classification of Platonic solids

Proposition..

The regular tetrahedron, regular octahedron, regular icosahedron, cube, and regular dodecahedronare the only Platonic solids.

Proof.

Each vertex of a Platonic solid is incident with at least threefaces. The interior angles incident with that vertex must sum toless than $2\pi$, for otherwise the solid would be flat at thatvertex. Since all faces of the solid have the same number of sides,this implies bounds on the number of faces which could meet at avertex.

The interior angle of an equilateral triangle has measure $\frac{\pi }{3}$,so a Platonic solid could only have three, four, or five trianglesmeeting at each vertex. By similar reasoning, a Platonic solid couldonly have three squares or three pentagons meeting at each vertex. Butthe interior angle of a regular hexagon has measure $\frac{2\pi }{3}$.To avoid flatness a solid with hexagons as faces would thus haveto have only two faces meeting at each vertex, which is impossible. Forpolygons with more sides it only gets worse.

Since a Platonic solid is uniquely determined by the number and kind offaces meeting at each vertex, there are at most five Platonic solids,with the numbers and kinds of faces listed above. Butthese correspond to the five known Platonic solids. Hence there areexactly five Platonic solids.∎