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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 diagram via
McKay correspondence
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A nA_ncyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
D n+4D_{n+4}dihedron,
hosohedron
dihedral group
D n+2D_{n+2}
binary dihedral group
2D n+22 D_{n+2}
special orthogonal group
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

References

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 April 15, 2018 at 14:01:33. See the history of this page for a list of all contributions to it.