A Relationship between Geometry and Algebra

TL;DR

This paper explores the deep connection between geometry and algebra, exemplified through the projective plane, highlighting the Erlangen program's ongoing influence in understanding geometric structures.

Contribution

It demonstrates the relationship between geometry and algebra using the projective plane as an example, emphasizing the Erlangen program's role in modern geometric study.

Findings

The Erlangen program remains a fundamental guideline for studying geometry.

Group theory provides a powerful algebraic framework for understanding geometric transformations.

The projective plane exemplifies the interplay between geometric concepts and algebraic structures.

Abstract

The three key documents for study geometry are: 1) "The Elements" of Euclid, 2) the lecture by B. Riemann at G\"ottingen in 1854 entitled "\"Uber die Hypothesen welche der Geometrie zu Grunde liegen" (On the hypotheses which underlie geometry) and 3) the "Erlangen Program", a document written by F. Klein (1872) on his income as professor at the Faculty of Philosophy and the Senate of the Erlangen University. The latter document F. Klein introduces the concept of group as a tool to study geometry. The concept of a group of transformations of space was known at the time. The purpose of this informative paper is to show a relationship between geometry and algebra through an example, the projective plane. Erlangen program until today continues being a guideline of how to study geometry.