Non-Euclidean geometries

Abstract

Euclidean geometry is founded on five postulates that are presented as self-evident truths, which cannot be doubted. For a long time it was thought that there could be no geometry other than Euclid's. However, doubts about the independence of the fifth postulate with respect to the other four, opened the way for the development of new geometries. The philosophical impact of non-Euclidean geometries caused a revolution in mathematical thought, transforming mathematics into an even more abstract science and opening the possibility of conceiving more complex spaces. The development of the theory of relativity, for example, would not have been possible without the antecedent of non-Euclidean geometries. At the present time the applications of non-Euclidean geometries cover many fields of human endeavor, such as science, mechanical engineering, construction, architecture, art and in the mathematic itself.