The Shape of Space (Pure and Applied Mathematics)

Book Description

Maintaining the standard of excellence set by the previous edition, this textbook covers the basic geometry of two- and three-dimensional spaces Written by a master expositor, leading researcher in the field, and MacArthur Fellow, it includes experiments to determine the true shape of the universe and contains illustrated examples and engaging exercises that teach mind-expanding ideas in an intuitive and informal way. Bridging the gap from geometry to the latest work in observational cosmology, the book illustrates the connection between geometry and the behavior of the physical universe and explains how radiation remaining from the big bang may reveal the actual shape of the universe.

Customer Reviews:

Magic book on Topology for educated commons.......2007-08-05

This is a great book for anyone who is interest in Mathematical Topology and Cosmology Topology. This book does not require a reader to have strong mathematics knowledge. It only requires a reader to have patience to think and solve some problems in the book. The most brilliant point in this book is using diagrams to illustrate the Topology concepts, such as Manifold. This help the reader to get a "feeling" of some really difficult concepts in Topology. This book should be a classic like "Flatland".

chris tam
hong kong

The joy of math.......2007-08-02

I have a bachelors degree in Math.

As Feynman said, what we really mean by math is careful reasoning. This book brings you the joy of careful reasoning, guided by an expert.

Perhaps what turns some people off math in school is that the supreme example of careful reasoning is the mathematical PROOF. (Or perhaps it's just that most math teachers are so poor.) A proof tends to look dull and ponderous on the outside, and a student can easily miss the beauty of the underlying ideas. On the other hand, for your own amusement you can figure something out to your own satisfaction, without necessarily constructing a watertight proof. This book helps you do just that.

Many newspapers contain Sudoku problems, often with the reassuring claim that no math is required! People who hated math in school can be seen working happily on Sudoku puzzles, for the sheer joy of exercising their ability to reason carefully. The same ability would bring them far more joy while reading this book and answering the puzzles/exercises spinkled throughout.

Excellent Introduction, No Assumptions.......2007-07-05

This text is non-intimidating as an introduction to topology. Weeks carefully guides the reader through the building blocks of torii, Moebius strips, projective planes, and other surfaces. After working appropriate exercises, the reader gets a chance to visualize 3-manifolds and connected sums. Some aspects of these two topics can be difficult to explain, but analogies are applied to make understanding attainable. Further, figures and illustrations exist throughout the text, and these are definitely helpful for visualizing connected sums and non-orientable surfaces (both one-sided and two-sided).

(I especially like the approach to the Gauss-Bonet theorem using double lunes. It is a carefully crafted derivation with plenty of illustrations to avoid confusion.)

Some may think this text is too simple, but it is a "must read" for anyone who has not encountered topology and who wants to do individual research on the topic. Many texts claim to be introductory texts, but they are actually designed for those who already have a degree in math and who have seen similar subject matter. However, this one is definitely for "newbies." So don't worry.

Interesting book.......2006-11-26

This is a painless way to learn some advanced topology--or at least to gain insight. It's almost a picture book. Most problems include solutions and require only a few minutes of thought. They are also worth solving.

Now that I understand what is meant be a certain topology of the cosmos, I'm astounded that anybody actually considers it possible. Fascinating.

Easy Reading.......2006-02-20

This is a very good book for people whom have a light background in math. It is a readable book and great introduction into manifolds and torus. As a mathematican I am amazed with the quality of material, examples, and thus provide one with the ability to understand the topics. I plan to use this book and some of its topics in future teachings. Thus I recommend this book for anyone especially for people who struggle with math.

Schaum's Outline of Differential Geometry (Schaum's)

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Differential Geometry review
Good as a basic textbook and a source of solve problems
Differential Geometry - A Schaum's Outline Series

Schaum's Outline of Differential Geometry (Schaum's)
Martin M. Lipschutz
Manufacturer: McGraw-Hill
ProductGroup: Book
Binding: Paperback

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Schaum's Outline of Tensor Calculus (Schaum's)
Schaum's Outline of General Topology
Schaum's Outline of Vector Analysis
Schaum's Outline of Complex Variables
Differential Geometry

ASIN: 0070379858

Book Description

Students will find all the information covered in the standard textbooks--and more--explained clearly and concisely in this powerful study tool. Unusually detailed, it elucidates all the most difficult-to-grasp concepts that class studies and texts sometimes gloss over. The hundreds of problems with fully explained solutions illuminate important points and teach students sound problem-solving skills. Ideal, also, for independent study.

Customer Reviews:

Differential Geometry review.......2007-01-04

I have found this to be an excellent addition to my library.

Good as a basic textbook and a source of solve problems.......2006-06-05

This book is intended to assist upper level undergraduate and graduate students in their understanding of differential geometry, which is the study of geometry using calculus. Usually students study differential geometry in reference to its use in relativity. I personally have a rather oddball application for the subject - modeling of curved geometry for computer graphics applications. The fundamental concepts are presented for curves and surfaces in three-dimensional Euclidean space to add to the intuitive nature of the material.
The book presumes very little in the way of background and thus starts out with the basic theory of vectors and vector calculus of a single variable in the first two chapters. The following three chapters discuss the concept and theory of curves in three dimensions including selected topics in the theory of contact.
Great care is given to the definition of a surface so that the reader has a firm foundation in preparation for further study in modern differential geometry. Thus, there is some background material in analysis and in point set topology in Euclidean spaces presented in chapters 6 and 7. The definition of a surface is detailed in chapter eight. Chapters 9 and 10 are devoted to the theory of the non-intrinsic geometry of a surface. This includes an introduction to tensor methods and selected topics in the global geometry of surfaces. The last chapter of the outline presents the basic theory of the intrinsic geometry of surfaces in three-dimensional Euclidean space.
Exercises are primarily in the form of proofs, and there are plenty of worked examples. Since the examples are kept to no more than three dimensions, the outline contains plenty of good instructive diagrams that illustrate key concepts. This Schaum's outline has quite a bit of instruction in it past the bare required minimum, but you might still want a good primary textbook. My personal favorite is Pressley's "Elementary Differential Geometry". Overall I find this to be a very good outline and source of solved problems on the subject and I highly recommend it.

Differential Geometry - A Schaum's Outline Series.......2000-06-25

As with all of the Schaum's Outline Series, this book is particularly useful if the readers intent is to gain a working knowledge of the subject. The subject of Differential Geometry is no exception. Dr. Lipschultz has done an excellent job of communicating the essential aspects of differential geometry to the reader. The book assumes a fairly low level of mathematical ability having calculus as the primary prerequisite. From this humble beginning, Dr. Lipschultz takes the reader through the necessary discussions of vector functions, curvature, fundamental forms, and tensor analysis. Given the theoretical nature of the subject, Dr. Lipschultz has included most of the theorems and associated proofs necessary for a general understanding of the subject. However, this book is not a substitute for a serious study of differential geometry. In addition most of the problems are limited to two dimensional surfaces and this reader would have enjoyed a more adventurous investigation of higher dimensional spaces. Like all Schaum's series, the text is chock full of problems and their solution. I recommend this book for anyone interested in quickly gaining a working knowledge of the subject.

Knots and Surfaces: A Guide to Discovering Mathematics (Mathematical World, Vol. 6) (Mathematical World)

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EXCELLENT INTRODUCTION TO UNUSUAL MATHEMATICAL TOPICS
Intellectual Treat

Knots and Surfaces: A Guide to Discovering Mathematics (Mathematical World, Vol. 6) (Mathematical World)
David W. Farmer , and Theodore B. Stanford
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

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Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5) (Mathematical World)
Intuitive Topology (Mathematical World, Vol 4) (Mathematical World)
The Knot Book
Why Knot?: An Introduction to the Mathematical Theory of Knots
The Shape of Space (Pure and Applied Mathematics)

ASIN: 0821804510

Book Description

In most mathematics textbooks, the most exciting part of mathematics--the process of invention and discovery--is completely hidden from the reader. The aim of Knots and Surfaces is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you.

Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirement.

Customer Reviews:

EXCELLENT INTRODUCTION TO UNUSUAL MATHEMATICAL TOPICS.......2005-12-09

This book (and the author's companion book GROUPS AND SYMMETRY) are both worth getting. This one introduces Graph Theory, Surface Topology and Knots while the other one introduces Groups, Border Patterns and Wallpaper Patterns. Both books provided a guided approach through exercises, and both books have exceptional bibliographies, and suggestions for further experimentation. (My inability to visualize or deal with Knots is not the author's fault, however.)

Intellectual Treat.......2002-05-22

This is such a wonderful book. If you are interested in mathematics but aren't a mathematician this is the book for you. While reading it and working through the problems I really had the feeling that I was doing real mathematics vs just walking the dog type problems. I think this book is just as good if not better in some regards to Jeffery Weeks popular and excellent book The Shape of Space. After reading this book you will really understand some Topology,Graph Theory, and Knot Theory.

Surgery on Contact 3-Manifolds and Stein Surfaces (Bolyai Society Mathematical Studies)

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Surgery on Contact 3-Manifolds and Stein Surfaces (Bolyai Society Mathematical Studies)
Burak Ozbagci , and András I. Stipsicz
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Similar Items:

Floer Homology, Gauge Theory, and Low Dimensional Topology: Proceedings of the Clay Mathematics Institute 2004 Summer School, Alfred Renyi Institute of Mathematics, Budapest, Hungary, June 5-26, 2004 (Clay Mathematics Proceedings, Vol. 5)
$J$-holomorphic Curves and Symplectic Topology (Colloquium Publications (Amer Mathematical Soc))
4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics) (Graduate Studies in Mathematics)
Introduction to Symplectic Topology (Oxford Mathematical Monographs)

ASIN: 3540229442

Book Description

This book is about an investigation of recent developments in the field of symplectic and contact structures on four and three dimensional manifolds, respectively, from a topologist’s point of view. The level of the book is appropriate for advanced graduate students.

There is no doubt that symplectic and contact structures are in the center of attention nowadays for low-dimensional geometers and topologists. In this volume there are two main issues that are addressed: what kind of symplectic and contact structures we can construct via surgery theory and what kind of symplectic and contact structures are not allowed via gauge theory and newly-invented Heegaard-Floer theory. It turns out that interesting results about contact structures can be obtained for example when the "classical" surgery theory is coupled with the Heegaard-Floer theory. The close relationship between symplectic and contact structures is another theme in the volume which naturally arises when one wants to perform symplectic cut and paste operation.

The material in the volume is based on two groundbreaking results of the near past Donaldson's result on the existence of Lefschetz pencils on symplectic four manifolds and Giroux' correspondence between contact structures and open book decompositions on three manifolds. The volume makes an attempt to illustrate some consequences of these results and incorporate them with the new developments in the Heegaard-Floer theory, especially the Ozsvath-Szabo contact invariants.

Plateau's Problem (Student Mathematical Library, V. 13)

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An elegant application of differential geometry

Plateau's Problem (Student Mathematical Library, V. 13)
Frederick J., Jr. Almgren
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

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Book Description

There have been many wonderful developments in the theory of minimal surfaces and geometric measure theory in the past 25 to 30 years. Many of the researchers who have produced these excellent results were inspired by this little book--or by Fred Almgren himself.

The book is indeed a delightful invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behavior of soap films. When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This new tool makes possible much exciting new analysis and many new results.

Plateau's problem and varifolds live in the world of geometric measure theory, where differential geometry and measure theory combine to solve problems which have variational aspects. The author's hope in writing this book was to encourage young mathematicians to study this fascinating subject further. Judging from the success of his students, it achieves this exceedingly well.

Customer Reviews:

An elegant application of differential geometry.......2006-04-18

This book is well-illustrated and motivates all of its theorems. In order to get anything out of this text, one should at least be familiar with basic algebraic topology of surfaces (I recommend Kinsey's "Topology of Surfaces" for beginners) and should know the calculus of differential forms and chains in Spivak's "Calculus on Manifolds". It makes for a good read aftwerwards.

Effective Computational Geometry for Curves and Surfaces (Mathematics and Visualization)

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Effective Computational Geometry for Curves and Surfaces (Mathematics and Visualization)

Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540332588

Book Description

The intent of this book is to settle the foundations of non-linear computational geometry. It covers combinatorial data structures and algorithms, algebraic issues in geometric computing, approximation of curves and surfaces, and computational topology.

Each chapter provides a state of the art, as well as a tutorial introduction to important concepts and results. The focus is on methods which are both well founded mathematically and efficient in practice.

References to open source software and discussion of potential applications of the presented techniques are also included.

This book can serve as a textbook on non-linear computational geometry. It will also be useful to engineers and researchers working in computational geometry or other fields, like structural biology, 3-dimensional medical imaging, CAD/CAM, robotics, and graphics.

Explorations in Topology: Map Coloring, Surfaces and Knots

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Explorations in Topology: Map Coloring, Surfaces and Knots
David Gay
Manufacturer: Academic Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0123708583

Book Description

This book gives students a rich experience with low-dimensional topology, enhances their geometrical and topological intuition, empowers them with new approaches to solving problems, and provides them with experiences that would help them make sense of a future, more formal topology course. The innovative story-line style of the text models the problems-solving process, presents the development of concepts in a natural way, and through its informality seduces the reader into engagement with the material. The end-of-chapter Investigations give the reader opportunities to work on a variety of open-ended, non-routine problems, and, through a modified Moore method, to make conjectures from which theorems emerge. The students themselves emerge from these experiences owning concepts and results. The end-of-chapter Notes provide historical background to the chapters ideas, introduce standard terminology, and make connections with mainstream mathematics. The final chapter of projects provides opportunities for continued involvement in research beyond the topics of the book.

* Students begin to solve substantial problems right from the start
* Ideas unfold through the context of a storyline, and students become actively involved
* The text models the problem-solving process, presents the development of concepts in a natural way, and helps the reader engage with the material

How Surfaces Intersect in Space: An Introduction to Topology (Series on Knots and Everything - Vol. 2)

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A Good Self Teaching Text for Beginners in Topology
This book taught me how to see.

How Surfaces Intersect in Space: An Introduction to Topology (Series on Knots and Everything - Vol. 2)
J. Scott Carter
Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Paperback

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The Shape of Space (Pure and Applied Mathematics)
A Geometric Approach to Differential Forms
Advanced Calculus: A Differential Forms Approach
Basic Topology (Undergraduate Texts in Mathematics)
Differential Forms (Dover Books on Mathematics)

ASIN: 9810220669

Book Description

This is a book of marvelous pictures that illustrates standard examples in low dimensional topology. The text starts at the most basic level (the intersection of coordinate planes) and gives hands on constructions of the most beautiful examples in topology: the projective plane, Poincare's example of a homology sphere, lens spaces, knotted surfaces, 2-sphere eversions, and higher dimensional manifolds. The text carefully explains the importance of the examples and the techniques without being bogged down in a morass of technicalities.

Chapter 1 opens with the classification of orientable surfaces, and the meaning of space. Chapter 2 discusses examples of non-orientable surfaces including models of the projective plane and the Klein bottle. Chapter 3 discusses how curves fit on surfaces and gives a general discussion of knotted strings in space. In Chapter 4, some examples of other 3-dimensional spaces are described. These include the 3-dimensional sphere, lens spaces, and the quaternionic projective space. In Chapter 5, the author reviews the movie techniques of studying surfaces in 4-dimensions. He shows how to move among the standard examples of Klein bottles, and he gives a "movie move" decomposition of turning the 2-sphere inside out. In the final Chapter, higher dimensional spaces are examined from the same elementary point of view.

The book is a guide book to a wide variety of topics. It will be of value to undergraduates who want to learn geometric topology and to graduate students who want examples with which they can make computations and who need an elementary description of topological spaces. Finally, the book should be interesting to other scientists and mathematicians who want to learn some examples of topological spaces.

Customer Reviews:

A Good Self Teaching Text for Beginners in Topology.......2001-04-01

I have been trying to teach myself topology and just finished the Weeks book on the shape of space. This book in comparison was a bit tougher going and a step towards teaching myself with a standard textbook. I think the previous review must have been a friend of the author because I can't say I was as enthusiastic about the book (and anyway it is a feat in itself to create paper models from the primative rapidograph drawings the author provides). I found it hard to follow the author's style and would have preferred more precision in his language. The drawings may be clear to the author but they were very confusing to me. I am not a bad artist and would find it impossible to create real life models from those drawings, many of which are a mess. On the positive side, Professor Carter relates the flavor of topology like the appetizers before a sumptuous banquet. I learned a little bit more about the subject and he shares the results of his personal research which is always good. In sum, it was a good read and probably deserves a reread. It was not a great read.

This book taught me how to see........1997-12-29

There are hundreds of diagrams of surfaces in the book. I (with assistance from my wife and two teenage children) built virtually every one with scissors, tape, and glue. We were amazed at the number of interesting questions about the shape of space that were raised in the process. In answering them, we have come to look at the world differently. Even an ordinary chair is seen as a movie with elementary changes from frame to frame. My wife made connections with sewing. We got ideas for dozens of science fair projects. We developed the beginnings of an intuition about objects in four dimensions. I will always look back on the period when I read this book as one of the exciting times of my life. The book earns my very highest recommendation.

Average customer rating:

Pretty amusing stuff
a neat book

Space-Filling Curves (Universitext)
Hans Sagan
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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Accessories:

Problem-Solving Strategies (Problem Books in Mathematics)
The Heart of Mathematics: An invitation to effective thinking
Agnesi to Zeno: Over 100 Vignettes from the History of Math

ASIN: 0387942653

Book Description

The subject of space-filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by Peano. Cantor, Hilbert, Moore, Knopp, Lebesgue, and Polya are among the prominent mathematicians who have contributed to the field. However, there have been no comprehensive treatments of the subject since Siepinsky's in 1912. Cantor showed in 1878 that the number of points on an interval is the same as the number of points in a square (or cube, or whatever), and in 1890 Peano showed that there is indeed a continuous curve that continuously maps all points of a line onto all points of a square, though the curve exists only as a limit of very convoluted curves. This book discusses generalizations of Peano's solution and the properties that such curves must possess and discusses fractals in this context. The only prerequisite is a knowledge of advanced calculus.

Customer Reviews:

Pretty amusing stuff.......2000-06-28

I really liked this book and especially enjoyed reading some parts. But I cannot give the book more than 4 stars. Several times in the text, his proofs could be much better written. Part of the difficulty, it seems to me, is his reluctance to rely on basic topology; I guess since he said you only need a background in analysis in his preface, he felt obliged to do everything with epsilons and deltas.

The beginning was very nice, introducing the Hilbert Curve the way Hilbert thought of it(geometric), and then generalizing to an arithmetic representation. A similar thing is done with the Peano Curve and the theme of the dual perspective of geometry and analysis continues throughout. For a beginner, I think this book goes a long way to explaining space-filling curves and self-similar fractals.

a neat book.......2000-05-03

Space-filling curves, such as the Hilbert and Sierpinski curves, have found applications in many domains. (I happened to become interested in them through the literature on spatial clustering.) This book gives a nice historical treatment of the development of space-filling curves, relating many of the families together. It also tries to provide a reasonable discussion of the related algorithms, e.g., for curve generation. Even if you can't follow all of the math (and I certainly can't), I think it helps to have this book around if you're trying to understand why and when these techniques work.

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CRC Standard Curves and Surfaces
David H. von Seggern
Manufacturer: CRC
ProductGroup: Book
Binding: Hardcover

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Book Description

CRC Standard Curves and Surfaces is a comprehensive illustrated catalog of curves and surfaces of geometric figures and algebraic, transcendental, and integral equations used in elementary and advanced mathematics. More than 800 graphics images are featured. Based on the successful CRC Handbook of Mathematical Curves and Surfaces, this new volume retains the easy to use "catalog" format of the original book. Illustrations are presented in a common format organized by type of equation. Associated equations are printed in their simplest form along with any notes required to understand the illustrations. Equations and graphics appear in a side-by-side format, with figures printed on righthand pages and text on lefthand pages. Most curves and surfaces are plotted with several parameter selections so that the variation of the mathematical functions are easily understandable. Coverage on algebraic surfaces and transcendental surfaces has been expanded by 30% over the original edition; material on functions in mathematical physics has expanded by 50%. New material on functions of random processes and functions of complex variable surfaces has been added. A complementary software program (see the next title listed in this catalog) enables you to plot all of the functions found in this book.

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