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
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 09:44:35. See the history of this page for a list of all contributions to it.