As defined by Plato, the essential properties of a Platonic solid are: all sides are convex regular polygons of equal size, all of the faces intersect at their edges and nowhere else, and the same number of faces meet at each of the solid's vertices. Only five polygons fit all of these criteria: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.
Euclid wrote a geometric proof for the criteria of Platonic solids in his work, "Elements." The proof includes four rules as follows: each vertex must be in contact with at least three sides; the sum of the angles at each vertex must be less than 360 degrees; the angles at all vertices must be equal; and the common face can only be a triangles, square, or pentagon, as faces with six or more sides have angles that are too great to be valid.
Platonic solids are lauded for their symmetry and beauty, and Plato himself described the solids as being symbolic of the fundamental forces of the world. He matched the tetrahedron with the sharpness of fire, the cube with the rigidity of earth, the octahedron with the lightness of air, the dodecahedron with "what god used to craft the heavens" (later labelled "aether" by Aristotle), and the icosahedron with the flowing nature of water.