Author: G.E. Martin
Publisher: Springer Science & Business Media
ISBN: 1461257255
Category : Mathematics
Languages : en
Pages : 525
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Book Description
This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref erence if necessary.
Author: Maria Helena Noronha
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 440
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Book Description
This book develops a self-contained treatment of classical Euclidean geometry through both axiomatic and analytic methods. Concise and well organized, it prompts readers to prove a theorem yet provides them with a framework for doing so. Chapter topics cover neutral geometry, Euclidean plane geometry, geometric transformations, Euclidean 3-space, Euclidean n-space; perimeter, area and volume; spherical geometry; hyperbolic geometry; models for plane geometries; and the hyperbolic metric.
Author: Patrick J. Ryan
Publisher: Cambridge University Press
ISBN: 0521127076
Category : Mathematics
Languages : en
Pages : 237
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Book Description
This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic.
Author: David Hilbert
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 190
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Book Description
Author: Ellery B. Golos
Publisher:
ISBN:
Category : Geometry
Languages : en
Pages : 248
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Book Description
Author: Nikolai I. Lobachevsky
Publisher:
ISBN: 9781927763247
Category : Mathematics
Languages : en
Pages : 212
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Book Description
Neither general relativity (which revealed that gravity is merely manifestation of the non-Euclidean geometry of spacetime) nor modern cosmology would have been possible without the almost simultaneous and independent discovery of non-Euclidean geometry in the 19th century by three great mathematicians - Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss (whose ideas were later further developed by Georg Friedrich Bernhard Riemann).This volume contains three works by Lobachevsky on the foundations of geometry and non-Euclidean geometry: "Geometry", "Geometrical investigations on the theory of parallel lines" and "Pangeometry". It will be of interest not only to experts and students in mathematics, physics, history and philosophy of science, but also to anyone who is not intimidated by the magnitude of one of the greatest discoveries of our civilization and would attempt to follow (and learn from) Lobachevsky's line of thought, helpfully illustrated by over 130 figures, that led him to the discovery.
Author: Marvin J. Greenberg
Publisher: Macmillan
ISBN: 9780716724469
Category : Mathematics
Languages : en
Pages : 512
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Book Description
This classic text provides overview of both classic and hyperbolic geometries, placing the work of key mathematicians/ philosophers in historical context. Coverage includes geometric transformations, models of the hyperbolic planes, and pseudospheres.
Author: I.M. Yaglom
Publisher: Springer Science & Business Media
ISBN: 146126135X
Category : Mathematics
Languages : en
Pages : 326
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Book Description
There are many technical and popular accounts, both in Russian and in other languages, of the non-Euclidean geometry of Lobachevsky and Bolyai, a few of which are listed in the Bibliography. This geometry, also called hyperbolic geometry, is part of the required subject matter of many mathematics departments in universities and teachers' colleges-a reflec tion of the view that familiarity with the elements of hyperbolic geometry is a useful part of the background of future high school teachers. Much attention is paid to hyperbolic geometry by school mathematics clubs. Some mathematicians and educators concerned with reform of the high school curriculum believe that the required part of the curriculum should include elements of hyperbolic geometry, and that the optional part of the curriculum should include a topic related to hyperbolic geometry. I The broad interest in hyperbolic geometry is not surprising. This interest has little to do with mathematical and scientific applications of hyperbolic geometry, since the applications (for instance, in the theory of automorphic functions) are rather specialized, and are likely to be encountered by very few of the many students who conscientiously study (and then present to examiners) the definition of parallels in hyperbolic geometry and the special features of configurations of lines in the hyperbolic plane. The principal reason for the interest in hyperbolic geometry is the important fact of "non-uniqueness" of geometry; of the existence of many geometric systems.
Author: John Ratcliffe
Publisher: Springer Science & Business Media
ISBN: 1475740131
Category : Mathematics
Languages : en
Pages : 761
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Book Description
This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. Particular emphasis has been placed on readability and completeness of ar gument. The treatment of the material is for the most part elementary and self-contained. The reader is assumed to have a basic knowledge of algebra and topology at the first-year graduate level of an American university. The book is divided into three parts. The first part, consisting of Chap ters 1-7, is concerned with hyperbolic geometry and basic properties of discrete groups of isometries of hyperbolic space. The main results are the existence theorem for discrete reflection groups, the Bieberbach theorems, and Selberg's lemma. The second part, consisting of Chapters 8-12, is de voted to the theory of hyperbolic manifolds. The main results are Mostow's rigidity theorem and the determination of the structure of geometrically finite hyperbolic manifolds. The third part, consisting of Chapter 13, in tegrates the first two parts in a development of the theory of hyperbolic orbifolds. The main results are the construction of the universal orbifold covering space and Poincare's fundamental polyhedron theorem.
Author: Harold E. Wolfe
Publisher: Courier Corporation
ISBN: 0486498506
Category : Mathematics
Languages : en
Pages : 274
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Book Description
One of the first college-level texts for elementary courses in non-Euclidean geometry, this volumeis geared toward students familiar with calculus. Topics include the fifth postulate, hyperbolicplane geometry and trigonometry, and elliptic plane geometry and trigonometry. Extensiveappendixes offer background information on Euclidean geometry, and numerous exercisesappear throughout the text.Reprint of the Holt, Rinehart & Winston, Inc., New York, 1945 edition
