Jean-Paul Brasselet
I2M, CNRS, Marseille
/user/jean-paul.brasselet/
Date(s) : 02/04/2020 iCal
14 h 00 min - 15 h 00 min
La « formule d’Euler » fait l’objet de multiples controverses : Qui a exprimé le premier cette formule ? Qui en a donné la première (correcte) démonstration ? La démonstration de Cauchy a fait l’objet de critiques sévères, en particulier de la part de Imre Lakatos et de Elon Lima. Dans cet exposé, après avoir fait un rapide survol de l’historique, nous donnons une démonstration élémentaire de la formule, avec pour seul outil la méthode de Cauchy. L’exposé est élémentaire et accessible aux étudiants de Maîtrise.
Avec Nguyen Thi Bich Thuy (UNESP, São José do Rio Preto).
An elementary demonstration of Euler’s Formula – a rehabilitation of the Cauchy method
Euler’s formula says that for any sphere triangulation, the alternate sum: n0−n1+n2= 2 where the numbers ni are respectively the number of vertices n0, the number of edges n1 and the number of triangles n2 from the triangulation. There is much controversy about the paternity of the formula, also about who gave the first correct proof. We provide precisely some elements on the history of the formula and also about Cauchy’s first topological proof. Some authors criticize Cauchy’s proof, saying the proof needs deep topology results that were proven after Cauchy. We show that with a technique of “stretching” and the use of sub-triangulations only, Cauchy’s proof works without using the other results. We use the same tools to show that for surfaces such as the torus, the projective plane and the Klein, the alternate sum does not depend on the surface. The lecture, elementary but original, is a way of rehabilitating Cauchy’s method. The lecture can be assisted by master students.
With Nguyen Thi Bich Thuy (UNESP, São José do Rio Preto).
https://arxiv.org/abs/2003.07696
Emplacement
FRUMAM, St Charles
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