This specific research is concentrated on the subject of the Non Euclidean Geometries. It aims to describe the historical course until their discovery, their scientific documentation and systematization, their expansion and connection with other sciences and  with all, finally, the radical consequences that came forward.

The research consists of 2 parts.

The first part is historical - geometric and consists of 5 chapters.

·    On the first chapter are shortly described the basic postulates of the Euclidean Geometry, among which is the parallel postulate or the notorious "fifth postulate".

·     The second chapter unveils the historical course of the efforts of various scientists, to prove the "non utility" of the fifth postulate, until the beginnings of the 18nth century.

·    The third chapter narrates the discovery and regularization of the Non Euclidean Geometry by three mathematicians, namely, Lobatchewsky, Bolyai and, partly, Gauss, despite of course, one another.

·    On the forth chapter, comes forth the confirmation of the correctness of the opinions of Lobatchewsky, Bolyai and Riemann.

·    On the fifth chapter are presented the first models of the Non Euclidean Geometries, documenting, thus, the coherence and competence of the theory of Lobatchewsky-Bolyai, by Beltrami, Klein and Poincare. Here, the chapter alludes to the notion of the projective Geometry, which connects inextricably the Non Euclidean Geometries with the Euclidean Geometry.

On the second part is analyzed how the theories of the Non Euclidean Geometries have influenced foundationally the radical discoveries of the 20nth century, on the area of Physics and Astrophysics. It consists of 5 chapters.

·    On the first chapter of this part, as a sequel to the first, is presented the notion of n-dimensional manifold of Riemann and deals with its application on the Physical Sciences.

·    The second chapter describes the human senses, featuring the inability of man to imagine and depict the multidimensional spaces. On the same chapter, there are references to various graphical models of the multidimensional spaces like the hypercube and the hypersphere.

·    The third chapter narrates the solution of Minkowski, about a four-dimensional geometric space, upon which was built the Special Relativity Theory of Einstein.

·    On the forth chapter, is approached the possible Geometries of the Universe, quoting various scientific views from the area of Astrophysics.

·    The  fifth chapter is developed upon the opinions of E. Danezis and Str. Theodosiou, referring to the Geometry of the Universe and the universal material.