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. It is known that there exist five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 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.
