Euclidean and Non-Euclidean Geometries

Development and History

Author: Marvin J. Greenberg

Publisher: Macmillan

ISBN: 9780716724469

Category: Mathematics

Page: 483

View: 4038

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.
Posted in Mathematics

Non-Euclidean Geometry

Author: Roberto Bonola

Publisher: Courier Corporation

ISBN: 048615503X

Category: Mathematics

Page: 448

View: 7977

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.
Posted in Mathematics

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: 6034

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.
Posted in Mathematics

Using History to Teach Mathematics

An International Perspective

Author: Victor J. Katz

Publisher: Cambridge University Press

ISBN: 9780883851630

Category: Mathematics

Page: 261

View: 6813

This volume examines how the history of mathematics can find application in the teaching of mathematics itself.
Posted in Mathematics

Das Zebra-Buch zur Geometrie

Author: Ferdinand Verhulst,Sebastian Walcher

Publisher: Springer-Verlag

ISBN: 3642052487

Category: Mathematics

Page: 296

View: 9275

In den Niederlanden erscheint seit ca. zehn Jahren die erfolgreiche Zebra-Buchreihe, die durch eine einzigartige Kooperation von Schulpraktikern, Mathematikdidaktikern und Fachwissenschaftlern entstanden ist. Ausgewählt für eine Übersetzung ins Deutsche wurde der Band mit dem Schwerpunktthema Geometrie. Die zahlreichen Facetten der Geometrie und ihre Querverbindungen innerhalb und außerhalb der Mathematik werden in dieser Sammlung erfahrbar. Leser werden zudem angeregt, selbst mehr herauszufinden. Im Internet stehen dafür weitere Materialien bereit.
Posted in Mathematics

Enzyklopädie Philosophie und Wissenschaftstheorie

Bd. 3: G–Inn

Author: Jürgen Mittelstraß

Publisher: Springer-Verlag

ISBN: 3476001350

Category: Philosophy

Page: 620

View: 2411

Philosophie und Wissenschaftstheorie in über 4.400 Artikeln von A bis Z. Lückenlos belegt das größte allgemeine Lexikon zur Philosophie in deutscher Sprache den heutigen Kenntnisstand: alle zentralen Begriffe, alle wichtigen Theorien, alle prägenden Philosophen. Was ist neu in der 2. Auflage? Über 400 zusätzliche Artikel dokumentieren die jüngste Entwicklung in Logik, Erkenntnis- und Wissenschaftstheorie sowie Sprachphilosophie. In der Neuauflage, erweitert auf acht Bände, liefert die Enzyklopädie zusätzlich aktuelle und ausführliche Literaturhinweise und vollständige Werkverzeichnisse auf dem jüngsten Stand. Jetzt legt der Herausgeber den 3. Band der Enzyklopädie vor. Rund 100 neue Artikel sind im Band G bis J enthalten. Darunter die Einträge: Stephen Hawking, Alexander von Humboldt, Roman Jakobson, Theodor Gomperz, Genetik, Gesundheit, Hirnforschung und Interdisziplinarität. Gehört auf den Schreibtisch eines jeden, der sich mit Philosophie und Wissenschaftstheorie beschäftigt.
Posted in Philosophy

János Bolyai

Die ersten 200 Jahre

Author: Tibor Weszely

Publisher: Springer-Verlag

ISBN: 3034600461

Category: Mathematics

Page: 283

View: 788

Biographie des ungarischen Mathematikers János Bolyai (1802-1860), der etwa gleichzeitig mit dem russischen Mathematiker Nikolai Lobatschewski und unabhängig von ihm die nichteuklidische Revolution eingeleitet hat. Diese erbrachte den Nachweis, dass die euklidische Geometrie keine Denknotwendigkeit ist, wie Kant irrtümlicherweise annahm. Das Verständnis für die kühnen Gedankengänge verbreitete sich allerdings erst in der zweiten Hälfte des 19. Jahrhunderts durch die Arbeiten von Riemann, Beltrami, Klein und Poincaré. Die nichteuklidische Revolution war eine der Grundlagen für die Entwicklung der Physik im 20. Jahrhundert und für Einsteins Erkenntnis, dass der uns umgebende reale Raum gekrümmt ist. Tibor Weszely schildert das wechselvolle Leben des Offiziers der K.u.K.-Armee, der krank und vereinsamt starb. Bolyai hat sich auch intensiv mit den komplexen Zahlen und mit Zahlentheorie befasst, ebenso auch mit philosophischen und sozialen Fragen („Allheillehre“) sowie mit Logik und Grammatik.
Posted in Mathematics

Mathematics of Physics and Engineering

Author: Edward K. Blum,Sergey V. Lototsky

Publisher: World Scientific

ISBN: 981256621X

Category: Mathematics

Page: 482

View: 1702

Aimed at scientists and engineers, this book is an exciting intellectual journey through the mathematical worlds of Euclid, Newton, Maxwell, Einstein, and Schrodinger-Dirac.While similar books present the required mathematics in a piecemeal manner with tangential references to the relevant physics and engineering, this textbook serves the interdisciplinary needs of engineers, scientists and applied mathematicians by unifying the mathematics and physics into a single systematic body of knowledge but preserving the rigorous logical development of the mathematics.The authors take an unconventional approach by integrating the mathematics with its motivating physical phenomena and, conversely, by showing how the mathematical models predict new physical phenomena.
Posted in Mathematics

Non-Euclidean Geometry

Author: H. S. M. Coxeter

Publisher: Cambridge University Press

ISBN: 9780883855225

Category: Mathematics

Page: 336

View: 4123

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.
Posted in Mathematics

5000 Jahre Geometrie

Geschichte, Kulturen, Menschen

Author: Christoph J. Scriba,Peter Schreiber

Publisher: Springer-Verlag

ISBN: 3540271864

Category: Mathematics

Page: 630

View: 970

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.
Posted in Mathematics

Is God a Mathematician?

Author: Mario Livio

Publisher: Simon and Schuster

ISBN: 9781416594437

Category: Mathematics

Page: 320

View: 9521

Bestselling author and astrophysicist Mario Livio examines the lives and theories of history’s greatest mathematicians to ask how—if mathematics is an abstract construction of the human mind—it can so perfectly explain the physical world. Nobel Laureate Eugene Wigner once wondered about “the unreasonable effectiveness of mathematics” in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that—mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is “a product of human thought that is independent of experience,” how can it so accurately describe and even predict the world around us? Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them.
Posted in Mathematics

Euclidean and Non-euclidean Geometries

Author: Maria Helena Noronha

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 409

View: 1570

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
Posted in Mathematics

A Course in Modern Geometries

Author: Judith Cederberg

Publisher: Springer Science & Business Media

ISBN: 9780387989723

Category: Mathematics

Page: 441

View: 7920

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".
Posted in Mathematics

Unvergängliche Geometrie

Author: H.S. Coxeter

Publisher: Springer-Verlag

ISBN: 3034851510

Category: Juvenile Nonfiction

Page: 558

View: 9383

Posted in Juvenile Nonfiction

Euclidean and Non-Euclidean Geometry International Student Edition

An Analytic Approach

Author: Patrick J. Ryan

Publisher: Cambridge University Press

ISBN: 0521127076

Category: Mathematics

Page: 232

View: 2156

This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic.
Posted in Mathematics

Mechanical Theorem Proving in Geometries

Basic Principles

Author: Wen-tsün Wu

Publisher: Springer Science & Business Media

ISBN: 370916639X

Category: Computers

Page: 288

View: 3501

There seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, "The objective of mathematics is the study of space forms and quantitative relations of the real world. " Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's "Elements," purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations.
Posted in Computers

Fashionable Nonsense

Postmodern Intellectuals' Abuse of Science

Author: Alan Sokal,Jean Bricmont

Publisher: Picador

ISBN: 1466862408

Category: Science

Page: 272

View: 9418

In 1996 physicist Alan Sokal published an essay in Social Text--an influential academic journal of cultural studies--touting the deep similarities between quantum gravitational theory and postmodern philosophy. Soon thereafter, the essay was revealed as a brilliant parody, a catalog of nonsense written in the cutting-edge but impenetrable lingo of postmodern theorists. The event sparked a furious debate in academic circles and made the headlines of newspapers in the U.S. and abroad. Now in Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science, Sokal and his fellow physicist Jean Bricmont expand from where the hoax left off. In a delightfully witty and clear voice, the two thoughtfully and thoroughly dismantle the pseudo-scientific writings of some of the most fashionable French and American intellectuals. More generally, they challenge the widespread notion that scientific theories are mere "narrations" or social constructions.
Posted in Science