A dihedron is a degenerate Platonic solid with only two (identical) faces, which can be any polygon (including possibly the degenerate 2-gon, which would make the dihedron very degenerate!)
ADE classification and McKay correspondence
Dynkin diagram/ Dynkin quiver | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|
$A_{n \geq 1}$ | 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, chapter I.4 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, Dihedron
Last revised on April 16, 2018 at 02:06:02. See the history of this page for a list of all contributions to it.