Counterexamples to Euler's Formula for Nonconvex Polyhedra
The Demonstration shows three counterexamples to Euler's formula, , where , , and are the number of vertices, edges, and faces, respectively. The formula holds for all polyhedra, so the exercise here is to see how these figures fail the proper definition of being a polyhedron.
The first example is due to L'Huilier, and the next two are modifications of Hessel's examples.
 P. R. Cromwell, Polyhedra, Cambridge: Cambridge University Press, 1997 pp. 202–205.
"Counterexamples to Euler's Formula for Nonconvex Polyhedra"
Wolfram Demonstrations Project
Published: January 5 2012