The dawn of non‐Euclidean geometry

The emergence of non‐Euclidean geometries in the beginning of the 19th century represents one of the dramatic episodes in the history of mathematics. It may be an often told tale, but although it entails a lot of important and useful geometrical ideas and constructions, it is seldom told with the necessary mathematical details in contemporary teaching of geometry. The purpose of this article is to provide a short, but relatively complete, exposition of the geometry in the Poincare disc model of the hyperbolic plane within the historical perspective of Euclid's postulates. We also describe the isometries in the hyperbolic plane and tilings of the hyperbolic plane with congruent regular hyperbolic polygons.