nLab
Platonic solid

Contents

Idea

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

tetrahedron, cube, octahedron, dodecahedron, icosahedron

Platonic symmetry groups

Regarding a Platonic solid, determined uniquely up to isometry on 3\mathbb{R}^3 as a regular convex polyhedron, as a metric subspace SS of 3\mathbb{R}^3. Then is symmetry group may be defined as the group of isometries of SS.

The groups arising this way are called the groups of ADE-type:

ADE classification

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)finite subgroup of SU(2)SU(2)simple Lie group
A lA_lcyclic groupcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupbinary tetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupbinary octahedral groupE7
E 8E_8dodecahedron/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

See also

Revised on July 28, 2016 12:10:45 by Urs Schreiber (131.220.184.222)