The Platonic solid reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

# Platonic solid

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A Platonic solid is a convex polyhedron whose faces all share the same regular polygon and such that the same number of faces meet at all its vertices. Compare with the Kepler solids, which are not convex, and the Archimedean and Johnson solids, which while made of regular polygons are not themselves regular.

There are five Platonic solids, all known to the ancient Greeks:
Name and picture Face polygon Faces Edges Vertices Faces meeting at each vertex Symmetry group
tetrahedron

()
triangle4643Td
cube (hexahedron)

()
square61283Oh
octahedron

()
triangle81264Oh
dodecahedron

()
pentagon1230203Ih
icosahedron

()
triangle2030125Ih

That there are only five such three-dimensional solids is easily demonstrated. To have vertices, there must be at least three of the faces meeting at a point, and the total of their angles must be less than 360 degrees; i.e the corners of the face must be less than 120 degrees: this rules out all the regular polygons except triangles, squares, and pentagons.

• triangular faces: each vertex of a triangle is 60 degrees, so a shape should be possible with 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
• square faces: each vertex of a square is 90 degrees, so there is only one arrangement possible with three faces at a vertex, the cube.
• pentagonal faces: each vertex is 108 degrees; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron, and that exhausts the list of regular 3-dimensional solids.

Note that if you connect the centers of the faces of a tetrahedron, you get another tetrahedron. If you connect the centers of the faces of an octahedron, you get a cube, and vice versa. If you connect the centers of the faces of a dodecahedron, you get an icosahedron, and vice versa. These pairs are said to be dual polyhedra.

The Platonic solids are named after Plato, who wrote about them in Timaeus. Plato learned about these solids from his friend Theaetetus. The constructions of the solids are included in Book XIII of Euclid's Elements. Proposition 13 describes the construction of the tetrahedron, proposition 14 of the octahedron, proposition 15 of the cube, proposition 16 of the icosahedron, and proposition 17 of the dodecahedron.

Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, (based on ideas regarding the music of the spheres etc.) and identified the five platonic solids with the five planets - Mercury, Venus, Mars, Jupiter, Saturn and the five classical elements. (The Earth, moon and sun were not considered to be planets.)

## Uses

The shapes are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as D followed by the number of faces (d4, d8, etc.)

The tetrahedron, cube, and octahedron, are found naturally in crystal structures. The dodecahedron is combinatorially identical to the pyritohedron (in that both have twelve pentagonal faces), which is one of the possible crystal structures of pyrite. However, the pyritohedron is not a regular dodecahedron, but rather has the same symmetry as the cube.