Transformation Geometry

An Introduction to Symmetry

Author: George E. Martin

Publisher: Springer Science & Business Media

ISBN: 1461256801

Category: Mathematics

Page: 240

View: 9087

Transformation Geometry: An Introduction to Symmetry offers a modern approach to Euclidean Geometry. This study of the automorphism groups of the plane and space gives the classical concrete examples that serve as a meaningful preparation for the standard undergraduate course in abstract algebra. The detailed development of the isometries of the plane is based on only the most elementary geometry and is appropriate for graduate courses for secondary teachers.

Geometric Constructions

Author: George E. Martin

Publisher: Springer Science & Business Media

ISBN: 1461206294

Category: Mathematics

Page: 206

View: 2593

Geometric constructions have been a popular part of mathematics throughout history. The first chapter here is informal and starts from scratch, introducing all the geometric constructions from high school that have been forgotten or were never learned. The second chapter formalises Plato's game, and examines problems from antiquity such as the impossibility of trisecting an arbitrary angle. After that, variations on Plato's theme are explored: using only a ruler, a compass, toothpicks, a ruler and dividers, a marked rule, or a tomahawk, ending in a chapter on geometric constructions by paperfolding. The author writes in a charming style and nicely intersperses history and philosophy within the mathematics, teaching a little geometry and a little algebra along the way. This is as much an algebra book as it is a geometry book, yet since all the algebra and geometry needed is developed within the text, very little mathematical background is required. This text has been class tested for several semesters with a master's level class for secondary teachers.

Geometry and Symmetry

Author: Paul B. Yale

Publisher: Courier Corporation

ISBN: 0486169324

Category: Mathematics

Page: 288

View: 4432

DIVIntroduction to the geometry of euclidean, affine and projective spaces with special emphasis on the important groups of symmetries of these spaces. Many exercises, extensive bibliography. Advanced undergraduate level. /div

Continuous Symmetry

From Euclid to Klein

Author: William H. Barker,Roger Howe

Publisher: American Mathematical Soc.

ISBN: 0821839004

Category: Mathematics

Page: 546

View: 9969

The fundamental idea of geometry is that of symmetry. With that principle as the starting point, Barker and Howe begin an insightful and rewarding study of Euclidean geometry. The primary focus of the book is on transformations of the plane. The transformational point of view provides both a path for deeper understanding of traditional synthetic geometry and tools for providing proofs that spring from a consistent point of view. As a result, proofs become more comprehensible, as techniques can be used and reused in similar settings. The approach to the material is very concrete, with complete explanations of all the important ideas, including foundational background. The discussions of the nine-point circle and wallpaper groups are particular examples of how the strength of the transformational point of view and the care of the authors' exposition combine to give a remarkable presentation of topics in geometry. This text is for a one-semester undergraduate course on geometry. It is richly illustrated and contains hundreds of exercises.


An Introduction to Group Theory and Its Applications

Author: Roy McWeeny

Publisher: Courier Corporation

ISBN: 0486138801

Category: Science

Page: 256

View: 9934

Well-organized volume develops ideas of group and representation theory in progressive fashion. Emphasis on finite groups describing symmetry of regular polyhedra and of repeating patterns, plus geometric illustrations.

Transformation Groups for Beginners

Author: Sergeĭ Vasilʹevich Duzhin,Boris Dmitrievich Chebotarevskiĭ

Publisher: American Mathematical Soc.

ISBN: 0821836439

Category: Mathematics

Page: 246

View: 8503

This book is intended for undergraduate students and all those interested in mathematics. Its goal is to give an easy introduction to the concept of a transformation group using examples from different areas of mathematics. The warm-up of the first two chapters includes a discussion of algebraic operations on points in the plane, and of Euclidean plane movements. Then the notions of a transformation group and of an abstract group are introduced. Group actions, orbits, and invariants constitute the subject of the next chapter. The book concludes with an elementary exposition of the basic ideas of Sophus Lie about symmetries of differential equations. The book contains plenty of figures, as well as many exercises with hints and solutions, which help the reader to master the material.

Linear Algebra Through Geometry

Author: Thomas Banchoff,John Wermer

Publisher: Springer Science & Business Media

ISBN: 1461243904

Category: Mathematics

Page: 308

View: 826

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.

The Four Pillars of Geometry

Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 0387255303

Category: Mathematics

Page: 228

View: 1954

This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises

Conics and Cubics

A Concrete Introduction to Algebraic Curves

Author: Robert Bix

Publisher: Springer Science & Business Media

ISBN: 1475729758

Category: Mathematics

Page: 292

View: 7692

Algebraic curves are the graphs of polynomial equations in two vari 3 ables, such as y3 + 5xy2 = x + 2xy. By focusing on curves of degree at most 3-lines, conics, and cubics-this book aims to fill the gap between the familiar subject of analytic geometry and the general study of alge braic curves. This text is designed for a one-semester class that serves both as a a geometry course for mathematics majors in general and as a sequel to college geometry for teachers of secondary school mathe matics. The only prerequisite is first-year calculus. On the one hand, this book can serve as a text for an undergraduate geometry course for all mathematics majors. Algebraic geometry unites algebra, geometry, topology, and analysis, and it is one of the most exciting areas of modem mathematics. Unfortunately, the subject is not easily accessible, and most introductory courses require a prohibitive amount of mathematical machinery. We avoid this problem by focusing on curves of degree at most 3. This keeps the results tangible and the proofs natural. It lets us emphasize the power of two fundamental ideas, homogeneous coordinates and intersection multiplicities.

Symmetries and Group Theory in Particle Physics

An Introduction to Space-Time and Internal Symmetries

Author: Giovanni Costa,Gianluigi Fogli

Publisher: Springer

ISBN: 3642154824

Category: Science

Page: 291

View: 6855

Symmetries, coupled with the mathematical concept of group theory, are an essential conceptual backbone in the formulation of quantum field theories capable of describing the world of elementary particles. This primer is an introduction to and survey of the underlying concepts and structures needed in order to understand and handle these powerful tools. Specifically, in Part I of the book the symmetries and related group theoretical structures of the Minkowskian space-time manifold are analyzed, while Part II examines the internal symmetries and their related unitary groups, where the interactions between fundamental particles are encoded as we know them from the present standard model of particle physics. This book, based on several courses given by the authors, addresses advanced graduate students and non-specialist researchers wishing to enter active research in the field, and having a working knowledge of classical field theory and relativistic quantum mechanics. Numerous end-of-chapter problems and their solutions will facilitate the use of this book as self-study guide or as course book for topical lectures.

Groups and Symmetry

A Guide to Discovering Mathematics

Author: David W. Farmer

Publisher: American Mathematical Soc.

ISBN: 9780821872642

Category: Mathematics

Page: 102

View: 5258

Mathematics is discovered by looking at examples, noticing patterns, making conjectures, and testing those conjectures. Once discovered, the final results get organized and put in textbooks. The details and the excitement of the discovery are lost. This book introduces the reader to the excitement of the original discovery. By means of a wide variety of tasks, readers are led to find interesting examples, notice patterns, devise rules to explain the patterns, and discover mathematics for themselves. The subject studied here is the mathematics behind the idea of symmetry, but the methods and ideas apply to all of mathematics. The only prerequisites are enthusiasm and a knowledge of basic high-school math. The book is only a guide. It will start you off in the right direction and bring you back if you stray too far. The excitement and the discovery are left to you.

Euclidean Geometry and Transformations

Author: Clayton W. Dodge

Publisher: Courier Corporation

ISBN: 0486138429

Category: Mathematics

Page: 304

View: 6645

This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.

An Introduction to Twistor Theory

Author: S. A. Huggett,None,K. P. Tod

Publisher: Cambridge University Press

ISBN: 9780521456890

Category: Mathematics

Page: 178

View: 3442

This book is an introduction to twistor theory and modern geometrical approaches to space-time structure at the graduate or advanced undergraduate level. It will be valuable also to the physicist as an introduction to some of the mathematics that has proved useful in these areas, and to the mathematician as an example of where sheaf cohomology and complex manifold theory can be used in physics.

Introduction to Mechanics and Symmetry

A Basic Exposition of Classical Mechanical Systems

Author: J.E. Marsden,Tudor Ratiu

Publisher: Springer Science & Business Media

ISBN: 0387217924

Category: Science

Page: 586

View: 3598

A development of the basic theory and applications of mechanics with an emphasis on the role of symmetry. The book includes numerous specific applications, making it beneficial to physicists and engineers. Specific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering. This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available.

Mathematics and Technology

Author: Christiane Rousseau,Yvan Saint-Aubin

Publisher: Springer Science & Business Media

ISBN: 0387692169

Category: Mathematics

Page: 582

View: 6240

This book introduces the student to numerous modern applications of mathematics in technology. The authors write with clarity and present the mathematics in a clear and straightforward way making it an interesting and easy book to read. Numerous exercises at the end of every section provide practice and reinforce the material in the chapter. An engaging quality of this book is that the authors also present the mathematical material in a historical context and not just the practical one. Mathematics and Technology is intended for undergraduate students in mathematics, instructors and high school teachers. Additionally, its lack of calculus centricity as well as a clear indication of the more difficult topics and relatively advanced references make it suitable for any curious individual with a decent command of high school math.

A Course in Modern Geometries

Author: Judith Cederberg

Publisher: Springer Science & Business Media

ISBN: 1475734905

Category: Mathematics

Page: 441

View: 7859

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".

Algebra and Geometry

Author: Alan F. Beardon

Publisher: Cambridge University Press

ISBN: 9781139443494

Category: Mathematics

Page: N.A

View: 1256

Describing two cornerstones of mathematics, this basic textbook presents a unified approach to algebra and geometry. It covers the ideas of complex numbers, scalar and vector products, determinants, linear algebra, group theory, permutation groups, symmetry groups and aspects of geometry including groups of isometries, rotations, and spherical geometry. The book emphasises the interactions between topics, and each topic is constantly illustrated by using it to describe and discuss the others. Many ideas are developed gradually, with each aspect presented at a time when its importance becomes clearer. To aid in this, the text is divided into short chapters, each with exercises at the end. The related website features an HTML version of the book, extra text at higher and lower levels, and more exercises and examples. It also links to an electronic maths thesaurus, giving definitions, examples and links both to the book and to external sources.

A Primer for Undergraduate Research

From Groups and Tiles to Frames and Vaccines

Author: Aaron Wootton,Valerie Peterson,Christopher Lee

Publisher: Birkhäuser

ISBN: 3319660659

Category: Mathematics

Page: 313

View: 5732

This highly readable book aims to ease the many challenges of starting undergraduate research. It accomplishes this by presenting a diverse series of self-contained, accessible articles which include specific open problems and prepare the reader to tackle them with ample background material and references. Each article also contains a carefully selected bibliography for further reading. The content spans the breadth of mathematics, including many topics that are not normally addressed by the undergraduate curriculum (such as matroid theory, mathematical biology, and operations research), yet have few enough prerequisites that the interested student can start exploring them under the guidance of a faculty member. Whether trying to start an undergraduate thesis, embarking on a summer REU, or preparing for graduate school, this book is appropriate for a variety of students and the faculty who guide them.

Geometries and Transformations

Author: Norman W. Johnson

Publisher: Cambridge University Press

ISBN: 1107103401

Category: Mathematics

Page: 350

View: 6363

A readable exposition of how Euclidean and other geometries can be distinguished using linear algebra and transformation groups.