Analytic Hyperbolic Geometry

Mathematical Foundations and Applications

Author: Abraham A Ungar

Publisher: World Scientific

ISBN: 9814479594

Category: Mathematics

Page: 484

View: 9736

' This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the resulting “gyrolanguage” of the book one attaches the prefix “gyro” to a classical term to mean the analogous term in hyperbolic geometry. The prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share. The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics. As such, it is naturally compatible with the special theory of relativity and, particularly, with the nonassociativity of Einstein velocity addition law. Along with analogies with classical results that the book emphasizes, there are remarkable disanalogies as well. Thus, for instance, unlike Euclidean triangles, the sides of a hyperbolic triangle are uniquely determined by its hyperbolic angles. Elegant formulas for calculating the hyperbolic side-lengths of a hyperbolic triangle in terms of its hyperbolic angles are presented in the book. The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and non-gyrocommutative, abound in group theory. Surprisingly, the seemingly structureless Einstein velocity addition of special relativity turns out to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, some gyrocommutative gyrogroups of gyrovectors become gyrovector spaces. The latter, in turn, form the setting for analytic hyperbolic geometry just as vector spaces form the setting for analytic Euclidean geometry. By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Möbius) gyrovector spaces form the setting for Beltrami-Klein (Poincaré) ball models of hyperbolic geometry. Finally, novel applications of Möbius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented. Contents: GyrogroupsGyrocommutative GyrogroupsGyrogroup ExtensionGyrovectors and CogyrovectorsGyrovector SpacesRudiments of Differential GeometryGyrotrigonometryBloch Gyrovector of Quantum ComputationSpecial Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint Readership: Undergraduates, graduate students, researchers and academics in geometry, algebra, mathematical physics, theoretical physics and astronomy. Keywords:Analytic Hyperbolic Geometry;Gyrogroup;Gyrovector Space;Hyperbolic Geometry;Relativistic Mass;Special RelativityKey Features:Develops an elegant conversion formula from the hyperbolic triangle hyperbolic angles to its hyperbolic side lengthsIntroduces hyperbolic vectors, called “gyrovectors", and demonstrates that Einstein velocity addition is nothing but a gyrovector addition in a gyrovector space just as Newton velocity addition is a vector addition in a vector spaceShows that Einstein's relativistic mass meshes extraordinarily well with analytic hyperbolic geometry, where it captures remarkable analogies with Newton's classical mass and the analytic Euclidean geometry of the center of momentumReviews:“This new book by Ungar is very well-written, with plenty of references and explanatory pictures. Almost all chapters include exercises which ensure that the book will reach a large audience from undergraduate and graduate students to researchers and academics in different areas of mathematics and mathematical physics. In this book, the author sets out his improved gyrotheory, capturing the curiosity of the reader with discernment, elegance and simplicity.”Mathematical Reviews “This book under review provides an efficient algebraic formalism for studying the hyperbolic geometry of Bolyai and Lobachevsky, which underlies Einstein special relativity … It is of interest both to mathematicians, working in the field of geometry, and the physicists specialized in relativity or quantum computation theory … It is recommended to graduate students and researchers interested in the interrelations among non-associative algebra, hyperbolic and differential geometry, Einstein relativity theory and the quantum computation theory.”Journal of Geometry and Symmetry in Physics “This book represents an exposition of the author's single-handed creation, over the past 17 years, of an algebraic language in which both hyperbolic geometry and special relativity find an aesthetically pleasing formulation, very much like Euclidean geometry and Newtonian mechanics find them in the language of vector spaces.”Zentralblatt MATH '

A Course in Modern Geometries

Author: Judith Cederberg

Publisher: Springer Science & Business Media

ISBN: 9780387989723

Category: Mathematics

Page: 441

View: 2775

Designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. The first chapter presents several finite geometries in an axiomatic framework, while Chapter 2 continues the synthetic approach in introducing both Euclids and ideas of non-Euclidean geometry. There follows a new introduction to symmetry and hands-on explorations of isometries that precedes an extensive analytic treatment of similarities and affinities. Chapter 4 presents plane projective geometry both synthetically and analytically, and the new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Throughout, each chapter includes a list of suggested resources for applications or related topics in areas such as art and history, plus this second edition points to Web locations of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions are available for "Cabri Geometry" and "Geometers Sketchpad".

Euclidean and Non-euclidean Geometries

Author: Maria Helena Noronha

Publisher: N.A


Category: Mathematics

Page: 409

View: 9359

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


A Metric Approach with Models

Author: Richard S. Millman,George D. Parker

Publisher: Springer Science & Business Media

ISBN: 9780387974125

Category: Mathematics

Page: 372

View: 2905

Geometry: A Metric Approach with Models, imparts a real feeling for Euclidean and non-Euclidean (in particular, hyperbolic) geometry. Intended as a rigorous first course, the book introduces and develops the various axioms slowly, and then, in a departure from other texts, continually illustrates the major definitions and axioms with two or three models, enabling the reader to picture the idea more clearly. The second edition has been expanded to include a selection of expository exercises. Additionally, the authors have designed software with computational problems to accompany the text. This software may be obtained from George Parker.

Was ist Mathematik?

Author: Richard Courant,Herbert Robbins

Publisher: Springer-Verlag

ISBN: 3642137016

Category: Mathematics

Page: 400

View: 7684

"Was ist Mathematik?" lädt jeden ein, das Reich der Mathematik zu betreten, der neugierig genug ist, sich auf ein Abenteuer einzulassen. Das Buch richtet sich an Leser jeden Alters und jeder Vorbildung. Gymnasiallehrer erhalten eine Fülle von Beispielen, Studenten bietet es Orientierung, und Dozenten werden sich an den Feinheiten der Darstellung zweier Meister ihres Faches erfreuen.

Mathematics of the 19th Century

Geometry, Analytic Function Theory

Author: Andrei N. Kolmogorov,Adolf-Andrei P. Yushkevich

Publisher: Springer Science & Business Media

ISBN: 9783764350482

Category: Mathematics

Page: 291

View: 4577

The general principles by which the editors and authors of the present edition have been guided were explained in the preface to the first volume of Mathemat ics of the 19th Century, which contains chapters on the history of mathematical logic, algebra, number theory, and probability theory (Nauka, Moscow 1978; En glish translation by Birkhiiuser Verlag, Basel-Boston-Berlin 1992). Circumstances beyond the control of the editors necessitated certain changes in the sequence of historical exposition of individual disciplines. The second volume contains two chapters: history of geometry and history of analytic function theory (including elliptic and Abelian functions); the size of the two chapters naturally entailed di viding them into sections. The history of differential and integral calculus, as well as computational mathematics, which we had planned to include in the second volume, will form part of the third volume. We remind our readers that the appendix of each volume contains a list of the most important literature and an index of names. The names of journals are given in abbreviated form and the volume and year of publication are indicated; if the actual year of publication differs from the nominal year, the latter is given in parentheses. The book History of Mathematics from Ancient Times to the Early Nineteenth Century [in Russian], which was published in the years 1970-1972, is cited in abbreviated form as HM (with volume and page number indicated). The first volume of the present series is cited as Bk. 1 (with page numbers).

Metamathematische Methoden in der Geometrie

Author: W. Schwabhäuser,W. Szmielew,A. Tarski

Publisher: Springer-Verlag

ISBN: 3642694187

Category: Mathematics

Page: 484

View: 8115

Das vorliegende Buch besteht aus zwei Teilen. Teil I enthält einen axiomatischen Aufbau der euklidischen Geometrie auf Grund eines Axiomensystems von Tarski, das in einem gewissen Sinne (auch für die absolute Geometrie) gleichwertig ist mit dem Hilbertschen Axiomensystem, aber formalisiert ist in einer Sprache, die für die Betrachtungen in Teil II besonders geeignet ist. Mehrere solche Axio mensysteme wurden schon vor langer Zeit von Tarski veröffentlicht. Hier wird nun die Durchführung eines Aufbaus der Geometrie auf Grund eines solchen Axiomensystems - unter Benutzung von Resultaten von H. N. Gupta - allgemein zugänglich gemacht. Die vorliegende Darstel lung wurde vom zuerst genannten Autor allein geschrieben, aber sie beruht zum Teil auf unveröffentlichten Resultaten von Alfred Tarski und Wanda Szmielew; daher gebührt ihnen ein Teil der Autorschaft. Mehr über Entstehung und Inhalt von Teil I sowie über die Geschichte der Tarskischen Axiomensysteme wird in der Einleitung (Abschnitt I.O) gesagt. Teil II enthält metamathematische Untersuchungen und Ergebnisse über verschiedene Geometrien, was vielfac~ auf eine Anwendung von Methoden und Sätzen der mathematischen Logik auf Geometrien hinausläuft (vgl.

The Elements of Non-Euclidean Geometry

Author: D. M.Y. Sommerville

Publisher: Courier Corporation

ISBN: 0486154580

Category: Mathematics

Page: 288

View: 3730

Renowned for its lucid yet meticulous exposition, this classic allows students to follow the development of non-Euclidean geometry from a fundamental analysis of the concept of parallelism to more advanced topics. 1914 edition. Includes 133 figures.

Differentialgeometrie von Kurven und Flächen

Author: Manfredo P. do Carmo

Publisher: Springer-Verlag

ISBN: 3322850722

Category: Technology & Engineering

Page: 263

View: 7948

Inhalt: Kurven - Reguläre Flächen - Die Geometrie der Gauß-Abbildung - Die innere Geometrie von Flächen - Anhang

College Geometry

A Unified Development

Author: David C. Kay

Publisher: CRC Press

ISBN: 1439895228

Category: Mathematics

Page: 652

View: 3019

Designed for mathematics majors and other students who intend to teach mathematics at the secondary school level, College Geometry: A Unified Development unifies the three classical geometries within an axiomatic framework. The author develops the axioms to include Euclidean, elliptic, and hyperbolic geometry, showing how geometry has real and far-reaching implications. He approaches every topic as a fresh, new concept and carefully defines and explains geometric principles. The book begins with elementary ideas about points, lines, and distance, gradually introducing more advanced concepts such as congruent triangles and geometric inequalities. At the core of the text, the author simultaneously develops the classical formulas for spherical and hyperbolic geometry within the axiomatic framework. He explains how the trigonometry of the right triangle, including the Pythagorean theorem, is developed for classical non-Euclidean geometries. Previously accessible only to advanced or graduate students, this material is presented at an elementary level. The book also explores other important concepts of modern geometry, including affine transformations and circular inversion. Through clear explanations and numerous examples and problems, this text shows step-by-step how fundamental geometric ideas are connected to advanced geometry. It represents the first step toward future study of Riemannian geometry, Einstein’s relativity, and theories of cosmology.

Geometry, Relativity, and the Fourth Dimension

Author: Rudy von Bitter Rucker

Publisher: Courier Corporation

ISBN: 9780486234007

Category: Science

Page: 133

View: 8216

Exposition of fourth dimension, concepts of relativity as Flatland characters continue adventures. Topics include curved space time as a higher dimension, special relativity, and shape of space-time. Includes 141 illustrations.

Linear Algebra Through Geometry

Author: Thomas Banchoff,John Wermer

Publisher: Springer Science & Business Media

ISBN: 9780387975863

Category: Mathematics

Page: 308

View: 5280

This book introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space.

Non-Euclidean Geometry

Author: Stefan Kulczycki

Publisher: Courier Corporation

ISBN: 0486155013

Category: Mathematics

Page: 208

View: 8993

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.

Vorlesungen über Grundlagen der Geometrie

Author: Kurt Reidemeister

Publisher: Springer-Verlag

ISBN: 3662400782

Category: Mathematics

Page: 150

View: 8088

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Modern Geometries

Non-Euclidean, Projective, and Discrete

Author: Michael Henle

Publisher: Pearson College Division

ISBN: 9780130323132

Category: Mathematics

Page: 389

View: 9316

Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.

5000 Jahre Geometrie

Geschichte, Kulturen, Menschen

Author: Christoph J. Scriba,Peter Schreiber

Publisher: Springer-Verlag

ISBN: 3540271864

Category: Mathematics

Page: 630

View: 5000

Lange bevor die Schrift entwickelt wurde, hat der Mensch geometrische Strukturen wahrgenommen und systematisch verwendet: ob beim Weben oder Flechten einfacher zweidimensionaler Muster oder beim Bauen mit dreidimensionalen Körpern. Das Buch liefert einen faszinierenden Überblick über die geometrischen Vorstellungen und Erkenntnisse der Menschheit von der Urgesellschaft bis hin zu den mathematischen und künstlerischen Ideen des 20. Jahrhunderts.