nLab Platonic solid

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 $\mathbb{R}^3$ 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	dihedron, 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 $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$, $Spin(2n)$
$E_6$	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

Related concepts

• Johannes Kepler, Music of the Spheres

• Plato’s Timaeus

• 120-cell

References

The Platonic solids are named after their discussion in

• Plato, Timaeus dialogue

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

• Euclid, Book XIII of the Elements, $\sim$300 BC

Modern textbook accounts:

• Klaus Lamotke, section 1 of: Regular Solids and Isolated Singularities, Vieweg (1986)

• Mark A. Armstrong, chapter 8 in: Groups and Symmetry, Undergraduate Texts in Mathematics, Springer (1988) [doi:10.1007/978-1-4757-4034-9, pdf]

• Elmer Rees, from p. 23 (32 of 124) on in: Notes on Geometry, Springer (2005)

See also

• Wikipedia, Platonic solid