The Flaw in Euler's Proof of His Polyhedral Formula
The American Mathematical Monthly
Leonhard Euler, in the course of his long and productive career, published 866 mathematical works, enough mathematics to ﬁll seventy-four volumes. On the theory of polyhedra, however, Euler did not contribute many pages of new material. Nevertheless, with the following theorem he made the most signiﬁcant contribution to the theory since the foundational work of the ancient Greeks, perhaps the most important contribution ever [6, p. 156]:
In every solid enclosed by plane faces, the number of faces along with the number of solid angles exceeds the number of edges by two.
This theorem, which we refer to as Euler’s polyhedral formula, typically has the form
V − E + F = 2,
where V, E, and F denote the number of vertices, edges, and faces of a polyhedron.
Francese, Christopher, and David Richeson. "The Flaw in Euler's Proof of His Polyhedral Formula." The American Mathematical Monthly 114, no. 4 (2007): 286-296. https://www.tandfonline.com/doi/abs/10.1080/00029890.2007.11920417