Platonic solid

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The Platonic solids (named after the Greek philosopher Plato) are a family of five convex polyhedra which exhibit a particularly high symmetry. They can be characterized by the following two properties: All its sides (faces) are regular polygons of the same shape, and the same number of sides meet in all its corners (vertices). However, they also satisfy the following much stronger symmetry condition: Each flag, i.e. each sequence consisting of a corner of an edge of a face, "looks the same". i.e., cannot be distinguished from the others by its position on the polyhedron. Because of this latter property they are called regular polyhedra.

The Greek names of the Platonic solids are derived from the number of sides:

Though these names only address the number of sides, they usually are used for the regular (Platonic) solids only.

[edit] Enumeration

It is easy to see that there are at most five Platonic solids:

The corner of a convex polyhedron is formed by three or more sides, and the sum of the angles cannot exceed 2π.

[edit] Duality

The central points of the sides of a Platonic solid are also the corners of a Platonic solid (the dual polyhedron):

[edit] Properties

name type of face volume surface
properties image
4 regular tetrahedron
(or regular triangular pyramid)
equilateral triangle \tfrac{\sqrt{2}}{12} s^3 \sqrt{3} s^2 4 vertices, 6 edges, self-dual Tetrahedron.png
6 cube square s3 6s2 8 vertices, 12 edges, dual to octahedron Hexahedron.png
8 regular octahedron equilateral triangle \tfrac{\sqrt{2}}{3} s^3 2\sqrt{3} s^2 6 vertices, 12 edges, dual to cube Octahedron.png
12 regular dodecahedron regular pentagon \tfrac{15+7\sqrt{5}}{4}s^3 3\sqrt{25+10\sqrt{5}}s^2 20 vertices, 30 edges, dual to icosahedron Dodecahedron.png
20 regular icosahedron equilateral triangle \tfrac{15+5\sqrt{5}}{12}s^3 5\sqrt{3}s^2 12 vertices, 30 edges, dual to dodecahedron Icosahedron.png

A sphere circumscribed about any of the Platonic solids will touch all the vertices, and a sphere inscribed within any will touch all the faces at the center of the face.

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