WOLFRAM|DEMONSTRATIONS PROJECT

Counterexamples to Euler's Formula for Nonconvex Polyhedra

​
solid
1
2
3
n
3
Vertices, V = 36
Edges, E = 54
Faces, F = 24
V - E + F = 6
The Demonstration shows three counterexamples to Euler's formula,
V-E+F=2
, where
V
,
E
, and
F
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.