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:
- Regular Tetrahedron
-
It has 6 edges and 4 vertices and 4 faces, each one being an equilateral triangle. Its symmetry group is .
- Regular Hexahedron
-
Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is .
- Regular Octahedron
-
It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is .
- Regular Dodecahedron
-
It has 20 vertices, 30 edges and 12 faces, each one being a regular pentagon. Its symmetry group is .
- Regular Icosahedron
-
It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is .
Note: is the alternating group of order , is the symmetric group of order and is the cyclic group with order .
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 | tetrahedron |
Defines | octahedron |
Defines | dodecahedron |
Defines | icosahedron |
Defines | regular tetrahedron |
Defines | regular octahedron |
Defines | regular dodecahedron |
Defines | regular icosahedron |