The regular icosahedron is the unique regular convex polyhedron with 20 (triangular) faces. One of the Platonic solids.
Its symmetry group is called the icosahedral group.
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ADE classification and McKay correspondence
Dynkin diagram/ Dynkin quiver | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
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$A_{n \geq 1}$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group $SU(n+1)$ | |
D4 | Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | quaternion group $2 D_4 \simeq$ Q8 | SO(8) | |
$D_{n \geq 4}$ | dihedron, hosohedron | dihedral group $D_{2(n-2)}$ | binary dihedral group $2 D_{2(n-2)}$ | special orthogonal group $SO(2n)$ |
$E_6$ | tetrahedron | tetrahedral group $T$ | binary tetrahedral group $2T$ | E6 |
$E_7$ | cube, octahedron | octahedral group $O$ | binary octahedral group $2O$ | E7 |
$E_8$ | dodecahedron, icosahedron | icosahedral group $I$ | binary icosahedral group $2I$ | E8 |
Felix Klein, chater I.8 of Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, 1884, translated as Lectures on the Icosahedron and the Resolution of Equations of Degree Five by George Morrice 1888, online version
Wikipedia, Icosahedron
Last revised on October 9, 2018 at 08:41:30. See the history of this page for a list of all contributions to it.