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Description of  Geometries: Selected Math Themes, #1

In the present text, we present a brief introduction to non-Euclidean geometries. We begin by presenting the foundations of classical geometry, and the discussion that gave rise to other geometries. There is an incredibly old example of an attempt to demonstrate the fifth postulate of Euclid due to Proclus, and part of Sacheri's work is retaken, to continue later with the achievements of Lobachevsky. In simple terms, the geometry to which we are accustomed, which is what we are taught from the basic studies, is the so-called classical geometry, which follows certain ideas that are taken as true facts and from there builds the veracity of other facts. In particular, the fifth postulate establishes the existence of lines that are called parallel, which are those that are never intercepted. Geometry arose from the need to understand the world around us, and from there arose the axioms of Euclidean geometry, but they were not entirely accurate. For example, in art we see that parallel lines are sometimes represented as intercepting lines on the horizon, at the so-called vanishing point. Apart from the non-Euclidean geometries we see other examples of geometry, the so-called friezes and mosaics. We conclude the book with a brief notion of topology and algebraic geometry.