# Contents

## Idea

The five regular convex polyhedron in 3-dimensional Cartesian space:

## Platonic symmetry groups

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:

Dynkin diagramPlatonic solidfinite subgroup of $SO(3)$finite subgroup of $SU(2)$simple Lie group
$A_l$cyclic groupcyclic groupspecial unitary group
$D_l$dihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
$E_6$tetrahedrontetrahedral groupbinary tetrahedral groupE6
$E_7$cube/octahedronoctahedral groupbinary octahedral groupE7
$E_8$dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8

## References

The Platonic solids are named after their discussion in

Their construction and the proof that there is exactly five of them appears in