Book Description

This textbook demonstrates the excitement and beauty of geometry. The approach is that of Klein in his Erlangen program: a geometry is a space together with a set of transformations of that space. The authors explore various geometries: affine, projective, inversive, non-Euclidean and spherical. In each case they carefully explain key results and discuss the relationship among geometries. This richly illustrated and clearly written text includes full solutions to over 200 problems and is suitable both for undergraduate courses on geometry and as a resource for self study. ... Read more

Reviews (3)

Good and enjoyable for a wide range of readers
A quarter century ago I noticed that some of the graduate physics students in my university were carrying around copies of Scientific American. Armed with that clue, I dug out every article on the newly discovered fundamental particles. Within the space of a week of fairly easy reading I was able to acquire a good sense of what this subject was all about. These articles explained the basic stuff our professors assumed we must know (but most of us surely didn't).

Brannan, Esplen, and Gray's Geometry accomplish for math what those Scientific American articles did for physics: speaking at a level accessible to anyone with a good high school education, they bring the interested reader up to speed in affine, projective, hyperbolic, inversive, and spherical geometry. They provide the simple explanations, diagrams, and computational details you are assumed to know-but probably don't-when you take advanced courses in topology, differential geometry, algebraic geometry, Lie groups, and more. I wish I had had a book like this when I learned those subjects.

Individual chapters of about 50 pages focus on distinct geometries. Each one is written to be studied in the course of five evenings: a week or two of work apiece. Although they build sequentially, just about any of them can be read after mastering the basic ideas of projective geometry (chapter 3) and inversive geometry (chapter 5). This makes the latter part of the book relatively accessible even to the less-committed reader and an effective handbook for someone looking for just an overview and basic formulas.

The approach is surprisingly sophisticated. The authors do not shy away from introducing and using a little bit of group theory, even at the outset. (Scientific American, even in its heyday, never dared do that.) They present all geometries from a relatively modern point of view, as the study of the invariants of a transitive group of transformations on a set. Many explanations and proofs are based on exploiting properties of these transformations. This brings a welcome current of rigor and elegance to a somewhat static subject long relegated to out of date or sloppy authors (with the exception of a few standouts, such as Lang & Murrow's "Geometry").

One nice aspect is the authors' evident awareness of and appreciation for the history of mathematics. Marginal notes begin at Plato and wind up with Felix Klein's Erlangen program some 2300 years later. Although the text does not necessarily follow the historical development of geometry, its references to that development provide a nice context for the ideas. This is an approach that would improve the exposition of many math texts at all levels.

The authors are British and evidently write for students with slightly different backgrounds than American undergraduates. Obvious prerequisites are a mastery of algebra and a good high school course in Euclidean geometry. Synopses of the limited amounts of group theory and linear algebra needed appear in two brief appendices. However, readers had better be intuitively comfortable with matrix operations, including diagonalization and finding eigenspaces, because matrices and complex numbers are used throughout the book for performing computations and developing proofs. A knowledge of calculus is not needed. Indeed, calculus is not used in the first two-thirds of the book, appearing only briefly to derive a distance formula for hyperbolic geometry (a differential equation for the exponential map is derived and solved). During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses). Definitions of these functions are not routinely provided, but algebraic identities appear in marginal notes where they are needed.

Now for the quibbles. The book has lots of diagrams, but not enough of them. The problems are usually trivial, tending to ask for basic calculations to reinforce points in the text. The text itself does not go very deeply into any one geometry, being generally content with a few illustrative theorems. An opportunity exists here to create a set of gradually more challenging problems that would engage smarter or more sophisticated readers, as well as show the casual reader where the theories are headed.

This book is the work of three authors and it shows, to ill effect, in Chapter 6 ("non-euclidean geometry"). Until then, the text is remarkably clean and free of typographical and notational errors. This chapter contains some glaring errors. For example, a function s(z) is defined at the beginning of a proof on page 296, but the proof confusingly proceeds to refer to "s(0,c)", "s(a,b)", and so on.

The written-by-committee syndrome appears in subtler ways. There are few direct cross-references among the chapters on inversive, hyperbolic, and spherical geometry, despite the ample opportunities presented by the material. Techniques used in one chapter that would apply without change to similar situations in another are abandoned and replaced with entirely different techniques. Within the aberrant Chapter 6, some complex derivations could be replaced by much simpler proofs based on material earlier in the chapter.

The last chapter attempts to unify the preceding ones by exhibiting various geometries as sub-geometries of others. It would have been better to make the connections evident as the material was being developed. It is disappointing, too, that nothing in this book really hints at the truly interesting developments in geometry: differentiable manifolds, Lie groups, Cartan connections, complex variable theory, quaternion actions, and much more. Indeed, any possible hint seems willfully suppressed: the matrix groups in evidence, such as SL(2, R), SU(1,1, C), PSL(3, R), O(3), and so on, are always given unconventional names, for instance. Even where a connection is screaming out, it is not made: the function abstractly named "g" on pages 296-97 is the exponential map of differential geometry, for instance.

Despite these limitations, Brannan et al. is a good and enjoyable book for anyone from high school through first-year graduate level in mathematics.

A lovely Introduction to all kinds of Plane Geometries
This book gives a beautiful overview of geometry of 2 dimensions. All of the book is about many plane geometries I have heard of, but didn't really know. This book changed that.

The first chapter treats some basics about conics. The second chapter is on affine geometry. The third and fourth chapters are about projective geometry. In the fifth chapter you will be led through Inversive geometry which functions as a base for the sixth and seventh chapter. The sixth chapter has as itst title Non-Euclidean geometry, but it is in fact the Hyperbolic geometry of Boljay in a formulation of Henry Poincaré. The seventh chapter is about Spherical Geometry. In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself. It is clearly demonstrated that this is less trivial than you would expect.

I learned two things from this book. The first is, that you can, in principle, prove every theorem of geometry by just using Euclidean geometry. But if you do this, the amount of work it takes can be very huge indeed. It is a far better strategy to try to determine what geometry is best suited for the problem at hand, and solve it within that geometry.

Since the book gives a very clear picture not only of the particular geometries, but also to how the geometries relate to each other, you have, as an extra bonus, insight in the level of abstraction and the scope of your theorem.

The second thing I learned is how you can use geometry to make concepts as simple as 'triangle' precise. What I mean is this: a right angle triangle is not the same as an equilateral triangle. But both are the same in the sense that they are both triangles. The question is this: how can two 'things' be the same and at the same time not 'the same'? The book gives an answer to this 'question about the meaning of abstractions'. It gives the following solution. Take a triangle, ANY triangle. Consider the group of all affine transformations A (which consists of an uncountably infinite set of transformations.) If you subject this one triangle Tr to every affine transformation in this group A, you will have created a set consisting of exactly ALL triangles. In other words, the abstract idea of 'triangle' consists of ONE triangle Tr together with the set of ALL affine transformations. You can denote this as the pair (Tr, A). In the same way you can express the abstract idea of ellipse by the pair (El, A), and the abstract idea of parabola by the pair (Par, A). And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations.

The book presupposes some group theory and some knowledge of linear algebra. Furthermore you have to know a little calculus. I have very little knowledge of group theory, and I have just about enough knowledge and skill about linear algebra to know the difference between an orthogonal and unitary matrix, and to know what eigenvectors are. I have studied the first 5 chapters of CALCULUS from Tom M. Apostol, which does not go too deep into linear algebra. This proved to be enough.

I have only one point of critique. Virtually all problems in the book are of the 'plug in type', even those at the end of every chapter (from which, by the way, you cannot find the solutions at the end of the book, while the solutions of those in the text can be found in an appendix). If you have understood the text, you have no difficulties whatsoever to solve them. The problems are not challenging enough to give you a real skill in all of these geometries, although they do become more challenging in later chapters. They are only intended to help you to understand the basic principles of all of these geometries, no more, no less. So if you want to have a tool to help you in obtaining a greater skill in, say, the special theory of relativity by studying hyperbolic geometry, this is not a suitable book. That is why I have given it 4 stars, and not the full 5 stars.

I also have a piece of advise. Although the problems are, from a conceptual point of view, not challenging, a mistake is easily made. Therefore it is best to solve the problems by making use of a mathematical program like Maple or Mathematica. If you then have made a mistake, you can backtrack exactly where you have made it, and let the program take care of all of the tedious calculations. This has also stimulated to try to calculate some outcomes by following a different approach, and then to compare the results.

I have enjoyed studying this book immensely.

Best on affine transformations used in computer graphics
I'm trying to understand transformations used in computer graphics, for example world transformation used in Windows GDI API. And I found this book to be the best description on the topic, that is affine transformations ... Read more

Book Description

This introductory text explores the translation of geometric concepts into the language of numbers in order to define the position of a point in space (the orbit of a satellite, for example). The two-part treatment begins with discussions of the coordinates of points on a line, coordinates of points in a plane, and the coordinates of points in space. Part 2 examines geometry as an aid to calculation and the necessity and peculiarities of four-dimensional space. Written for systematic study, it features a helpful series of "road signs" in the margins, alerting students to passages requiring particular attention, and an abundance of ingenious problems--with solutions, answers, and hints--promote habits of independent work. 1967 edition.
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Reviews (1)

A fascinating introduction to coordinate geometry.
The authors of this slim volume demonstrate the power of coordinate geometry, which they describe as a means of translating geometric figures into algebraic formulas, through their lucid exposition, interesting examples, and well-chosen exercises.

The authors begin with the coordinate geometry of the real line. They discuss absolute value and define what distance means. Next the authors examine the coordinate geometry of the plane. They define distance in the plane, show how relations among the coordinates define geometric figures, and discuss different coordinate systems that can be used in the plane. Their examples illustrate how algebraic methods developed by Rene Descartes make it possible to solve geometric problems efficiently that would be quite difficult to solve using synthetic geometry. The authors then treat the coordinate geometry of three-dimensional space in a similar manner.

The second part of the book begins with a problem concerning lattice points in the plane. The authors use this example and its generalizations to justify exploring the coordinate geometry of four-dimensional space. They carefully treat the example of a four-dimensional unit hypercube, examining its properties by considering its analogues in lower dimensions: the segment [0, 1] of the real number line, the unit square in the coordinate plane, and the unit cube in space.

Since the book was initially written for a correspondence course for high school students in the Soviet Union, it is designed for self-study and accessible to students who have had high school courses in algebra and geometry. Since students in the Soviet Union were able to mail their solutions to the exercises to the authors when the authors were professors at the University of Moscow, answers to most of the exercises are not provided. The exercises are thought-provoking and some are quite challenging.

I also highly recommend that you explore the other volumes in the Gelfand School Outreach Program. They include Algebra, Functions and Graphs, and Trigonometry. ... Read more

Book Description

In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises.The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.The author emphasizes the geometric aspects of the subject, which helps students gain intuition.A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. ... Read more

Reviews (3)

It's worth your money!
This book is not just for topologists! If you're like me, then you've spent countless nights sans Hatcher's book trying to figure out the fundamental group of a beer can. Look no further, the answers are here!

Be sure to check out the vivid detail Hatcher brings to the Van Kampen theorem. I've not actually read that part myself, as I do not trust german mathematics.

You would not regret if you buy this.
There are many really good textbooks on algebraic topology and each has its own merit: Bredon for his effort in explaining everything that can be dealt without using spectral equences, Fomenko & Novikov for their effort in unifying differential geometry and algebraic/differential topology.
Hatcher's book is intended as one of the series that cover every aspect of the subject. Separate books on vector bundles and K-theory, and spectral sequences respectively, are to appear sometime in the future. Thus this one covers ordinary homology/cohomology and homotopy theory only. His writing style is helpful and user-friendly, not demanding extensive "mathematical maturity".
I am not sure if this is "the" textbook on algebraic topology, but I bet this is among the best ones. You would not regret if you buy this, even when an electronic version is available online (for free) from the author's home page.

The Last Text on Introductory Algebraic Topology
No serious introductory text on basic algebraic topology has ever achieved this level of clarity, readability and depth. Its richness in examples (in both the main text and the problems) exposes a beginner to the underlying mechanisms of geometry in algebraic topology; its choice and arrangement of topics strike a perfect balance between accesibility and substantiveness; its lively and motivating exposition makes a student reluctant to attend the often boring topology classes. For a novice, this should be the first reading on the subject before (s)he is ruined by the many existing daunting texts; for a veteran, this can be very nourishing, especially if (s)he is already ruined by those either unreadable or shallow 'introduction's. ... Read more

Book Description

Geometry is hard. This book makes it easier. You do the math. This is the fourth title in the series designed to help high school and college students through a course they'd rather not be taking. A non-intimidating, easy-to-understand companion to their textbook, this book takes students through the standard curriculum of topics, including proofs, polygons, coordinates, topology, and much more. ... Read more

Book Description

This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods. ... Read more

Reviews (3)

Ingenious Compilation of Essential Fractals
The Geometry of Fractal Sets by Falconer is an elegant composition of many necessary fractals, measures, projections, and dimensions. Included in the monograph are the most inspiring and applicable Besicovitch fractal sets, Kakeya fractal sets, the Appolonian packing fractal, osculatory packings, horseshoe fractals, Perron trees, hypercycloids, the Nikodym set, Lebesgue measure, Hausdorff dimension, sets of integral and non-integral dimension, sets in higher-dimensions, Borel measure, binary sets, Vitali coverings, polar reciprocity, Souslin sets, sigma-fields, tangents, net measures, the semicontinuity theorems of Golab and Vishtukin, osculatory packings, diophantine approximations, Fourier series, transforms and multipliers, Brownian motion, Grassmanian manifolds.......you name it this book explains and connects it all.

The text is written in full proper-fonting and contains many illustrations. Qualitatively the book should be of high value to researchers, graduates, and Phd's with the finest tastes.

Introduction to geometric measure theory
This book is devoted to the hausdorf measure and Hausdorff dimension of subsets of R^n and to an extensive study of their geometry: existence of tangency, projection, etc. One chapter deals with Besicovich sets used for constructing counter-examples, especially in Harmonic analysis.

The book finish with a magnificent list of examples of haussdorff dimension computation: self-similar sets, Apollonian packings, number theory, Feigenbaum logistic map and Brownian motion.
The bibliography, of incredible quality, achieves to make the book a reference for anyone interested in fractals.

Advanced treatise on fractal geometry.
This text is a must-reading for anyone seeking advanced knowledge on fractal geometry. It is dense and deep, but clear and concise. It includes a lot of interesting material ranging from basic measure-theoretic concepts up to the disprove of Vitushkin's conjecture. It's got an extensive list of references, mostly to the original papers, making it a fundamental research tool.

As it can be inferred from the preceeding paragraph, the book is not for begineers; it was designed for graduate level courses. Undergrads and laymen should start with Edgar's "Measure, Topology, and Fractal Geometry" and Falconer's "Fractal Geometry: Mathematical Foundations and Applications".

Please check my other reviews (just click on my name above). ... Read more

Book Description

From the reviews: This book is devoted to the study of sheaves by microlocal methods..(it) may serve as a reference source as well as a textbook on this new subject. Houzel's historical overview of the development of sheaf theory will identify important landmarks for students and will be a pleasure to read for specialists. Math. Reviews 92a (1992). The book is clearly and precisely written, and contains many interesting ideas: it describes a whole, largely new branch of mathematics.(...)The book can be strongly recommended to a younger mathematician enthusiastic to assimilate a new range of techniques allowing flexible application to a wide variety of problems. Bull. L.M.S. (1992) ... Read more

Book Description

Polyhedra have cropped up in many different guises throughout recorded history. Recently, polyhedra and their symmetries have been cast in a new light by combinatorics and group theory. This unique text comprehensively documents the many and varied ways that polyhedra have come to the fore throughout the development of mathematics. The author strikes a balance between covering the historical development of the theory surrounding polyhedra and rigorous treatment of the mathematics involved. Attractively illustrated--including 16 color plates--Polyhedra elucidates ideas that have proven difficult to grasp. Mathematicians, as well as historians of mathematics, will find this book fascinating. ... Read more

Reviews (3)

Comprehensive masterpiece!
This is the best book about polyhedra! But it's not always easy to read. He has chosen to take a chronological approach. That means that sometimes you have to look around a bit.

I picked up the book wanting to understand two things.

1. What are the exact definition of the Platonic and Archimedian solids, i.e., how to destinguish the Platonic from the the Deltahedra and the 13 Archimedian from their isomeric forms and the pyramids.

3. What's the reason behind the names for the Kepler-Poinsot solids. Why is the great stellated dodecahedron called the great stellated dodecahedron?

Cromwell answers the first question beautifully in Chapter 2. The second question is first discussed in Chapter 4, but I was still confused. It was only in Chapter 7 that it started to make sense.

I believe the book will answer most of your questions, but you may have to look around for it.

The _Best_ Polyhedra Book
I've read many books on polyhedra, and this is the best I have seen. It covers the history and mathematics of many different polyhedra; the Platonic and Archimedean solids are just the beginning. Kepler's rhombic polyhedra, stellated polyhedra, Miller's solid, etc. -- it's all here. The diagrams are exceptional. I teach high school geometry, and have found this book to be an essential resource in class. The level of detail is quite high, making the book useful as a straight-through read (for someone who is really into math) or a book to flip around in (for those who find heavy math intimidating, but still like polyhedra). Includes helpful tips for model-making. Buy it!

You should buy this!
It's a wonderful book for learning history of polyhedra, but I think it has too little 'mathematics' in. All in all, it's a masterpiece in my mathbook collection. ... Read more

Reviews (1)

Random fractals
From the standpoint of deterministic dynamics and multifractals defined by a generating partition, I do not like this book. However, for affine fractals and methods like R/S analysis it provides a very nice introduction! Like Hull's book on options, stochastic calculus is unfortunately not formulated using Ito calculus. Also nice, the gambler's ruin is presented as an example of affine scaling. What is missing is an introductory discussion of multiaffine scaling, which is of interest for soft turbulence in fluids (an introduction to multiaffine scaling with simple examples can be found in the book on surface fluctuations by Barabasi and Stanley). ... Read more

Book Description

The complete works of antiquity's great geometer appear here in a highly accessible English translation by a distinguished scholar. Remarkable for his range of thought and his mastery of treatment, Archimedes addressed such topics as the famous problems of the ratio of the areas of a cylinder and an inscribed sphere; the measurement of a circle; the properties of conoids, spheroids, and spirals; and the quadrature of the parabola. This edition offers an informative introduction with many valuable insights into the ancient mathematician's life and thought as well as the views of his contemporaries. Modern mathematicians, physicists, science historians, and logicians will find this volume a source of timeless fascination. Unabridged reprint of the classic 1897 edition, with supplement of 1912.
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Book Description

Incorporated in this volume are the first two books in Mukai's series on Moduli Theory.The notion of a moduli space is central to geometry. However, its influence is not confined there; for example, the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. Researchers and graduate students working in areas ranging from Donaldson or Seiberg-Witten invariants to more concrete problems such as vector bundles on curves will find this to be a valuable resource. Among other things this volume includes an improved presentation of the classical foundations of invariant theory that, in addition to geometers, would be useful to those studying representation theory. This translation gives an accurate account of Mukai's influential Japanese texts. ... Read more

Book Description

Master Math: Basic Math and Pre-Algebra teaches you in a very user-friendly and accessible manner the principles and formulas for establishing a solid math foundation. ... Read more

Reviews (2)

Table of Contents
Master Math: Pre-Calculus Table of Contents

Introduction

Chapter 1 Geometry

1.1. Lines and angles 1.2. Polygons 1.3. Triangles 1.4. Quadrilaterals (four sided polygons) 1.5. Circles 1.6. Perimeter and area of planar two-dimensional shapes 1.7. Volume and surface area of three-dimensional objects 1.8. Vectors

Chapter 2 Trigonometry

2.1. Introduction 2.2. General trigonometric functions 2.3. Addition, subtraction and multiplication of two angles 2.4. Oblique triangles 2.5. Graphs of cosine, sine, tangent, secant, cosecant and cotangent 2.6. Relationship between trigonometric and exponential functions 2.7. Hyperbolic functions

Chapter 3 Sets and Functions 3.1. Sets 3.2. Functions

Chapter 4 Sequences, Progressions and Series

4.1. Sequences 4.2. Arithmetic progressions 4.3. Geometric progressions 4.4. Series 4.5. Infinite series: convergence and divergence 4.6. Tests for convergence of infinite series 4.7. The power series 4.8. Expanding functions into series 4.9. The binomial expansion

Chapter 5 Limits

5.1. Introduction to limits 5.2. Limits and continuity

Chapter 6 Introduction to the Derivative

6.1. Definition 6.2. Evaluating derivatives 6.3. Differentiating multivariable functions 6.4. Differentiating polynomials 6.5. Derivatives and graphs of functions 6.6. Adding and subtracting derivatives of functions 6.7. Multiple or repeated derivatives of a function 6.8. Derivatives of products and powers of functions 6.9. Derivatives of quotients of functions 6.10. The chain rule for differentiating complicated functions 6.11. Differentiation of implicit vs. explicit functions 6.12. Using derivatives to determine the shape of the graph of a function (minimum and maximum points) 6.13. Other rules of differentiation 6.14. An application of differentiation: curvilinear motion

Chapter 7 Introduction to the Integral

7.1. Definition of the antiderivative or indefinite integral 7.2. Properties of the antiderivative or indefinite integral 7.3. Examples of common indefinite integrals 7.4. Definition and evaluation of the definite integral 7.5. The integral and the area under the curve in graphs of functions 7.6. Integrals and volume 7.7. Even functions, odd functions and symmetry 7.8. Properties of the definite integral 7.9. Methods for evaluating complex integrals; integration by parts, substitution and tables

Index

Appendix Tables of Contents of First and Second Books in the Master Math Series

Great book. Lots of good trig.
This book is pretty small but it gives great explanations of geometric shapes, angles, and trig functions like tan, sin, cos, and others. It's straight to the point and I learned from it very quickly. I highly suggest getting this book before moving on to a more advanced, or even just a regular geometry or trig book. ... Read more

Book Description

This practical introduction to the techniques needed to produce high-quality mathematical illustrations is suitable for anyone with basic knowledge of coordinate geometry. Bill Casselman combines a completely self-contained step-by-step introduction to the graphics programming language PostScript with an analysis of the requirements of good mathematical illustrations. The many small simple graphics projects can also be used in courses in geometry, graphics, or general mathematics. Code for many of the illustrations is included, and can be downloaded from the book's web site: www.math.ubc.ca/~cass/graphics/manualMathematicians, ... Read more

Book Description

Geometry is probably the most accessible branch of mathematics, and can provide an easy route to understanding some of the more complex ideas that mathematics can present. This book is intended to introduce readers to the major geometrical topics taught at undergraduate level, in a manner that is both accessible and rigorous. The author uses world measurement as a synonym for geometry - hence the importance of numbers, coordinates and their manipulation - and has included over 300 exercises, with answers to most of them. The text includes such topics as:- Coordinates- Euclidean plane geometry- Complex numbers- Solid geometry- Conics and quadratic surfaces- Spherical geometry- QuaternionsIt is suitable for all undergraduate geometry courses, but it is also a useful resource for advanced sixth formers, research mathematicians, and those taking courses in physics, introductory astronomy and other science subjects. ... Read more

Book Description

Applications from condensed matter physics, statistical mechanics and elementary particle theory appear in the book. An obvious omission here is general relativity--we apologize for this. We originally intended to discuss general relativity. However, both the need to keep the size of the book within the reasonable limits and the fact that accounts of the topology and geometry of relativity are already available, for example, in The Large Scale Structure of Space-Time by S. Hawking and G. Ellis, made us reluctantly decide to omit this topic. ... Read more

Reviews (5)

Excellent overview and graphical explanation
This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

Good attempt
When reading this book one can both admire these authors and feel sympathy with them. They have made an honest effort to explain the concepts of differential geometry and topology in a way that is understandable and appreciated by the physicist reader. But the book falls short in many places, although there are some places where they do a fine job. They have taken on a very difficult project in this book, for it is quite straightforward to expound on the formalism of mathematics, but explaining it in a way that grants insight into its conceptual meaning is another matter altogether. Many physicists complain, with justification, that the way mathematics is presented in textbooks is not sufficient for giving them a deep appreciation of the underlying ideas involved. This, they argue, is what is needed for devising new physical theories and results based on these ideas. Physicists must assimilate very complex mathematical ideas very quickly in order to formulate these theories in a reasonable time frame. This is especially true in high energy physics, which in the last two decades has used mathematics like it has never been used before. Indeed, the mathematical complexity of high energy physics is dizzying, and if progress is going to be made in this field by the students of the 21st century, they are going to need mathematics books and documents that are more than just formal expositions. But, again, writing these kinds of books is very hard to do, and has yet to be done in a book to this date, although there are helpful discussions scattered throughout the mathematical literature.

Some of the concepts that need more in-depth explanation include: the theory of characteristic classes, sheaf theory, the theory of schemes in algebraic geometry, and spectral sequences in algebraic topology. There are of course many others, and some of the ones that the authors do a fairly good job of explaining in this book include: 1. the reason that the continuity of a function is defined in terms of inverses of open sets; 2. The orientability of a manifold; 3. The fundamental group and its relation with the first homology group. 4. The discussion on Morse theory.

Covers a lot of ground . . . but not always well
Unlike many physics students, I grant a lot of leeway to books on mathematics for physicists. I think it's all right for an author to engage in hand-waving arguments if this enhances physical intuition or even to make the occasional statements without proof if this allows more ground to be covered. However, if a proof actually is presented, I expect this proof to be correct. In this book, proofs are sometimes only for special cases of theorems stated more generally and often contain logical errors.

flawed and incomplete
Nash's book commits the sin many mathematical physics textbooks out there commit: "oh, we're writing for dimwit physicists, lets just give them a few scrawny examples and assure them everything else works alright." I'm sorry but writing for physicists is NOT an excuse for writing a sloppy textbook. Would you feel alright not knowing how an integral is defined? Would you use a numerical evaluation software to calculate integrals in serious research without understanding the algorithm it uses? If you do then you're a pretty shoddy physicist. I'm not saying this out of some "macho" sentiment many purist physicists have - I'm simply saying this because I feel the way this book teaches you diff. geometry is wrong - it teaches you to draw pictures and go by the pictures. When the pictures run out, so does your understanding.

This book is supposed to teach differential geometry. However, very little can be learned from it unless one already knows differential geometry: definitions are sometimes not general and sometimes not present at all, theorems are often stated only for special cases and even more often than that not proved at all. Sure, the book offers nice geometrical intuition, but this is not enough. An example: the book "proves" Stoke's theorem around page 40. Now, even a rigorous and condensed book would have problems doing that, considering the amount of "machinery" one needs to build up for it (tensors, differential forms, manifolds and so forth). This means the book makes a mess of it - big time.
There are many fine diff. geometry books out there, some for physicists, some not, which you should check out - Nakahara's text is so much better. For geometrical intuition I suggest picking up Schutz's book. Several books from the GTM (Graduate texts in mathematics series, the yellow ones) are really very accessible, such as Introduction to Topological Manifolds/Smooth Manifolds. Another good one is Allen Hatcher's Algebraic Topology for homotopy, homology and cohomology. For a good and responsible exposition, do yourself a favor and look for something else.

Great introduction to mathematical physics.
This book is written by physicists. Like a book by M. Nakahara it describes basics of diff geometry and topology. Though it stresses physical intuition more than formal definitions. I especially liked discussion of fiber bundles and characteristic classes. Highly recommended. ... Read more

Book Description

This elegant book by distinguished mathematician John Milnor, provides a clear and succinctintroduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem. ... Read more

Reviews (5)

Take full advantage of the clear, encompassing exposition:
Do the exercises. Many were Ph.D. dissertation-level problems in the 1960s; today, they're aptly described as "elementary"- because Milnor MADE them elementary.

This book forms part of the toolkit you will need to fully explore the more modern work in dynamics, complexity, and applications (e.g., economics, physics).

The clarity of the exposition also forms an ideal example of how to communicate mathematics powerfully and simply.

Compact, readable text on the topology of manifolds
This book is very short, which is quite an asset for a math book to have. It also requires little knowledge of math beyond advanced calculus and point-set topology. I found it extremely readable, and I greatly enjoyed it. I recommend it highly, and especially enjoyed the proof on page 8 of the fundamental theorem of algebra. How far we've come since Gauss' first proof...

A good place to start
This book is exceptionally well written and easy to read. Milnor proves a major result on almost every page. One learns a lot per unit time spent on this book. Despite being less than 80 pages, the book covers a significant amount of material in a clear concise manner.

Excellent starting point for differential topology
One of the best points of this little book is its brevity and clear exposition of the basic ideas. It makes a great reference guide because it's so short and well-organized. Written by a distinguished mathematician, it's no wonder that other graduate-level texts such as Guillemin & Pollacks "Differential Topology" highly recommend reading it alongside their book. Milnor's booklet is a classic, whose style and ideas surely pervade other texts.

An excellent introduction to topology of manifolds.
Perfect for a first-year graduate or advanced undergraduate course, Milnor takes us on a brief stroll through elementary differential topology. Elegant and self-contained, this book serves as an excellent first taste of the subject. Milnor is a master expositor, and is at his best in this book. ... Read more

Book Description

Models of the regular and semiregular polyhedral solids have fascinated people for centuries. The Greeks knew the simplest of them. Since then the range of figures has grown; 75 are known today and are called, more generally, 'uniform' polyhedra. The author describes simply and carefully how to make models of all the known uniform polyhedra and some of the stellated forms. Fully illustrated with drawings and photographs, this is the first practical guide to making these intricate and beautiful solids. ... Read more

Reviews (3)

The definitive guide to uniform polyhedra
This book contains a detailed description of how to make (paper) models of each of the 75 uniform polyhedra, as well as some stellated polyhedra. Wenninger's descriptive and precise writing style is invaluable--his advice on construction methods and techniques are right on the mark. In nearly all cases, he provides sufficient data to allow the reader to draw his/her own templates, but especially in the most complex polyhedra the facial plane data is lacking, most likely because it is too lengthy to include. I would like to have seen more math in this text, and larger photographs. Otherwise, this is a must-have book for anyone interested in polyhedra.

The most definitive modern work of polyhedra I've seen
I have been fascinated with these structures since my disciovery of this book in 1980. Magnus is a fine builder of models and a competent teacher. The explanations for building each model are concise. I also compliment the photographer. As one begins to understand the underlying principles of these solids, a vast array of options present themselves as topics of further study.

Let patience (and beautiful models) be your reward-
Since I discovered this book, I've easily spent hundreds ofhours building these wonderful polyhedra. With a goodgeneral guide to model building and clear instructions, patterns and pictures of the completed model -- the book provides the raw material for a great hobby. The completed models are a continuing fascination, the relations between polyhedra and their symmetry can really only be appreciated when you have the models in your hands. While I lovedbuilding some of the over 120 models, it requires patience, a steady hand and practice. ... Read more

Book Description

This book gives the reader a thorough knowledge of the basic topological ideas necessary for studying differential manifolds. These topics include immersions and imbeddings, approach techniques, and the Morse classification of surfaces and their cobordism. The author keeps the mathematical prerequisites to a minimum; this and the emphasis on the geometric and intuitive aspects of the subject make the book an excellent and useful introduction for the student. There are numerous excercises on many different levels ranging from practical applications of the theorems to significant further development of the theory and including some open research problems. ... Read more

Reviews (1)

Nice introduction to differential topology
This book introduces the basic concepts in differential topology, a field that has taken on particular importance in medical imaging, game theory, and network optimization. Although written for mathematicians, and therefore somewhat formal, a good course in multivariable calculus should prepare the reader for this book. The most difficult chapter is probably Chapter 2, where Hirsch studies manifolds by means of function spaces and jets. He does do a good job in this chapter though of explaining the origin and need for partitions of unity and gives examples. He also gives the reader good insight into why analytic maps are more difficult to handle than the C-r case, and wets the readers appetite for further reading on the analytic case. The important notion of transversality is discussed in Chapter 3, which would be good reading for one interested in applications of differential topology to dynamical systems. A more detailed discussion of vector bundles would have been helpful in Chapter 4, which discusses these important objects and the idea of a tubular neighborhood. Sring theorists or those learning the mathematics should get a lot out of Chapter 5, wherein intersection theory in differential topology is discussed. The most important chapter of the book is Chapter 6, which discusses Morse theory. The applications of Morse theory are immense, and cover not only mathematics, but physics via quantum field theory and string theory, economics, and even computer graphics. A short chapter on cobordism follows, which is very nicely written, but a few more words would have been nice on this topic. After discussing isotopy in Chapter 8, Hirsch gives a good proof of the classification for surfaces in the last chapter of the book. A nice book to have for reference if one is interested in the subject for its own sake or for its many applications. It should prepare one for further advanced reading in differential topology, such as the work of Freedman and Smale on the Poincare conjecture in dimesions 4 and above. Those interested in applications of differential topology will be amply prepared to apply these results to the relevant areas, which are many. ... Read more

Book Description

Quasicrystals and Geometry brings together for the first time the many strands of contemporary research in quasicrystal geometry and weaves them into a coherent whole. The author describes the historical and scientific context of this work, and carefully explains what has been proved and what is conjectured. This, together with a bibliography of over 250 references, provides a solid background for further study. The discovery in 1984 of crystals with 'forbidden' symmetry posed fascinating and challenging problems in many fields of mathematics, as well as in the solid state sciences. Increasingly, mathematicians and physicists are becoming intrigued by the quasicrystal phenomenon, and the result has been an exponential growth in the literature on the geometry of diffraction patterns, the behaviour of the Fibonacci and other nonperiodic sequences, and the fascinating properties of the Penrose tilings and their many relatives. ... Read more

Reviews (1)

Worthwhile with lots of cool pictures
I bought the book on the strength of the pretty pictures scattered throughout the book (much much more interesting than the front cover).The text of the book is as impressive.

"Quasicrystals and Geometry" is a bit between an overview and a cookbook.Sections of the book are historical, parts are practical.Explicit and easy to follow instructions for generation of penrose tilings (and many other really cool tilings) are included.

Despite what the introduction says, this book is not written for a lay audience.A quarter or two of college level math seems to be assumed. ... Read more

Book Description

Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space -- in the living room or in some other galaxy -- have been the hidden engine of the highest achievements in science and technology.

Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same. ... Read more

Reviews (44)

from Dallas Morning News
"Euclid's work is a work [is] a work of beauty whose impact rivaled that of the bible, whose ideas were as radical as those of Marx and Engels. For with his book, Elements Euclid opened a window thgrough which the nature of our universe has been revealed." Strong words, but Mlodinow backs them up with this surprisingly exciting history of how mathematicians and physicists discovered geometric space beyond Euclid's three dimensions. Each advance in mathematical geometry has been followed by unexpected discoveries proving that the strange mathematics actually describe measurable physical properties. Mlodinow, a physicist and former faculty member of the California Institute of Technology, has also written TV screenplays for Star Trek: the Next Generation and other shows. He has a good sense of popular science writing, and he personalizes geometric abstractions by endowing them with personalities of his adolescent sons Alexei and Nicolai. Euclid, Descartes, Gauss, Einstein, and Witten are among the mathematicians profiled, and each of them also emerges with a distinct personality based on the style of their writing and historical anecdotes. This engaging history does an excellent job of explaining the importance of the study of geometry without requiring the reader to be a mathematician.

A MUST BUY FOR ANYONE INTERESTED IN MATHEMATICS OR PHYSICS!!
Euclid's Window by Leonard Mlodinow is an outstanding book. From the discoveries of Pythagoras and Isaac Newton to John Schwarz's String Theory, you can learn so much about the history of mathematics and physics through Euclid's Window. Mlodinow basically provides the reader with a summary of the evolution of mathematics and science, yet he does it in such a way that it is like reading a novel. The genius of Mlodinow is seen through his ability to take a topic that would take most authors thousands of pages to cover and convert it into a concise, easy to read story. Most people turn and run when they see a math history book, but this is no ordinary math book. Mlodinow's use of real life examples, graphic images, and stories from his own experiences with his children turn complex, abstract math concepts into concrete ideas that the ordinary person walking down the street can understand. Also, this is the first math/science book that I have ever read that actually provides some background information about the men who invented the formulas and theories. Most books either do one or the other. They either discuss the theories and formulas or they talk about the life of the person who invented them. Mlodinow does both. For example, Mlodinow not only discusses the mathematical discoveries made by Carl Gauss, but he gives an overview of his childhood, schooling, and life.
However, if there is one draw back to this book it is the physic's side of the story. I come from a mathematical background and even I found it difficult to understand the physic's theories like String Theory and M-theory. The author continually throws out new theories and new terms like quarks and positrons without much explanation. On the other hand, I think you have to give Mlodinow some credit for trying to discuss these extremely complex ideas in layman terms. Most authors would just leave out the part about String Theory altogether. According to Mlodinow, we all live in a giant puzzle known as the Earth, and since the Egyptian and Greek civilizations we have been trying to use reason, observation, and experimentation to piece the puzzle together. Through Euclid's Window, Mlodinow shows just how far humanity has come in its search to complete the puzzle.

Amazing Book
Anyone who thought geometry was boring or dry should prepare to be amazed. Despite its worthy cover this book is exactly what its title says - a story - and the plot of this story involves life, death and revolutions of understanding and belief, and stars the some of the most famous names in history.

The book opens with Aristotle watching ships at sea disappearing hull first over the horizon. "On a flat earth, ships should dwindle evenly until they disappear", and so he came to the realisation that the earth must be curved. This sets the scene for Mlodinow's tale of how geometry has shaped human history - "to observe the large scale structure of our planet, Aristotle had looked through the window of geometry." The book recounts how we have continued to look through this window to understand the reality we live in, and how the window has changed along the way.

The book is arranged as a series of five tales of the "five geometric revolutions of world history". These are told as the story of their main figures - Euclid, Descartes, Gauss, Einstein and Witten - in the context of their time, place and culture. This is one of the things that makes this book stand apart from others on the history of mathematics and science. It is told as a series of personal stories, of discoveries and leaps of understanding made by human beings. And this perhaps unexpectedly human side of geometry is enhanced by Mlodinow's accessible style. He is able to bring historical situations and mathematical concepts to life with the language of the present day. For example he explains the importance of applied geometry to Egyptians: "In building a pyramid, just a degree off from true, and thousands of tons of rocks, thousands of person-years later, hundreds of feet in the air, the triangular faces of your pyramid miss, forming not an apex by a sloppy four pointed spike. The Pharaohs, worshipped as gods, with armies who cut the phalluses off enemy dead just to help them keep count, were not the kind of all-powerful deities you would want to present with a crooked pyramid."

This book also contains some of the clearest explanations of relativity and string theory that I have ever read. Placed in the context of the evolution of geometry, and told as human triumphs of discovery by Einstein and Witten and their peers, these theories offer answers to obvious questions arising from our struggle to understand our reality. They also contain some very amusing examples such as Mlodinow explaining the entropy of black holes in terms of the messiness of his son, Alexei's bedroom. "Before Hawking, black holes, thought to have no internal structure, were thought to be something like an empty room. But now it seems they are like Alexei's actual room. Had Hawking asked, I could have confirmed this: I have always told Alexei that his room was like a black hole."

This is an excellent book not just for those select few fascinated by geometry, but for anyone interested in history of science, philosophy and humanity. In fact I would recommend it to anyone who enjoys a good story. Who would have thought that the story of geometry would include tales of life, death, sex and taxes?

can curved space be a good read?
In the past few years I have read several books about Einstein's theories, the beginning of the universe, and string theory. Though they were understandable at times, they were also often obtuse, and in a couple cases too dry. But most important, the one thing they didn't explain well was "what is curved space anyway?" Then, recently I read Mlodinow's book, Feynman's Rainbow, and I thought - he really writes well. So next I purchased Euclid's Window. I wasn't disappointed! In Euclid's Window Mlodinow finally gives all us non-scientists a good feeling for what physicists are all talking about in modern physics and astronomy as he tells an entertaining tale of the development of human ideas about space from the early Greeks puzzling over parallel lines to Einstein's theory and even the extra dimensions of string theory. His accounts of the mathematical developments are punctuated with wit and humor, and with tales of the times, the people, and the cultural history surrounding the advances. But, best of all, I understood it, and kept wanting to read more!

gross historical error makes one question entire book
I was enjoying all of Mr. M's anecdotes of Ancient Greek mathematicians and then I got to the part on Charlemagne. I am no religious scholar, but when M refers to Dominicans and Franciscans as providing teachers to Charlemagne's church schools (page 61) I started to wonder if most of Mr. M's book is fiction, albeit a nice readable fiction. Mr. M (and his editors) failed to grasp that Dominicans and Franciscans were orders founded in the 13th century and of course Charlemagne lived in 8th and 9th centuries. That is only a small tiny error of about 4-500 years.
Well this is another example of what happens when one tries too hard to popularize material best left to nerds. I don't object to all the "made up stuff". It makes for a good story. I only wish authors like Mr. M would be clear that their work belongs in the fiction not the non-fiction section. Mr. M should refer back to his comments on recidivism on page 46. Like Topographia Christiana maybe Mr. M is shooting for the 500 year best seller list. ... Read more