  The statement of the Gauss-Bonnet theorem brings up an unexpected form of
reflexivity (major concept of philosophy of mathematics), so that geometry
contemplates itself in it. It is therefore the revolutionary and multifaceted
concept of Gaussian curvature that triggers a new conceptuality above Euclidean
geometry. Here, the equality between integral of total curvature and Euler
characteristic indicates that a concept of topological nature is equal to a
number which expresses a concept of geometric nature. This further demonstrates
that mathematics develops through the intervention of different disciplines on
top of each other, as observation tools, formal structuring, new unifying
points of view.
