Format: Paperback

Language: English

Format: PDF / Kindle / ePub

Size: 9.50 MB

Downloadable formats: PDF

Pages: 62

Publisher: CreateSpace Independent Publishing Platform (February 9, 2014)

ISBN: 1495495922

The Elements Of Non Euclidean Geometry

Introduction to non-Euclidean geometries. Functions of a Complex Variable I. 4 Credits. Prereq: MATH 281; one from MATH 232, 262, 307. Functions of a Complex Variable II. 4 Credits. Differentiation and integration on the real line and in a dimensional Euclidean space; normed linear spaces and metric spaces; vector field theory and differential forms Plane Ans Solid Geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry __The Fourth Dimension and Non-Euclidean Geometry in Modern Art (Leonardo Book Series)__. In the elliptic model, for any given line ℓ and a point A, which is not on ℓ, all lines through A will intersect ℓ. Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for hyperbolic geometry ?". The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.) In the hyperbolic model, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there are infinitely many lines through A that do not intersect ℓ Projective Geometry: Creative Polarities in Space and Time. The students are also expected to meet once a week by themselves, for a self-study tutorial session. They will each take responsibility for leading the session, based on guidance from the lecturer, supplemented as necessary. There will also be office-hour support from the lecturer. The assessment will be based on a portfolio of solved problems submitted periodically during the module Analytical and Geometric Aspects of Hyperbolic Space (London Mathematical Society Lecture Note Series).

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