Non-Euclidean Geometry


Author: H. S. M. Coxeter
Publisher: Cambridge University Press
ISBN: 9780883855225
Category: Mathematics
Page: 336
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A reissue of Professor Coxeter's classic text on non-Euclidean geometry. It surveys real projective geometry, and elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases. This is essential reading for anybody with an interest in geometry.

Introductory Non-Euclidean Geometry


Author: Henry Parker Manning
Publisher: Courier Corporation
ISBN: 0486154645
Category: Mathematics
Page: 112
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This fine and versatile introduction begins with the theorems common to Euclidean and non-Euclidean geometry, and then it addresses the specific differences that constitute elliptic and hyperbolic geometry. 1901 edition.

NonEuclidean Geometry


Author: Herbert Meschkowski
Publisher: Academic Press
ISBN: 1483259218
Category: Mathematics
Page: 112
View: 9498

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Noneuclidean Geometry focuses on the principles, methodologies, approaches, and importance of noneuclidean geometry in the study of mathematics. The book first offers information on proofs and definitions and Hilbert's system of axioms, including axioms of connection, order, congruence, and continuity and the axiom of parallels. The publication also ponders on lemmas, as well as pencil of circles, inversion, and cross ratio. The text examines the elementary theorems of hyperbolic geometry, particularly noting the value of hyperbolic geometry in noneuclidian geometry, use of the Poincaré model, and numerical principles in proving hyperparallels. The publication also tackles the issue of construction in the Poincaré model, verifying the relations of sides and angles of a plane through trigonometry, and the principles involved in elliptic geometry. The publication is a valuable source of data for mathematicians interested in the principles and applications of noneuclidean geometry.

Euclidean and Non-euclidean Geometries


Author: Maria Helena Noronha
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: 409
View: 6293

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Designed for undergraduate juniors and seniors, Noronha's (California State U., Northridge) clear, no-nonsense text provides a complete treatment of classical Euclidean geometry using axiomatic and analytic methods, with detailed proofs provided throughout. Non-Euclidean geometries are presented usi

Introduction to Non-Euclidean Geometry


Author: Harold E. Wolfe
Publisher: Courier Corporation
ISBN: 0486320375
Category: Mathematics
Page: 272
View: 2985

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College-level text for elementary courses covers the fifth postulate, hyperbolic plane geometry and trigonometry, and elliptic plane geometry and trigonometry. Appendixes offer background on Euclidean geometry. Numerous exercises. 1945 edition.

A History of Non-Euclidean Geometry

Evolution of the Concept of a Geometric Space
Author: Boris A. Rosenfeld
Publisher: Springer Science & Business Media
ISBN: 1441986804
Category: Mathematics
Page: 471
View: 8377

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The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-Euclidean geometry. The importance of the discovery of non-Euclidean geometry goes far beyond the limits of geometry itself. It is safe to say that it was a turning point in the history of all mathematics. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes. " During the seventies of the last century there occurred another scientific revolution. By that time mathematicians had become familiar with the ideas of non-Euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arith metic and algebra of real and complex numbers, and, finally, to new mathe matical systems, i. e. , sets furnished with various structures having no classical analogues. Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathe matics.

Introduction to Non-Euclidean Geometry


Author: EISENREICH
Publisher: Elsevier
ISBN: 1483295311
Category: Mathematics
Page: 274
View: 321

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An Introduction to Non-Euclidean Geometry covers some introductory topics related to non-Euclidian geometry, including hyperbolic and elliptic geometries. This book is organized into three parts encompassing eight chapters. The first part provides mathematical proofs of Euclid’s fifth postulate concerning the extent of a straight line and the theory of parallels. The second part describes some problems in hyperbolic geometry, such as cases of parallels with and without a common perpendicular. This part also deals with horocycles and triangle relations. The third part examines single and double elliptic geometries. This book will be of great value to mathematics, liberal arts, and philosophy major students.

Non-Euclidean Geometry

A Critical and Historical Study of Its Development
Author: Roberto Bonola
Publisher: Courier Dover Publications
ISBN: N.A
Category: Mathematics
Page: 389
View: 1356

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Examines various attempts to prove Euclid's parallel postulate — by the Greeks, Arabs, and Renaissance mathematicians. It considers forerunners and founders such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, others. Includes 181 diagrams.

Non-Euclidean Geometries

János Bolyai Memorial Volume
Author: András Prékopa,Emil Molnár
Publisher: Springer Science & Business Media
ISBN: 0387295550
Category: Mathematics
Page: 506
View: 2172

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"From nothing I have created a new different world," wrote János Bolyai to his father, Wolgang Bolyai, on November 3, 1823, to let him know his discovery of non-Euclidean geometry, as we call it today. The results of Bolyai and the co-discoverer, the Russian Lobachevskii, changed the course of mathematics, opened the way for modern physical theories of the twentieth century, and had an impact on the history of human culture. The papers in this volume, which commemorates the 200th anniversary of the birth of János Bolyai, were written by leading scientists of non-Euclidean geometry, its history, and its applications. Some of the papers present new discoveries about the life and works of János Bolyai and the history of non-Euclidean geometry, others deal with geometrical axiomatics; polyhedra; fractals; hyperbolic, Riemannian and discrete geometry; tilings; visualization; and applications in physics.

Euclidean and Non-Euclidean Geometries

Development and History
Author: Marvin J. Greenberg
Publisher: Macmillan
ISBN: 9780716724469
Category: Mathematics
Page: 483
View: 2527

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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.

A Simple Non-Euclidean Geometry and Its Physical Basis

An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity
Author: I.M. Yaglom
Publisher: Springer Science & Business Media
ISBN: 146126135X
Category: Mathematics
Page: 307
View: 5841

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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.

Taxicab Geometry

An Adventure in Non-Euclidean Geometry
Author: Eugene F. Krause
Publisher: Courier Corporation
ISBN: 048613606X
Category: Mathematics
Page: 96
View: 5403

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Fascinating, accessible introduction to unusual mathematical system in which distance is not measured by straight lines. Illustrated topics include applications to urban geography and comparisons to Euclidean geometry. Selected answers to problems.

The “Golden” Non-Euclidean Geometry

Hilbert's Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant
Author: Alexey Stakhov,Samuil Aranson
Publisher: World Scientific
ISBN: 9814678317
Category: Mathematics
Page: 308
View: 641

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This unique book overturns our ideas about non-Euclidean geometry and the fine-structure constant, and attempts to solve long-standing mathematical problems. It describes a general theory of "recursive" hyperbolic functions based on the "Mathematics of Harmony," and the "golden," "silver," and other "metallic" proportions. Then, these theories are used to derive an original solution to Hilbert's Fourth Problem for hyperbolic and spherical geometries. On this journey, the book describes the "golden" qualitative theory of dynamical systems based on "metallic" proportions. Finally, it presents a solution to a Millennium Problem by developing the Fibonacci special theory of relativity as an original physical-mathematical solution for the fine-structure constant. It is intended for a wide audience who are interested in the history of mathematics, non-Euclidean geometry, Hilbert's mathematical problems, dynamical systems, and Millennium Problems. Contents:The Golden Ratio, Fibonacci Numbers, and the "Golden" Hyperbolic Fibonacci and Lucas FunctionsThe Mathematics of Harmony and General Theory of Recursive Hyperbolic FunctionsHyperbolic and Spherical Solutions of Hilbert's Fourth Problem: The Way to the Recursive Non-Euclidean GeometriesIntroduction to the "Golden" Qualitative Theory of Dynamical Systems Based on the Mathematics of HarmonyThe Basic Stages of the Mathematical Solution to the Fine-Structure Constant Problem as a Physical Millennium ProblemAppendix: From the "Golden" Geometry to the Multiverse Readership: Advanced undergraduate and graduate students in mathematics and theoretical physics, mathematicians and scientists of different specializations interested in history of mathematics and new mathematical ideas.

Non-Euclidean Geometry


Author: Stefan Kulczycki
Publisher: Courier Corporation
ISBN: 0486155013
Category: Mathematics
Page: 208
View: 365

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This accessible approach features stereometric and planimetric proofs, and elementary proofs employing only the simplest properties of the plane. A short history of geometry precedes the systematic exposition. 1961 edition.

The Foundations of Geometry and the Non-Euclidean Plane


Author: G.E. Martin
Publisher: Springer Science & Business Media
ISBN: 1461257255
Category: Mathematics
Page: 512
View: 6922

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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.

Geometry of Lengths, Areas, and Volumes: Two-Dimensional Spaces, Volume 1


Author: James W. Cannon
Publisher: American Mathematical Soc.
ISBN: 1470437147
Category: Geometry
Page: 119
View: 9179

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This is the first of a three volume collection devoted to the geometry, topology, and curvature of 2-dimensional spaces. The collection provides a guided tour through a wide range of topics by one of the twentieth century's masters of geometric topology. The books are accessible to college and graduate students and provide perspective and insight to mathematicians at all levels who are interested in geometry and topology. The first volume begins with length measurement as dominated by the Pythagorean Theorem (three proofs) with application to number theory; areas measured by slicing and scaling, where Archimedes uses the physical weights and balances to calculate spherical volume and is led to the invention of calculus; areas by cut and paste, leading to the Bolyai-Gerwien theorem on squaring polygons; areas by counting, leading to the theory of continued fractions, the efficient rational approximation of real numbers, and Minkowski's theorem on convex bodies; straight-edge and compass constructions, giving complete proofs, including the transcendence of and , of the impossibility of squaring the circle, duplicating the cube, and trisecting the angle; and finally to a construction of the Hausdorff-Banach-Tarski paradox that shows some spherical sets are too complicated and cloudy to admit a well-defined notion of area.

A New Perspective on Relativity

An Odyssey in Non-Euclidean Geometries
Author: Bernard H. Lavenda
Publisher: World Scientific
ISBN: 9814340499
Category: Geometry, Non-Euclidean
Page: 668
View: 1621

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Starting off from noneuclidean geometries, apart from the method of Einstein''s equations, this book derives and describes the phenomena of gravitation and diffraction. A historical account is presented, exposing the missing link in Einstein''s construction of the theory of general relativity: the uniformly rotating disc, together with his failure to realize, that the Beltrami metric of hyperbolic geometry with constant curvature describes exactly the uniform acceleration observed. This book also explores these questions: How does time bend? Why should gravity propagate at the speed of light? How does the expansion function of the universe relate to the absolute constant of the noneuclidean geometries? Why was the Sagnac effect ignored? Can Maxwell''s equations accommodate mass? Is there an inertia due solely to polarization? Can objects expand in elliptic geometry like they contract in hyperbolic geometry?