Topology of Surfaces
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The table below lists the number of edges, vertices, and faces of the five Platonic solids.
The table shows that for each Platonic solid we have the relation
Name of polyhedron
Number V of vertices quotation
Number E of edges
Number F of faces
tetrahedron
4
6
4
cube
8
12
6
octahedron
6
12
8
dodecahedron
20
30
12
icosahedron
12
30
20
$$V - E + F = 2$$
(6)
It is easy to show that relation (6) is also valid for pyramids, prisms, and other polyhedra. Euler was the first to recognize and prove this important property of polyhedra.