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regular polyhedron
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(Definition)
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A regular polyhedron is a polyhedron such that
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
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- Regular Hexahedron
- Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is
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- Regular Octahedron
- It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is
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- Regular Dodecahedron
- It has 20 vertices, 30 edges and 12 faces, each one being a regular pentagon. Its symmetry group is
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- Regular Icosahedron
- It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is
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Figure: The five Platonic solids - created in Blender 2.36. (Download the Blender file for this picture.)
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Note:  is the alternating group of order  ,  is the symmetric group of order  and  is the cyclic group with order  .
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"regular polyhedron" is owned by mathwizard. [ full author list (3) | owner history (1) ]
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See Also: regular polygon, grafix
| Other names: |
Platonic solid, regular polyhedra, regular |
| Also defines: |
tetrahedron, octahedron, dodecahedron, icosahedron, regular tetrahedron, regular octahedron, regular dodecahedron, regular icosahedron |
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Cross-references: cyclic group, symmetric group, order, alternating group, PlanetMath, pentagon, square, cube, group, symmetry, equilateral triangle, vertices, solids, sources, polyhedra, dihedral angle, edges, number, vertex, face, polyhedron
There are 22 references to this entry.
This is version 17 of regular polyhedron, born on 2002-02-19, modified 2007-07-28.
Object id is 2254, canonical name is RegularPolyhedron.
Accessed 26965 times total.
Classification:
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Pending Errata and Addenda
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![\includegraphics[scale=0.7]{platonic.ps}](http://images.planetmath.org:8080/cache/objects/2254/js/img6.png "\includegraphics[scale=0.7]{platonic.ps}")