engleski

Non-Euclidean Geometry: Facts, Features and Funs

The common theme that links Plato, Archimedes, Kepler, Einstein, the quantum theorists and present-day string theorists is the belief that an understanding of the basic stuff of the universe will be found using mathematics. Sometimes, math runs ahead of physics. Today, however, the mathematicians are behind and trying to catch up. This situation is nothing new. Newton’ s investigations into mechanics (planet’ s motions, etc.) and optics led him to develop calculus. More recently, Einstein’ s observations that gravity could be understood using a "strange" new kind of geometry developed earlier by Gauss, Lobachevsky, Bolyai, and in a unifying way by Riemann, led to the massive development of this geometry and its rapid incorporation into mainstream mathematics. In this talk we shall discuss Lobachevsky geometry (also called hyperbolic) and Riemannian (elliptic) geometry. Riemannian geometry locally (in the small) coincides with the geometry of the sphere in ordinary three-dimensional space, and theorems of spherical geometry can be interpreted as ordinary space geometry theorems. In Lobachevsky geometry, through any point not belonging to a line, one can draw many lines disjoint with the given line. For many centuries, people could not believe that this was possible. Euclid’ s fifth postulate (through any point not on a line there is only one parallel to that line) seemed to be a theorem following from the other axioms. This belief proved to be false. The basic theorems of hyperbolic geometry were proved by the above mentioned mathematicians, and then followed by Beltrami, Cayley, Klein who Poincare who constructed models of this geometry, and the reality of this non-Euclidean geometry became evident. Beside showing some fundamental facts about these geometries, we shall also present some original results concerning some basic triangle inequalities. The first is the cosine-law type inequality in all three geometries. We further present the non-Euclidean version of the well known Euler’ s inequality (R > 2r), Finsler-Hadwiger’ s inequality and the "fundamental triangle inequality".

non-Euclidean geometry

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