Magnetostatics and the volumes of the platonic solids

Abstract

The volume of the dodecahedron exceeds that of the icosahedron by about 10%, when inscribed within spheres of the same radius, contrary to what one would judge by eye. To establish this, one would like to have a formula for the volume of a platonic solid in terms of readily identifiable quantities, such as the number of faces and the number of sides to a face of a polyhedron. We present three methods for arriving at such a formula. The first involves simple geometry. The second follows from computing the solid angle subtended by one face of a polyhedron in terms of the magnetic scalarpotential due to a current flowing around the polygonal edge. The third follows from a formula, obtained from the literature, for the solid angle subtended by a triangle in terms of the position vectors of its vertices. The second and third methods are shown to produce essentially the same formula.