The Colours of Infinity: The Beauty, The Power and the Sense of Fractals

Book Description

Benoit Mandelbrot discovered what is now called the M-Set in the early seventies and coined the term ‘fractal’ to describe the geometry behind it. The power and the beauty of fractals were only capable of being seen with the advent of computers, which become psychedelic windows on the infinite when using simple fractal equations.

In 1992 Nigel Lesmoir-Gordon made the TV documentary, The Colors of Infinity about the Mandelbrot Set and fractals, which has since been seen right round the world. Nigel’s enthusiasm brought together a dream team of contributors for the film who all now contribute to the book tracking how fractals have developed since the film was made. Sir Arthur C Clarke presented the film and in the book gives a lucidly simple account of the mathematics of the M-Set. Benoit Mandelbrot, the Belgian mathematician explains how it began. Professor Michael Barnsley, the computer graphics researcher who developed fractal image compression technology, explains the applications of the breakthroughs. Professor Ian Stewart, author of Does God Play Dice? adds his insights into the beautifully simple equation that gives birth to fractals.

Two of the most interesting applications of fractal geometry, reflected by the two new contributors to the book, are to the Internet and to the Stock Market. Dr Gary Flake, Chief Technology Officer at Overture, the leading provider of commercial search on the Internet and just taken over by Yahoo for 1.6 billion dollars, discusses the profoundly fractal nature of the Web in his article: The Self-ish Web. Robert Prechter Jr is President of Elliott Wave International, Inc. and founder of the Socionomics Institute. His latest title is Socionomics: The Science of History and Social Prediction (2003). He writes about how fractals can help us understand the oscillations of stock markets.

In the back of the book is a DVD of the original documentary with soundtrack by David Gilmour of Pink Floyd PLUS a 30-minute fractal animation to the music of members of Quintessence.

Customer Reviews:

Amazing!.......2007-08-13

The book and the DVD are truly amazing and beautiful. It has introduced me to a wonderful world which is a marriage of art and mathematics. The price is almost too low for such a masterpiece.

The Colours of Enlightenment.......2007-03-09

Depending upon one's education, it is possible to gain either an art appreciation of the fractal geometry of the Mandelbott Set or a realization of how all life is ordered and the universe structured. Life most certainly would exist somewhere out there and it would resemble what we have here on Earth. While viewing it, I felt that this was at least fifty percent of the riddle of the universe explained in at least its basis. The other fifty percent would be what Stephen Hawkin called the other formulae that will reverse this one. He hopes we will find this one before it begins to act in its turn. That would mark the collapse of the universe and he seems to think that we might prevent that. But the collapse may be inevitable and part of the eternal operation of two formulae.

The Mystery remains; was this ordered and if so, by whom or what? We may never know, but for this devastating mystery, we have David Gilmour's compelling score to propel us along through an eternal race toward infinity.

My only complaint is that the film needs re-mastering so that the fractal geometry can be expressed better. Ironic that the documentary that defines the detail of the universe is so fuzzy! Also, there is a second section of fractal art expression on the DVD that will only play on a computer. That ought to be fixed. I think it is fitting that Arthur C. Clarke is the narrator for this film, so I hope this original can be cleaned up and not trashed for an updated version with all new presentation and cast.

This documentary should be shown in all high-school science classes. In fact, I think it ought to be shown to everyone regardless of partticipation in science curriculum because it also assists the refinement of questions like evolution and religion.

Amazing Fractals.......2007-01-12

What an illuminating, thought provoking book and dvd! I have watched it several times and each time it has opened my eyes more to the amazing possibilities in almost every aspect of our existance here on earth. And then who doesn't like David Gilmour's music ?

A great way to discover the colours of infinity.......2007-01-05

Just a perfect combination of top-level science outreach with the
fantastic music of David Gilmour.

It's a journey into the wonders of the fractal geometry explained in
a clear and easy-to-follow way.

Buy it now !!! You won't regret !!!

Fractal Visions.......2006-12-04

Fractals were just hitting the popular consciousness when I was in high school. I remember a flurry of pop science books that came out around that time, all of which presented the wonderful fractal images, while nevertheless dancing quite vaguely around the underlying ideas. The questions these mushheaded books immediately provoked in me (i.e. "what does 'self-similar' actually MEAN?" and "is complexity an objective quality or in the eye of the beholder?) were left totally unanswered. In my high school, these books (and the fractal posters, key-chains, videos, water-pipes...) commanded the attention of an intellectually philistine (and usually highly-intoxicated) crowd. People who use "party" as a verb. This "whoa dude-it's a fractal" factor was so powerful that it left the whole topic kind of disreputable in my book...until now.

This,too, is a "pop" science book, after a fashion, but unlike those other fractal books, the authors here refuse to dumb the subject down to a mere collection of pretty pictures. There is enough detail here to permit a diligent reader to truly understand the logic behind fractal geometry. If you can grasp the math, the basics are here. If, like me, you are a "liberal arts" type, the concepts are also explained in precise English.

The content of the video is similarly excellent. The interviews with Benoit Mandelbrot and Stephen Hawking are very cool indeed, but my favorite part is when Arthur C. Clarke states that, while he doesn't know about this PERSONALLY, "the ingestion of certain illegal substances produces fractal visions." Clad in his trademark blue utility shirt, and with a perfectly straight face, Clarke goes on to note the similarity between the words "Mandelbrot" and "Mandala." (Whoa, dude!)

Unfortunately, the video itself (i.e. formally, visually) is pretty dated. It seems to have been made for VHS, and has not been cleaned up for the DVD release at all. The colors are bled, and the titles look like they were made on a Commodore 64. The fractal images are quite low-resolution, which is unfortunate, since fractals are one topic for which high resolution would seem particularly essential. (Then again, NO level of resolution would really be sufficient.) This video is just begging to be remastered. With contemporary high-def video and computer technology it would be absolutely jaw-dropping. Still, this is totally worth it.

The Shape of Space (Pure and Applied Mathematics)

Average customer rating:

Magic book on Topology for educated commons
The joy of math
Excellent Introduction, No Assumptions
Interesting book
Easy Reading

The Shape of Space (Pure and Applied Mathematics)
Jeffrey R. Weeks
Manufacturer: CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 0824707095

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.

Average customer rating:

A refreshing style of writing
fun and geometric-intuition-minded

Three-Dimensional Geometry and Topology
William P. Thurston
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0691083045

Book Description

This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

This book was the origin of a grand scheme developed by Thurston that is now coming to fruition. In the 1920s and 1930s the mathematics of two-dimensional spaces was formalized. It was Thurston's goal to do the same for three-dimensional spaces. To do this, he had to establish the strong connection of geometry to topology--the study of qualitative questions about geometrical structures. The author created a new set of concepts, and the expression "Thurston-type geometry" has become a commonplace.

Three-Dimensional Geometry and Topology had its origins in the form of notes for a graduate course the author taught at Princeton University between 1978 and 1980. Thurston shared his notes, duplicating and sending them to whoever requested them. Eventually, the mailing list grew to more than one thousand names. The book is the culmination of two decades of research and has become the most important and influential text in the field. Its content also provided the methods needed to solve one of mathematics' oldest unsolved problems--the Poincaré Conjecture.

Thurston received the Fields Medal, the mathematical equivalent of the Nobel Prize, in 1982 for the depth and originality of his contributions to mathematics. In 1979 he was awarded the Alan T. Waterman Award, which recognizes an outstanding young researcher in any field of science or engineering supported by the National Science Foundation.

Customer Reviews:

A refreshing style of writing.......2001-06-21

Stanislaw Ulam once compared learning mathematics to learning a language, in that some people learn mathematics by "grammar" while other learn it by ear. Thurston's book is a bit like learning by ear.

fun and geometric-intuition-minded.......1998-12-23

A must for anyone entering the field of three-dimensional topology and geometry. Most of it is about hyperbolic geometry, which is the biggest area of research in 3-d geometry and topology nowdays.

Most of it is readable to undergraduates. Its target audience, though, is beginning graduate students in mathematics. If not already familiar with hyperbolic geometry, you might want to get an introduction to the subject first. Once with this background, though, you will discover there is another level of understanding of hyperbolic space you never realized was possible. One imagines Thurston able to skateboard around hyperbolic space with the kind of geometric understanding he conveys here.

Book Description

Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.

Customer Reviews:

Elegant and astonishing.......2006-03-17

Professor Luenberger unites many areas of optimization using a few principles from functional analysis. The explanations are clear and the proofs are compact and elegant. This book is your tool for understanding the deep connection between linear programming, convex optimization, game theory, optimal control and series approximation (e.g. Fourier series).

Luenberger's book has over 1300 citations as of March 2006. In my opinion, the material in this book is essential for any graduate student or professional who intends to contribute to the literature in optimization or optimal control.

A Good Book but Seriously Overpriced.......2005-01-15

The exposition is pretty clear and the book has a good number of worked non-trivial examples. At $40 this would be a great book, but $100 for a PAPERBACK book written 30 years ago is a bit ridiculous. The first 1/4 of the book is also a (very) basic introduction to functional analysis which, if you have had any contact with this subject before, you will probably skip making the book quite short.

This is a true classic.......2004-12-18

This book is a timeless classic, filled with extraordinarily powerful mathematics and applicable to just about every serious subject area. Luenberger did a masterful job of writing a book that will "unravel the spaghetti" seen in most other books. The visual perspectives he provides to seemingly abstract ideas are the key.

Simply the perfect math book.......2003-07-04

Optimization by Vector Space Methods, by David Luenberger, is one of the finest math texts I have ever read, and I've read hundreds. Many years ago this book sparked my interest in optimization and convinced me that the abstract mathematics I had been immersed in actually would be applicable to real problems. Since then, Luenberger's book has inspired several of my graduate students. I merely lent them my copy, and Luenberger did the rest; he drew them in by carefully laying the foundation for an elegant theory, with just the right mix of formalism and intuition, and opened their eyes to the beauty and practicality of abstract mathematics. Anyone with an interest in higher-level mathematics (beyond multi-variable calculus, say) would benefit from exposure to this finely-crafted book. I daresay, the rampant math anxiety that is so prevalent in the West would be substantially reduced if more authors would take such meticulous care in presenting their material.

The format of Luenberger's book is also extremely appealing in a way that I cannot quite put my finger on. The typography and illustrations are inherently crisp and inviting; they draw you in. There is nothing at all superfluous or gratuitous in this book. It is utterly to-the-point, methodical, and above all, clear. The techniques are developed starting from an elementary treatment of vector spaces, then proceeding on to Banach spaces and Hilbert spaces. Along the way, Luenberger introduces convexity, cones, basic topology, random variables, minimum-variance estimators, and least squares, among many other things. There is a recurring theme of duality, which can be used in a way analogous to the inner product of a Hilbert space. In particular, the familiar projection theorems of Hilbert spaces can be echoed in simpler normed linear spaces using duality, which Luenberger motivates and covers beautifully.

The book also covers some of the standard fare of functional analysis, such as the Han-Banach theorem, strong and weak convergence, and the Banach inverse theorem. However, Luenberger never wanders too far off into abstract nonsense; around every corner lay tantalizing application of these ideas to optimization. Luenberger first explores optimization of functionals then covers constrained optimization, which builds upon concepts such as positive cones and Lagrange multipliers. The optimization methods themselves have endless applications in fields such as computer vision, computer graphics, economics, and physics. Indeed, the list is effectively endless as optimization techniques pervade math and science.

I'm certain that the appeal of this book is helped immeasurably by the inherent beauty of the subject matter. Hilbert-space methods are lovely in themselves--they possess a structure that engages one's geometric intuition while at the same time admitting convenient algebraic properties. Once you are in the habit of phrasing problems in abstract settings such as Hilbert spaces, it forever changes how you look at things; you cannot help but look past the clutter to the essence of a problem (or, at least try very hard to do so). While this material is not nearly as abstract as, say, category theory, it nevertheless hits a high point in mathematics--a point more people ought to experience.

If you've had some exposure to optimization methods, or need to apply them in the context of computer vision, graphics, or finance, to mention just a few areas, then I urge you to take a look at Luenberger's fine book. It too hits a high point in clarity of mathematical writing. Combine beautiful theory with endless applications and lucid writing, and you have a winner of a book.

Thank You Dr. Luenberger.......2002-10-15

I owe Dr Luenberger a million thanks for writing this book. As his student, I think he is the master of putting complex issues in simple words. Your faithful student..Jayanth Krishnan

The Geometry of Physics: An Introduction, Second Edition

Average customer rating:

Fantastic - for the scientist
a book worth keeping
Phenomenal
You should buy this, despite its flaws
The perfect first book in differential geometry

The Geometry of Physics: An Introduction, Second Edition
Theodore Frankel
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521833302

Book Description

Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1

Customer Reviews:

Fantastic - for the scientist.......2007-07-18

A very good book: buy it. But only if you are a scientist or student of physics/mathematics. This is not popular-science-common-public level.

a book worth keeping.......2007-05-01

This book can be quite confusing if you start without any background on the idea of manifold or knows nothing about general relativity. However, it does have strong points:

1. The notation is very up-to-date, and is entirely coordinate-independant approach.

2. The author explains in great details of formulation of modern differential geometry, and the details are comparatively lacking in other reference books.

3. The author never hesitate to use graphs and diagrams to illustrate points, and stroke nice balance in between mathematics rigor and physical insight.

Although it appears quite verbose at some point, it is mainly because differential geometry is such a heavy subject. Another book nice to have as companion reading is Goldburg's "Tensor analysis on Manifold", a terse, well-written text book.

Phenomenal .......2006-11-13

I just finished reading this book and I found it phenomenal. The physical ideas are made very clear in a natural mathematical framework.

You should buy this, despite its flaws.......2006-03-03

The other reviews on this page give this book anywhere from 1 to 5 stars, and they are all correct in their own way. The book is inspired, deep and full of physics applications and insights. On the other hand, it skims over mathematical rigor to a large degree and focuses more on defining things, getting a feel for them and moving on to application.

My advice: buy the book for its strengths, and read other books in parallel if you need more rigor. But still, buy it.

Also, things can be confusing on the first two or three reads, but keep at it and you will be glad you did.

The perfect first book in differential geometry.......2005-01-28

Differential geometry can be a very intimidating subject due to its heavy formalism. There are complete books (such as Kobayashi& Nomizu) very good as reference books, and there very few books that show the reader the picture behind the formulas.

This is one such book. It tells you the intuition behind each construction and from this point of view it has many things in common with Arnold's famous book on Math. Methods in Classical Mechanics. But where as Arnold does not pay too much attention to formalism, this book achieves this task as well. It shows the reader how to do those impossible computations as well.

This is definitely the first place to look at if you want to really learn differential geometry. If it seems difficult it is only because the subject is so.

Average customer rating:

Great for self-study
If we make the assumption that "good book" means a book
Best Book Evar!!11!!11!
A must-have text for any grad student!
Great book

Introduction to Smooth Manifolds
John M. Lee
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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Accessories:

ASIN: 0387954481

Book Description

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. Along the way, the book introduces students to some of the most important examples of geometric structures that manifolds can carry, such as Riemannian metrics, symplectic structures, and foliations. The book is aimed at students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of two previous Springer books, Introduction to Topological Manifolds (2000) and Riemannian Manifolds: An Introduction to Curvature (1997).

Customer Reviews:

Great for self-study.......2007-09-28

I should say first that I was already familiar with manifold theory before picking up this book. I had already wrestled with some of the definitions, theorems, and whatnot, so I can't necesarily say I was a complete beginner before reading this book. Also, I'm not sure if I can say how great this book would be if you have no idea what a manifold (or tangent space, etc.) is. However, that stuff aside, this is an amazing text. I'm studying this book on my own, and it's great. The concepts are woven throughout the text instead of being lumped into chapters devoted to them (though some people might prefer the latter). Also, they're used to reinforce and build on each other.

As an example, Spivak doesn't treat Lie groups until the second to last chapter. Lee introduces them in the second chapter, uses them as examples throughout the text, builds up the theory of Lie groups as the book goes on, uses Lie groups (and their actions on other manifolds) in developing certain other areas (it really streamlines the development) and ends with a nice big chapter on them. Of course, this is just one example.

Lee developes manifold theory so that it would appeal to a physicist, geometer, algebraist, topologist, etc. Everything gets talked about! This means, however, that he can't treat any one subject in too much detail. For instance, he leaves curvature and other parts of Riemannian geometry to his other Riemannian Geometry text, but it's definitely worth the trade off. This book trashes Spivak. Buy it today!

If we make the assumption that "good book" means a book.......2007-07-24

that lends itself to self-studying then this is not a good book, but excellent. All complaints reported in other reviews are actually answered in the preface: the book is about the mathematical machinery ordinated under the title smooth manifold theory. It is not a book on riemannian geometry that's why there is no extensive treatment of metrics or any treatment of connections. Each topic comes up whenever the prerequisite tools are built and enough motivation can be given, that's why it is a pleasure to read this book. If you like encyclopedic expositions there are plenty of them out there. It is obvious that the author belongs to that group of people who like to excel in whatever they do. All books written by J.M. Lee not only teach you the subject of their titles but also how to write a book if it happens to reach that point in your mathematical career. They are in some sense both books and meta-books on mathematics :)
This review is not intended to comment on other reviews, but let us be honest and agree on the fact that an author never faces the danger of being too clear: as to the length and the pace of the book, I wish this book were only one volume of a series from the same author starting with topology and culminating with the interplay of differential geometry and pdes. There is a drawback however, reasonably not anticipated. Most math books are not written to be actually read (aphoristic but true). This book makes an exception and thus the usual binding proves insufficient quickly. A hardcover version would be convenient. Suggestion for "clever" math students: learn the stuff from Lee and then pretend you are reading Lang's "introduction"...

Best Book Evar!!11!!11!.......2007-03-30

I really like this book. Physically, it looks much like Lang's algebra book, but I assure you that it contains none of the snide remarks. Though, it does have a picture of the author in a berra which is odd. I'm sure I mis-spelled that, but it's the french hat that people like to use to make fun of artist types.

In any case, this book is long and contains a lot of problems for you to do. Unfortunately I do not do them, but that is a different story. I'm nowhere near finishing all the stuff this book has to tell me, but whenever I need to find something I don't know this book tends to have it. The index is great. It might be the best of any book I've used. The greatness of this book is a little surprising juxtaposed with Lee's book on Riemannian geometry which is not exceptional.

Since this book is so large, and it says it's a graduate math book right on the cover, I like to take it out with me when I go out on the town. I find it's a great ice breaker with the ladies. I only wish it was the nice burnt orange of the newer springer books.

All in all, this is a great book, and really puts Spivak to shame.

A must-have text for any grad student!.......2007-02-11

We're using Gullemin and Pollack's text for our differential topology course. I found it rather difficult to learn from it. A friend of mine strongly recommended this book by Lee (actually, he recommended the whole series.) The definitions are concrete, and the proofs are rigorous. Lee provides some great motivations for the ideas presented in this text. Ultimately, I find that it's a well written topology book and should be on any mathematicians bookshelf.

Great book.......2005-10-27

It's very readable. He has a good descriptive, conversational style. It's also very thorough. For example after he gives his definitions of the tangent space he copmares and it to the competitors and shows equivalence. There is plenty of work in coordinates but things are defined in the proper coordinate invariant ways. Nice coverage of vector bundles and a whole chaptor on the cotangent bundle which is nice.

Lots of Lie groups... he introduces symplectic manifolds and talks about Hamiltonian mechanics on the cotangent bundle. What I'm saying is all and all he talks about a lot of wicked good stuff.

One warning: The word transversality appears I believe once in the whole book and that's in an exercise. Intersection theory does not seem to be covered at all. That's not a complaint. That stuff is in lots of good books that don't go anywhere near a lot of the things that are in Lee's book. I'm just saying if you are thinking of using this as a reference for a course that has transversality on the syllabus you will need a second book. Let's say Hirsch's differential topology for the classic, or Guillemin and Pollack's book by the same name for something that doesn't have function spaces as it's second chapter.

So yeah. Good book. Thanks Dr. Lee.

Geometry and Topology (Lecture Notes in Pure and Applied Mathematics)

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Geometry and Topology (Lecture Notes in Pure and Applied Mathematics)
Mccrory
Manufacturer: CRC
ProductGroup: Book
Binding: Paperback

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fails to help
Good Book
A Geometrical Journey

Symmetry, Shape and Space
L.Christine Kinsey , and Teresa E. Moore
Manufacturer: Key College
ProductGroup: Book
Binding: Hardcover

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Accessories:

ASIN: 1930190093

Book Description

This text is suitable for introductory students, perhaps in programs such as education, art and architecture. The text contains some traditional material from geometry as well as more innovative topics. Throughout the text, the authors place strong emphasis on pedagogy, hands-on model building, a guided discovery method of learning, etc. Much of the material is written in such a way that it can be used in the classroom for enrichment projects, by prospective mathematics teachers.

Customer Reviews:

fails to help.......2007-09-10

Sorry, any text book without a glossary fails its primary goal of making its information accessible.

Good Book.......2007-02-08

The book does a good job of reintroducing basic geometry to the more remedial college mathematician. It also does a fine job of introducing higher order geometry. However, the best quality of the book is its ever present agenda to relate geometry to real life situations. This helped to give me a reason to study geometry for practical purposes. Too bad my professor is a douchebag.

A Geometrical Journey.......2005-12-03

This book will well serve any student who wants an interesting, visual approach to mathematics. A plethora of topics are explored including constructions, tessalations, other dimensions-the fourth dimension, polyhedra, three-dimensional symmetry, spiral growth, shape, graph theory, and topology.
All topics are presented, for the most part, in an intuitive, visual manner making it much easier for the average reader to grasp the concepts being presented. Students are required to analyze patterns thereby enhancing their analytical and visualization skills. Each chapter is presented in a stand alone manner. The book is well written and replete with numerous high quality drawings. The text would well serve
any instructor who is presenting a course for liberal arts students, a course in mathematics for teachers, or an enrichment course for high school students. The book would also provide an excellent, self-study guide for high school students who are interested in a gentle guide to mathematics outside the standard high school: algebra, plane geometry, and pre-calculus curriculum. An excellent and extensive biography is provided for the reader who wants to delve deeper into any of the many topics covered in the text. Overall, an excellent introductory text to the many facets of geometry.

The NURBS Book (Monographs in Visual Communication)

Average customer rating:

Good reference for implementers who are familiar with NURBS
An Essential Foundation
This is a definitely a math book, not a programming book !
Excellent introduction to splines for the non-mathematician
Start Point

The NURBS Book (Monographs in Visual Communication)
Les A. Piegl , and Wayne Tiller
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 3540615458

Book Description

The second, revised edition of this book covers all aspects of non-uniform rational B-splines necessary to design geometry in a computer-aided environment. Basic B-spline features, curve and surface algorithms, and state-of-the-art geometry tools are all discussed. Detailed code for design algorithms and computational tricks are covered, too, in a lucid, easy-to-understand style, with a minimum of mathematics and using numerous worked examples. The book is a must for students, researchers, and implementors whose work involves the use of splines.

Customer Reviews:

Good reference for implementers who are familiar with NURBS.......2003-07-24

This book is best for those who have at least become familiar with Bezier and NURBS curves and surfaces and want to start implementing some programs.

The illustrations are not as clear as they could be so it may make it a little difficult for first-timers to understand. If you're unfamiliar with Bezier and NURBS curves then "Curves and Surfaces for CAGD" by Farin is probably a better place to start. But if you get Farin's book make sure to get the 5th edition since earlier editions aren't as clear.

With that said there is a wealth of information about creating and manipulating NURBS geometry in this book. All explanations are followed by code implementation. The code is a little difficult to understand at first, but if you stick with it you'll understand their coding style.

4 stars since thier layout and illustrations could be a lot clearer.

An Essential Foundation.......2002-08-07

I own many books on the subject of NURBS. I always come back to this one while programming NURBS and reading research papers. It provides a fantastic foundation. It is an older book, so don't expect recent research to be included. For those that also want to know how to do ray tracing with NURBS surfaces look here:

...

Good luck!

This is a definitely a math book, not a programming book !.......2000-05-23

This book is complete, but there is too many math formula. The composition is appr. 80% formula explanation, 10% sample and 10% algorithm. So if you are a programmer you are wasting 3/4 pages of the book. But, since there's no other book about NURBS, as complete as this, you may want to buy this one or else you can take a look "Mathematical Element for Computer Graphics". It also contains 300 pages of NURBS.

Excellent introduction to splines for the non-mathematician.......2000-02-12

One of the greates assets of this book is that it is fairly comprehensive while still using a minimum amount of mathematics. The style is lucid, easy to follow; the numerous C-like algorithms can be adapted to one's own use, and deal with the most important aspect of NURBS. This is easily the best introductory NURBS text. The only problem I see is the lack of discussion for ray-surface intersection tests, which are necessary if NURBS are to be rendered by raytracing. I recommend "The Nurbs Book" to any engineers and graphics professionals interested in curved surfaces.

Start Point.......1999-12-29

Before owning this book i cassified NURBS as kind of mathematical definitions of curves and surfaces. I was wrong, now it opened my mind so i develop new NURBS surfaces based on well described primitives. Twisting, blending, extruding is good starting point to develop surface definitions you exactly need. How about lofting shapes along branch like path curves?

Average customer rating:

The best for the price.
Summer School REview
Terrible for Theory
Great for Trigononometry exams
Utterly and Deplorably Horrible

Schaum's Outline of Trigonometry
Robert Moyer , and Robert E. Moyer
Manufacturer: McGraw-Hill
ProductGroup: Book
Binding: Paperback

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ASIN: 0070068933

Book Description

Updated to match the emphasis in today's courses, this clear study guide focuses entirely on plane trigonometry. It summarizes the geometry properties and theorems that prove helpful for solving trigonometry problems. Also, where solving problems requires knowledge of algebra, the algebraic processes and the basic trigonometric relations are explained carefully. Hundreds of problems solved step by step speed comprehension, make important points memorable, and teach problem-solving skills. Many additional problems with answers help reinforce learning and let students gauge their progress as they go.

Customer Reviews:

The best for the price........2007-03-27

Before I bought the book, I had very little understanding of Trigonometry. After I bought it and received it, I worked my way through all the chapters. When I was done, I feel like I just mastered it and increased my self-confidence in my thorough knowledge in the subject of Trigonometry. Anyone out there who has absolutely no idea what is a sin, cos, or tangent and why we use them, get it! You'll get it after you work through the chapters. Not as painful as you think, but rather entertaining and informative. Just make sure you push yourself to memorize some of the formulas because that will give you a very strong push to doing the problems much more efficentive way. Best part of the book is pretty much everything included but more importantly, the De Moivre's Theorem. Worst part of the book is the practical application related to air navigation. That should be either elaborated deeper or eliminated. But it has absolutely nothing to do with the cumulative effect that you want out of this book. Suggestions for 4th edition: get rid of the geometry appendix and trigonometric functions tables (since it's all in the calculator), include more proofs and applications, elaborate section 5.5 further which relates to air navigation, and pretty much keep the rest of the whole book intact.

Summer School REview.......2006-07-31

I used Schaum's for a Summer School Trigonometry course. It doesn't replace a textbook but it covered all the necessary topics effectively and provided a good alternative to derivations and problems found in the text. I relied on it as a backup and a good check on the work being covered in class.

Terrible for Theory.......2005-09-17

I needed to brush up on my Trig for a calculus class that I'm currently in. While reading the text I found it hard to follow what the author was getting at.

At certain times magic equations would pop out of thin air and you would have to stew over them for hours at a time trying to figure out what the heck they meant.

This book is definately not for a beginner or someone who's looking for an easy quick overview after being away from the subject for abotu 6 years.

Great for Trigononometry exams.......2005-01-08

I am observing that my test scores on tests on tests involving trigonometery are increasing, thanks to this thin aid. It is thin, yet it is good. That is almost impossible. This book is one of the best created. Trigonometry is hard, and is mentioned and applied almost everywhere. This book is comphrehensive and is both easy and advanced. Everyone should have this. You can have knowledge of math, science, and computers with it.

Utterly and Deplorably Horrible.......2003-01-07

This book as mentioned in the title is horrible. It is incomplete in many areas, case and point, curve graphing. In many cases the book does little more than introduce the topic and give somewhat bland math questions. This book will not help you through a normal course because it is somewhat babified.
Now back to the incompleteness. Half-way through the book trigonometric function graphs are introduced (y=sinx and so on). The book very briefly describes aspects of each periodic function in a somewhat scattered manner. All of the information that is given fits on about one 8 1/2 X 11 piece of paper, somewhat terse isn't it?
This book is not for beginners and is most likely not even for people that would like to brush up on trigonometry. For a more comprehensive edition of a trigonometry tutorial you must turn elsewhere because this book will leave you asking what? huh? how? Perhaps one of the better trigonometry titles out there, and believe me I say this reluctantly because it is also deplorable, is Trigonometry the Easy Way. In conclusion if you have this book return it or if you can't use it only as a way to reinforce trigonometry ideas.

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