Fonaments de la geometria

Abstract

This Final Degree Project consists of six chapters. In the first chapter, we introduce the history of geometry starting from the lives and contributions of five mathematicians who transformed the way the geometry was understood with their researches. Chronologically arranged, they are Euclides, Descartes, Riemann, Klein and finally Hilbert. In each case, we’ve realized a short biography intending to contextualize.The second chapter is dedicated to the most famous Euclid’s work, the Elements, which is considered the basis on which modern mathematics were constructed. We first explain the methodology used by Euclid writing his work from characterizing the four types of statements in which the content can be classified and by delving into the method he devised to prove propositions, exemplified by the demonstration of the Pythagorean theorem. Then, we explain the content of the thirteen books that conform the Euclid’s Elements. Finally, we briefly comment which work parts generated controversy. In Chapter 3, we develop two-dimensional hyperbolic geometry from the Poincaré half-plane model. Then, we introduce the inversion and we prove that hyperbolic plane verifies the first four axioms, but not the fifth, thus proving the consistency of hyperbolic geometry. In the fourth chapter, we present Hilbert’s axiom system in plane geometry and we define absolute, Euclidean and hyperbolic geometry according to the criteria determined by Hilbert. Then we compare these geometries, showing some results that are validated in each geometry and finally, we proove in three ways a result that is verified in all three cases by using in each demonstration only the tools each geometry provides us. Chapter 5 is about comparing elliptic, parabolic and hyperbolic geometry from the projective geometry. We first comment which postulates are verified in each case and then we give a projective model of each geometry, briefly commentig some characteristics. The last chapter is about Riemannian geometry. We first give some basic definitions and then we introduce the homogeneity and isotropy concepts. After that, we comment the Riemannian interpretation of the first and second Euclid’s axioms and finally, we conclude with Cartan-Hadamard theorem’s demonstration, which determines the space forms in some kind of homogeneous isotropic Riemannian manifold.