Non-Euclidean geometry

The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton Univ. Press, 1983. 

Sommerville, D. M. Y. Bibliography of Non-Euclidean Geometry. Univ. of St. Andrews,... 

Cox98] H. S. M. Coxeter, Non-Euclidean Geometry, 6th edition, MAA Spectrum. Mathematical Association of America, 1998. 

All solutions of Einstein s equations are conditioned by the need of some ad hoc assumption about the geometry of space-time. The only indisputably valid assumption is that space-time is of absolute non-euclidean geometry. It is interesting to note that chiral space-time, probably demanded by the existence of antimatter and other chiral forms of matter, rules out the possibility of affine geometry, the standard assumption of modern TGR [7]. 

Application of Non-Euclidean Geometry Ideas in Polymer Topology... 

As was mentioned above, there are a lot of different ways of considering the Edwards-Frisch problem. However, from the methodological point of view and for the sake of a better clarification of non-euclidean geometry ideas for the description of topological constraints, we would like to present the method of conformal transformation. 

GEOMETRY OF COMPLEX NUMBERS, Hans Schwerdtfeger. Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and two-dimensional non-Euclidean geometries. 200pp. 54 x 84. 

Calculus deals with the relationship between changing quantities. In differential calculus, the problem is to find the rate at which a known but varying quantity changes. The problem in integral calculus is the reverse of this to find a quantity when the rate at which it is changing is known. Mathematics is the name for the broad area which is comprised of all these subject areas, and many others not included in the school curriculum, e.g., non-Euclidean geometry. 

A consistent logical system for which one of these postulates is modified in an essential way is non-Euclid-ean geometry. Although there are different types of Non-Euclidean geometry which do not use all of the postulates or make alterations of one or more of the postulates of Euclidean geometry, hyperbolic and elliptic are usually most closely associated with the term non-Euclidean geometry. 

Despite the general acceptance of Euclidean geometry, there appeared to be a problem with the parallel postulate as to whether or not it really was a postulate or that it could be deduced from other definitions, propositions, or axioms. The history of these attempts to prove the parallel postulate lasted for nearly 20 centuries, and after numerous failures, gave rise to the establishment of Non-Euclidean geometry and the independence of the parallel postulate. 

The attempt to solve this problem was made also by Farkas Bolyai, the father of Johann Bolyai, one of the founders of non-Euclidean geometry but his proof was also invalid. It is interesting to note that Johann s father cautioned his son not to get involved with the proof of the parallel postulate because of its complexity. 

It was at the University of Kazan, in the Russian province of Kazakhstan, that Nicolai Ivanovitch Lobachevsky made his contributions in Non-Euclidean geometry. In his early days at the university, he did try to find a proof of the parallel postulate, but later changed direction. As early as 1826, he made use of the hypothesis of the acute angle already developed by Saccheri and... 

The most important conclusions of Bolyai s research in non-Euclidean geometry were the following (1) The definition of parallels and their properties independent of... 

Bonola, R. Non-Euclidean Geometry. New York Dover Publications, 1911. 

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