Counterexamples to Euler's Formula for Nonconvex Polyhedra

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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.

Contributed by: Izidor Hafner (January 2012)
Open content licensed under CC BY-NC-SA



The first example is due to L'Huilier, and the next two are modifications of Hessel's examples.


[1] P. R. Cromwell, Polyhedra, Cambridge: Cambridge University Press, 1997 pp. 202–205.

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