A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.
Solution
Suppose it has E edges, F faces, V vertices. Every face must be a triangle. So E = 3F/2. Using Euler's relation E + 2 = V + F, gives V = F/2 + 2. So E = 3V - 6. But if every pair of vertices is connected by an edge then we have V(V-1)/2 ≤ E, or V2 - 7V + 12 ≤ 0, or (V-3)(V-4) ≤ 0, so V = 3 or 4. But there are no polyhedra with V = 3, so we must have V = 4 and the polyhedron is a tetrahedron.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03