An amazing property of
polytopes in
three dimensions. Suppose you have some
convex polyhedron. You may create a new (generally
different) polyhedron by the following procedure:
- Create a dual vertex from each face, by choosing some point on that face (of course, if you start from some "nice" solid, you'll choose the centre of each face). So there are as many vertices in the dual as there were faces in the original.
- Connect by a dual edge every pair of dual vertices which were derived from adjacent faces. So there are the same number of edges in the dual as in the original.
- This construction naturally gives rise to the dual faces, and there are as many faces in the dual as there were vertices in the original.
Note that switching V and F in V-E+F=2 preserves equality.
Interestingly, the duals of the platonic solids are also platonic solids!