nbsp 6 20 10 rotations around a vertex and h t s vertex D2 2 2 nbsp nbsp nbsp nbsp nbsp 222 222 D4 Z22 nbsp 4 30 15 halfturn around edge midpoint and h t s edge C5 5 nbsp nbsp nbsp 55 5 Z5 nbsp 5 24 6 rotations around a face centerC3 3 nbsp nbsp nbsp 33 3 Z3 A3 nbsp 3 40 10 rotations around a vertexC2 2 nbsp nbsp nbsp 22 2 Z2 nbsp 2 60 15 half turn around edge midpointC1 nbsp 11 1 Z1 nbsp 1 120 1 trivial groupVertex stabilizers edit 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 Ih 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 Ih give D3 1 displaystyle D 3 times pm 1 nbsp Edge stabilizers edit 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 Ih give Klein four groups Z2 Z2 displaystyle Z 2 times Z 2 nbsp stabilizers of a pair of edges in I give Klein four groups Z2 Z2 displaystyle Z 2 times Z 2 nbsp there are 5 of these given by rotation by 180 in 3 perpendicular axes stabilizers of a pair of edges in Ih give Z2 Z2 Z2 displaystyle Z 2 times Z 2 times Z 2 nbsp there are 5 of these given by reflections in 3 perpendicular axes Face stabilizers edit 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 Ih 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 Ih give D5 1 displaystyle D 5 times pm 1 nbsp Polyhedron stabilizers edit For each of these there are 5 conjugate copies and the conjugation action gives a map indeed an isomorphism I A5 lt S5 displaystyle I stackrel sim to A 5 lt S 5 nbsp stabilizers of the inscribed tetrahedra in I are a copy of T stabilizers of the inscribed tetrahedra in Ih 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 Ih are a copy of ThCoxeter group generators edit The full icosahedral symmetry group 5 3 nbsp nbsp nbsp nbsp nbsp 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 nbsp nbsp nbsp nbsp nbsp 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 ϕ 5 12 displaystyle phi tfrac sqrt 5 1 2 nbsp denotes the golden ratio 5 3 nbsp nbsp nbsp nbsp nbsp Reflections Rotations RotoreflectionName R0 R1 R2 S0 1 S1 2 S0 2 V0 1 2Group nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Order 2 2 2 5 3 2 10Matrix 100010001 displaystyle left begin smallmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end smallmatrix right nbsp 1 ϕ2 ϕ2 12 ϕ2121 ϕ2 121 ϕ2ϕ2 displaystyle left begin smallmatrix frac 1 phi 2 amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 phi 2 frac 1 2 amp frac 1 phi 2 amp frac phi 2 end smallmatrix right nbsp 1000 10001 displaystyle left begin smallmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end smallmatrix right nbsp ϕ 12ϕ212 ϕ2121 ϕ2 121 ϕ2ϕ2 displaystyle left begin smallmatrix frac phi 1 2 amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 phi 2 frac 1 2 amp frac 1 phi 2 amp frac phi 2 end smallmatrix right nbsp 1 ϕ2ϕ2 12 ϕ2 121 ϕ2 12ϕ 12ϕ2 displaystyle left begin smallmatrix frac 1 phi 2 amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 phi 2 frac 1 2 amp frac phi 1 2 amp frac phi 2 end smallmatrix right nbsp 1000 10001 displaystyle left begin smallmatrix 1 amp 0 amp 0 0 amp 1 amp 0 0 amp 0 amp 1 end smallmatrix right nbsp ϕ 12 ϕ212 ϕ2 121 ϕ2 12ϕ 12ϕ2 displaystyle left begin smallmatrix frac phi 1 2 amp frac phi 2 amp frac 1 2 frac phi 2 amp frac 1 2 amp frac 1 phi 2 frac 1 2 amp frac phi 1 2 amp frac phi 2 end smallmatrix right nbsp 1 0 0 n ϕ2 12 ϕ 12 displaystyle begin smallmatrix frac phi 2 frac 1 2 frac phi 1 2 end smallmatrix nbsp n 0 1 0 n 0 1 ϕ displaystyle 0 1 phi nbsp axis 1 ϕ 0 ϕ displaystyle 1 phi 0 phi nbsp axis 0 0 1 displaystyle 0 0 1 nbsp axisFundamental domain editFundamental domains for the icosahedral rotation group and the full icosahedral group are given by nbsp Icosahedral rotation groupI nbsp Full icosahedral groupIh nbsp Faces of disdyakis triacontahedron are the fundamental domainIn 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 editFurther information Solids with icosahedral symmetryExamples of other polyhedra with icosahedral symmetry include the regular dodecahedron the dual of the icosahedron and the rhombic triacontahedron Chiral polyhedra edit Class Symbols PictureArchimedean sr 5 3 nbsp nbsp nbsp nbsp nbsp nbsp Catalan V3 3 3 3 5 nbsp nbsp nbsp nbsp nbsp nbsp Full icosahedral symmetry edit Platonic solid Kepler Poinsot polyhedra Archimedean solids nbsp 5 3 nbsp nbsp nbsp nbsp nbsp nbsp 5 2 5 nbsp nbsp nbsp nbsp nbsp nbsp 5 2 3 nbsp nbsp nbsp nbsp nbsp nbsp t 5 3 nbsp nbsp nbsp nbsp nbsp nbsp t 3 5 nbsp nbsp nbsp nbsp nbsp nbsp r 3 5 nbsp nbsp nbsp nbsp nbsp nbsp rr 3 5 nbsp nbsp nbsp nbsp nbsp nbsp tr 3 5 nbsp nbsp nbsp 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