The five regular convex polyhedron in 3-dimensional Cartesian space:
tetrahedron, cube, octahedron, dodecahedron, icosahedron
Regarding a Platonic solid, determined uniquely up to isometry on $\mathbb{R}^3$ as a regular convex polyhedron, as a metric subspace $S$ of $\mathbb{R}^3$. Then is symmetry group may be defined as the group of isometries of $S$.
The groups arising this way are called the groups of ADE-type:
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 |
The Platonic solids are named after their discussion in
Their construction and the proof that there is exactly five of them appears in
Modern textbook accounts include
Klaus Lamotke, section 1 of Regular Solids and Isolated Singularities, Vieweg 1986
Elmer Rees, from p. 23 (32 of 124) on in Notes on Geometry, Springer 2005
See also
Last revised on December 7, 2018 at 04:44:35. See the history of this page for a list of all contributions to it.