Zoom Talk: Paolo Bussotti (Udine)

Please contact office.leitgeb@lrz.uni-muenchen.de for the password.

Title:

The meaning of “cross ratio”: mathematical meaning and historical origin

Abstract:

The seminary will be divided into two parts. In the first one, the mathematical meaning and importance of the concept of cross ratio will be highlighted. In particular, the invariance of the cross ratio by sections and projections will be proved as well as the existence of six classes of equivalence connoting the 24 possible combinations of the cross ratios of four elements. Two applications of the cross ratios to projective transformations will be shown. One of such applications will concern the very important case in which the cross ratio of four elements has the value -1. A group of four elements whose cross ratio holds such a value is called “harmonic group”. In the second part of the seminary, I will present the last developments of my research concerning the origin of the concept of cross ratio, which is connected to the foundational programmes for geometry carried out by two great mathematicians, at the beginning of the 30 of the 19th century, the French Michel Chasles (1793-1780) and the Swiss Jakob Steiner (1796-1863). I will also outline their foundational programmes.