# regular polyhedron

A regular polyhedron is a polyhedron such that

• Every face is a regular polygon.

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

• The dihedral angle 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 dodecahedron.

The five solids are:

It has 6 edges and 4 vertices and 4 faces, each one being an equilateral triangle. Its symmetry group is $S_{4}$.

Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is $S_{4}\times C_{2}$.

Regular Octahedron

It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is $S_{4}\times C_{2}$.

Regular Dodecahedron

It has 20 vertices, 30 edges and 12 faces, each one being a regular pentagon. Its symmetry group is $A_{5}\times C_{2}$.

Regular Icosahedron

It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is $A_{5}\times C_{2}$.

Note: $A_{n}$ is the alternating group of order $n$, $S_{n}$ is the symmetric group of order $n$ and $C_{n}$ is the cyclic group 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 Defines Defines dodecahedron Defines Defines regular tetrahedron Defines regular octahedron Defines regular dodecahedron Defines regular icosahedron