Applied and Computational Mathematics 2013; 2(5): 115-117 Published online September 10, 2013 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.20130205.11 Occurrence of Galilean geometry Abdullaaziz Artıkbayev 1 , Abdullah Kurudirek 2 , Hüseyin Akça 2 1 Department of Mathematics, Railway Inst., Tashkent, Uzbekistan 2 Department of Mathematics Education, Ishik University, Arbil, Iraq Email address: aartykbaev@mail.ru(A. Artıkbayev), drcebir@hotmail.com(A. Kurudirek), huseyinakcha@gmail.com(H. Akça) To cite this article: Abdullaaziz Artıkbayev, Abdullah Kurudirek, Hüseyin Akça. Occurrence of Galilean Geometry. Applied and Computational Mathematics. Vol. 2, No. 5, 2013, pp. 115-117. doi: 10.11648/j.acm.20130205.11 Abstract: The main difference of Galilean geometry is its relative simplicity, for it enables the student to study it in relative detail without losing a great deal of time and intellectual energy. In this paper, we introduce you with new geometric(non-Euclidean) ideas which exist in affine plane and more simple than Euclidean plane. Keywords: Non-Euclidean Geometry, Galilean Geometry, Affine Plane, Isotropic, Minkowski Space 1. Introduction The 19 th century was a period of rapid development in geometry. In 1854 the eminent German mathematician G. F. B. Riemann formulated, in a famous memoir [1], an extremely general view of geometry which greatly widened its scope. Riemann also noted that there are three related but distinct geometric systems the usual Euclidean geometry, hyperbolic geometry and so-called elliptic geometry which is closer spherical geometry. This list of geometries was extended in 1870 by the German mathematician F. Klein [2] ,[3]. According to Klein there are nine related plane geometries including Euclidean geometry, hyperbolic geometry and elliptic geometry. Klein's views, which were in a way a synthesis of the geometric views of his predecessors and of the work of the English algebraist A. Cayley appeared in 1872 in his Erlanger Program [4]. Klein's broad view of geometry has a universality comparable to that of Riemann. Thus, just as the fundamental discoveries of Lobachevsky, Bolyai, and Gauss destroyed the exclusive position of Euclidean geometry, so, too, the classical investigations of Riemann and Klein (1854-1872) destroyed the exclusive position of hyperbolic geometry. Nevertheless, even today the term "non-Euclidean geometry" frequently stands for just hyperbolic geometry (less frequently, the plural "non-Euclidean geometries" is used to denote just hyperbolic geometry) and elliptic geometry, and the existence of other geometric systems is known only to specialists. The views presented in those discussions have long ago lost all scientific significance. Thus, for example, even in Klein's "Non-Euclidean Geometry", we find the assertion that the geometry of our universe must be either Euclidean, hyperbolic, or elliptic; this in spite of the fact that the scientific unsoundness of this viewpoint, at least in its original formulation, followed from Einstein's special theory of relativity of 1905 and, even more decisively, from his general theory of relativity of 1916. The fact that hyperbolic geometry is linked to the issue of the independence of the parallel axiom and clarifies the role of that axiom in Euclidean geometry, is a strong argument in favor of its pedagogical value. On the other hand, hyperbolic geometry is rather complex—it is definitely more complex than Euclidean geometry—and yet the non-Euclidean nature of a geometry need not imply complexity. The main distinction of Galilean geometry is its relative simplicity, for it enables the student to study it in relative detail without losing a great deal of time and intellectual energy. Put differently, the simplicity of Galilean geometry makes its extensive development an easy matter, and extensive development of a new geometric system is a precondition for an effective comparison of it with Euclidean geometry. Also, extensive development is likely to give the student the psychological assurance of the consistency of the investigated structure. Another distinction of Galilean geometry is the fact that it exemplifies the fruitful geometric idea of duality. These reasons make me think that one should give serious thought to a mathematics program for teachers' colleges which would include a comparative study of three simple geometries, namely, Euclidean geometry, the geometry associated with the Galilean principle of relativity, and the geometry associated with Einstein's principle of relativity as well as an