Icosahedral symmetry

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.

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 . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters.

Description

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

As point group

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.

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)^, t^5(st)^\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.

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

Visualizations

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

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 ''I'' is of order 60. The group ''I'' is isomorphic to ''A''₅, 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 ''I_h'' has order 120. It has ''I'' as normal subgroup of index 2. The group ''I_h'' is isomorphic to ''I'' × ''Z''₂, or ''A''₅ × ''Z''₂, with the inversion in the center corresponding to element (identity,-1), where ''Z''₂ 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.

{, class="wikitable collapsible collapsed" align='center' style="font-family:'DejaVu Sans Mono','monospace'"
!width="25%", Rotation matrix
!width="25%", Permutation of 5
on 1 2 3 4 5
!width="50%", 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

The following groups all have order 120, but are not isomorphic:
* ''S''₅, the symmetric group on 5 elements
* ''I_h'', the full icosahedral group (subject of this article, also known as ''H''₃)
* 2''I'', 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

The 120 symmetries fall into 10 conjugacy classes.
{, class=wikitable
, + 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

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".

{, class="wikitable sortable"
! Schön., , colspan=2, Coxeter, , Orb., , H-M, , Structure, , Cyc., , Order, , Index, , Mult., , Description
, - align=center BGCOLOR="#e0f0f0"
, I_h, , ,3, , , *532, , 2/m, , A₅×Z₂, , , , 120, , 1, , 1, , full group
, - align=center BGCOLOR="#e0f0f0"
, D_2h, , ,2, , , *222, , mmm, , D₄×D₂=D₂³, , , , 8, , 15, , 5, , fixing two opposite edges, possibly swapping them
, -align=center BGCOLOR="#e0f0f0"
, C_5v , , , , , *55 , , 5m, , D₁₀, , , , 10 , , 12, , 6, , fixing a face
, -align=center BGCOLOR="#e0f0f0"
, C_3v , , , , , *33 , , 3m, , D₆=S₃, , , , 6 , , 20, , 10, , fixing a vertex
, -align=center BGCOLOR="#e0f0f0"
, C_2v , , , , , *22 , , 2mm, , D₄=D₂², , , , 4 , , 30, , 15, , fixing an edge
, -align=center BGCOLOR="#e0f0f0"
, C_s , , nbsp;, , , * , , or m, , D₂, , , , 2 , , 60, , 15, , reflection swapping two endpoints of an edge
, - align=center BGCOLOR="#f0f0e0"
, T_h, , ⁺,4, , , 3*2, , m, , A₄×Z₂, , , , 24, , 5, , 5, , pyritohedral group
, -align=center BGCOLOR="#f0f0e0"
, D_5d , , ⁺,10, , , 2*5 , , m2, , D₂₀=Z₂×D₁₀, , , , 20 , , 6, , 6, , fixing two opposite faces, possibly swapping them
, -align=center BGCOLOR="#f0f0e0"
, D_3d , , ⁺,6, , , 2*3 , , m, , D₁₂=Z₂×D₆, , , , 12 , , 10, , 10, , fixing two opposite vertices, possibly swapping them
, -align=center BGCOLOR="#f0f0e0"
, D_1d = C_2h , , ⁺,2, , , 2* , , 2/m, , D₄= Z₂×D₂, , , , 4 , , 30, , 15, , halfturn around edge midpoint, plus central inversion
, -align=center BGCOLOR="#e0e0e0"
, S₁₀ , , ⁺,10⁺, , , 5× , , , , Z₁₀=Z₂×Z₅, , , , 10 , , 12, , 6, , rotations of a face, plus central inversion
, -align=center BGCOLOR="#e0e0e0"
, S₆ , , ⁺,6⁺, , , 3× , , , , Z₆=Z₂×Z₃, , , , 6 , , 20, , 10, , rotations about a vertex, plus central inversion
, -align=center BGCOLOR="#e0e0e0"
, S₂ , , ⁺,2⁺, , , × , , , , Z₂, , , , 2 , , 60, , 1, , central inversion
, -align=center BGCOLOR="#f0e0f0"
, I, , ,3sup>+, , , , 532, , 532, , A₅, , , , 60, , 2, , 1, , all rotations
, - align=center BGCOLOR="#f0e0f0"
, T, , ,3sup>+, , , , 332, , 332, , A₄ , , , , 12, , 10, , 5, , rotations of a contained tetrahedron
, - align=center BGCOLOR="#f0e0f0"
, D₅, , ,5sup>+, , , , 522, , 522, , D₁₀, , , , 10, , 12, , 6, , rotations around the center of a face, and h.t.s.(face)
, - align=center BGCOLOR="#f0e0f0"
, D₃, , ,3sup>+, , , , 322, , 322, , D₆=S₃, , , , 6, , 20, , 10, , rotations around a vertex, and h.t.s.(vertex)
, - align=center BGCOLOR="#f0e0f0"
, D₂, , ,2sup>+, , , , 222, , 222, , D₄=Z₂², , , , 4, , 30, , 15, , halfturn around edge midpoint, and h.t.s.(edge)
, - align=center BGCOLOR="#f0e0f0"
, C₅, , sup>+, , , , 55, , 5, , Z₅, , , , 5, , 24, , 6, , rotations around a face center
, - align=center BGCOLOR="#f0e0f0"
, C₃, , sup>+, , , , 33, , 3, , Z₃=A₃, , , , 3, , 40, , 10, , rotations around a vertex
, - align=center BGCOLOR="#f0e0f0"
, C₂, , sup>+, , , , 22, , 2, , Z₂, , , , 2, , 60, , 15, , half-turn around edge midpoint
, - align=center BGCOLOR="#f0e0f0"
, C₁, , nbsp;sup>+, , , , 11, , 1, , Z₁, , , , 1, , 120, , 1, , trivial group

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 ''C''₃
* vertex stabilizers in ''I_h'' give dihedral groups ''D''₃
* stabilizers of an opposite pair of vertices in ''I'' give dihedral groups ''D''₃
* stabilizers of an opposite pair of vertices in ''I_h'' give $D_3 \times \pm 1$

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 ''Z''₂
* 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

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 ''C''₅
* face stabilizers in ''I_h'' give dihedral groups ''D''₅
* stabilizers of an opposite pair of faces in ''I'' give dihedral groups ''D''₅
* stabilizers of an opposite pair of faces in ''I_h'' give $D_5 \times \pm 1$

Polyhedron stabilizers

For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, $I \stackrel{\sim}\to A_5 < S_5$.
* 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 ,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 ,3sup>+ () 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.

{, class=wikitable
, + ,3
!
!colspan=3, Reflections
!colspan=3, Rotations
!Rotoreflection
, -
!Name
! R₀
! R₁
! R₂
! S_0,1
! S_1,2
! S_0,2
! V_0,1,2
, - align=center
!Group
,
,
,
,
,
,
,
, - align=center
!Order
, 2, , 2, , 2, , 5, , 3, , 2, , 10
, - align=center
!Matrix
, \left \begin{smallmatrix} -1&0&0\\ 0&1&0\\ 0&0&1\end{smallmatrix} \right/math>
, $\left[ \begin{smallmatrix} {\frac {1-\phi}{2&{\frac {-\phi}{2&{\frac {-1}{2\\ {\frac {-\phi}{2&{\frac {1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {1-\phi}{2&{\frac {\phi}{2\end{smallmatrix} \right]$
, $\left[ \begin{smallmatrix} 1&0&0\\ 0&-1&0\\ 0&0&1\end{smallmatrix} \right]$
, $\left[ \begin{smallmatrix} {\frac {\phi-1}{2&{\frac {\phi}{2&{\frac {1}{2\\ {\frac {-\phi}{2&{\frac {1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {1-\phi}{2&{\frac {\phi}{2\end{smallmatrix} \right]$
, $\left[ \begin{smallmatrix} {\frac {1-\phi}{2&{\frac {\phi}{2&{\frac {-1}{2\\ {\frac {-\phi}{2&{\frac {-1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {\phi-1}{2&{\frac {\phi}{2\end{smallmatrix} \right]$
, $\left[ \begin{smallmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\end{smallmatrix} \right]$
, $\left[ \begin{smallmatrix} {\frac {\phi-1}{2&{\frac {-\phi}{2&{\frac {1}{2\\ {\frac {-\phi}{2&{\frac {-1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {\phi-1}{2&{\frac {\phi}{2\end{smallmatrix} \right]$
, - align=center
!
, (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

Fundamental domains for the icosahedral rotation group and the full icosahedral group are given by:
{, class=wikitable width=580
, - align=center valign=top
,
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

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

Chiral polyhedra

{, class=wikitable
!Class
! Symbols
! Picture
, -align=center
! Archimedean
! sr{5,3}

, 50px
, -align=center
!Catalan
! V3.3.3.3.5

, 50px

Full icosahedral symmetry

{, class=wikitable
!colspan=1, Platonic solid, , colspan=2, Kepler–Poinsot polyhedra
!colspan=5, Archimedean solids
, - align=center
,
{5,3}

,
{5/2,5}

,
{5/2,3}

,
t{5,3}

,
t{3,5}

,
r{3,5}

,
rr{3,5}

,
tr{3,5}

, - align=center
!Platonic solid, , colspan=2, Kepler–Poinsot polyhedra
!colspan=5, Catalan solids
, - align=center
,
{3,5}
=
,
{5,5/2}
=
,
{3,5/2}
=
,
V3.10.10

,
V5.6.6

,
V3.5.3.5

,
V3.4.5.4

,
V4.6.10

Other objects with icosahedral symmetry

* Barth surfaces
* Virus structure, and Capsid
* In chemistry, the dodecaborate ion ( ₁₂H₁₂sup>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 articl
here

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

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.

This arose from his efforts to give a geometric setting for why icosahedral symmetry arose in the solution of the quintic equation, with the theory given in the famous ; a modern exposition is given in .

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

See also

* Tetrahedral symmetry
* Octahedral symmetry
* Binary icosahedral group
* Icosian calculus

References

* Translated in
*
*
*
* Peter R. Cromwell, ''Polyhedra'' (1997), p. 296
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,
* ''Kaleidoscopes: Selected Writings of H.S.M. Coxeter'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

* Norman Johnson (mathematician), N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups

External links

*

THE SUBGROUPS OF W(H3)


{{Webarchive, url=https://web.archive.org/web/20200802022826/http://schmidt.nuigalway.ie/subgroups/cox.html , date=2020-08-02 ) Gotz Pfeiffer

Finite groups
Rotational symmetry