Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 3G (Geometry)

The Euler formula for a sphere, applied to the surface of a convex polyhedron, is $V-E+F=2$. In a regular polyhedron let $q\ge 3$ be the common number of edges incident at a vertex. Counting vertex-edge and face-edge incidences, pentagonal faces give $qV=2E$ and $5F=2E$. Hencewhose right-hand side can be positive only if $q<10/3$. Thus $q=3$. Substitution then yields $2=E/15$, provingThese are the counts of the dodecahedron. Here regular polyhedron is understood in the convex, ordinary-face sense used by the stated Euler formula; self-intersecting star constructions do not share these hypotheses.