regular polyhedron

A regular polyhedronMathworldPlanetmathPlanetmath is a polyhedron such that

  • Every face is a regular polygonMathworldPlanetmath.

  • On each vertex, the same number of edges concur.

  • The dihedral angleMathworldPlanetmath between any two faces is always the same.

These polyhedra are also known as Platonic solids, since Plato described them in his work. There are only 5 regular polyhedra, as was first shown by Theaetetus, one of Plato’s students. Some sources ascribe to Theaetetus also the discovery of the dodecahedronMathworldPlanetmath.

The five solids are:

Regular TetrahedronMathworldPlanetmathPlanetmath

It has 6 edges and 4 vertices and 4 faces, each one being an equilateral triangleMathworldPlanetmath. Its symmetry group is S4.

RegularPlanetmathPlanetmath HexahedronMathworldPlanetmath

Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is S4×C2.

Regular Octahedron

It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is S4×C2.

Regular Dodecahedron

It has 20 vertices, 30 edges and 12 faces, each one being a regular pentagon. Its symmetry group is A5×C2.

Regular Icosahedron

It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is A5×C2.

Figure 1: The five Platonic solids – created in Blender 2.36. (Download the \htmladdnormallinkBlender file for this picture.)

Note: An is the alternating groupMathworldPlanetmath of order n, Sn is the symmetric groupMathworldPlanetmathPlanetmath of order n and Cn is the cyclic groupMathworldPlanetmath with order n.

Title regular polyhedron
Canonical name RegularPolyhedron
Date of creation 2013-03-22 12:24:17
Last modified on 2013-03-22 12:24:17
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 20
Author mathwizard (128)
Entry type Definition
Classification msc 51-00
Synonym Platonic solid
Synonym regular polyhedra
Synonym regular
Related topic RegularPolygon
Related topic Grafix
Defines tetrahedronMathworldPlanetmath
Defines octahedronMathworldPlanetmath
Defines dodecahedron
Defines icosahedronMathworldPlanetmath
Defines regular tetrahedron
Defines regular octahedron
Defines regular dodecahedron
Defines regular icosahedron