Euler's formula
Visual, surprising and rich in applications (the structure of the football is the most well-known), it features in the pantheon of the most beautiful mathematical formulas: V + F = E + 2. While the study of polyhedra already fascinated Plato, the relationship connecting the number of its vertices, its faces and its edges had been anticipated by Descartes before being rigorously expressed by Euler, the Swiss mathematician who gave it his name (it's only fitting!). Poincaré undertook to generalize it, but it is still far from having revealed all its secrets.
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The pursuit of Euler's formula for polyhedra | Tangente
In a polyhedron, the number of vertices, S, plus the number of faces, F, equals the number of edges, A, plus 2. In other words, S + F = A + 2. For many mathematicians, Euler's formula is the most beautiful formula of all! Above all, it has had an eventful history, to say the least...
An illustrious line of descendants
It all began with a football to which Euler's formula was applied. Why does the constant 2 appear on the right-hand side? To find out, we follow in the footsteps of Henri Poincaré, André Weil and… Alexander Grothendieck.
Following in Euler's footsteps
Euler's formula has led to many further developments, including its use in the study of polytopes.
Imre Lakatos and mathematics education | Tangente
The Hungarian mathematician Imre Lakatos set out his perspective in his best-known work, Proofs and Refutations. He illustrated his argument using Euler's formula.