Classifying the Platonic Solids Using Graph Theory

The Platonic solids are regular polyhedra, meaning each vertex has the same number of edges incident on it, and each face has the same number of edges on its border. Since antiquity, it has been known that there exist five Platonic solids: the tetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron. But how do we know that these are the only ones? By using Euler's formula for planar graphs, and solving a Diophantine equation with factoring and fudging, we can show that there are no further Platonic solids.

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