This chapter contains a rather spectacular proof by induction. The result weshall prove is a famous formula of Euler from the 18th century, concerning therelationship between the numbers of corners, edges and faces of a solid object.As an application of Euler’s formula we shall then study the five Platonic solids-the cube, regular tetrahedron, octahedron, icosahedron and dodecahedron.We shall call our solid objects polyhedra. A polyhedron is a solid whosesurface consists of a number of faces, all of which are polygons, such that anyside of a face lies on exactly one other face. The corners of the faces are calledthe vertices of the polyhedron, and their sides are the edges of the polyhedron.Here are some everyday examples of polyhedra. (1) Cube

This has 8 vertices, 12 edges and 6 faces. (2) Tetrahedron

This has 4 vertices, 6 edges and 4 faces.