1. A convex polyhedron is a 3 dimensional geometric figure made up of faces, each of which is a polygon. Thus a polyhedron has a number of faces, edges and vertices, just like a planar graph. In fact, you can represent every convex polyhedron as a planar graph. ("Convex" means that the interior angle between any two faces is less then 180°.) (a) An octahedron is a regular polyhedron made up of 8 equilateral triangles (it sort of looks like two square-based pyramids with their bases glued together). Draw a planar graph representation of an octahedron. How many vertices, edges and faces does an octahedron (and your graph) have? It might be helpful to decide on these numbers before drawing the graph. (b) The traditional design of a soccer ball is in fact a (spherical projection of a) truncated icosahedron. This consists of 12 regular pentagons and 20 regular hexagons. No two pentagons are adjacent (so the edges of each pentagon are shared only by hexagons). How many vertices, edges, and faces does a truncated icosahedron have? Explain how you arrived at your answers. Bonus: draw the planar graph representation of the truncated icosahedron. (c) Your "friend" claims that he has constructed a convex polyhedron out of 2 triangles, 2 squares, 6 pentagons and 5 octagons. Prove that your friend is lying. Hint: each vertex of a convex polyhedron must border at least three faces.