Customer Reviews

The Shape of Space (Chapman & Hall/CRC Pure and Applied Mathematics)

5 star:		(10)
4 star:		(1)
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What is the universe as a whole shaped like? Does it curve back on itself? Does it meet itself at the other side without curving? Is its Flatland analogy a plane, or a sphere, or a doughnut, or a Klein bottle? What other, stranger geometries become possible with the added dimension? And if the universe has one of these exotic shapes, how could astronomers ever know for sure?

Jeffrey Weeks, a MacArthur ("genius grant") fellow and a consultant to NASA on cosmological observations, believes that there's no reason why a liberal arts student or a high schooler shouldn't be able to have a solid understanding of the answers to these questions, even though some of them are at the edge of research in cosmology and three-manifolds, and others have traditionally not been part of the math curriculum before graduate school.

The math is presented at an elementary level, but it is genuine mathematics. Readers in the intended audience must be prepared to roll up their sleeves; there are exercises, and there are formulas, and their minds will be stretched. But there are no prerequisites other than a little first-year algebra, and the discussion stays at a vividly concrete level, with a plethora of diagrams to aid the swelling imagination. High schoolers will benefit from some guidance getting through it; it's appropriate for undergraduate self-study.

More mathematically sophisticated readers, even those who've taken a course in algebraic topology or differentiable manifolds, will find the book a lively read, but will still probably learn a thing or two. I, for one, was startled to be shown a Moebius strip that was two-sided! (The trick is to embed it in a non-orientable three-space.)

The payoff is in the final two chapters, which detail programs of astronomical observation that could well tell us the precise topology and geometry of the universe, and explain just how they would do it. One chapter is devoted to a technique based on correlating distances between galactic clusters, and the other to a statistical search for correlated arcs of great circles in the cosmic microwave background. Both observations will probably be completed within the next decade. It's an exciting prospect.

Buyers note: I believe the Amazon characterization of this as a paperback is in error. I bought the second edition in hardcover at the same list price. In its (successful) attempt to avoid intimidation, it uses a large typeface, so it would fill out some 200 pages in a more typical math format.

But this book can also be quite serious, although it may take someone with an extensive math background to see this. The book seems aimed primarily at high-schoolers, but graduate students in topology can definitely benefit from reading it.

Weeks starts out by explaining surfaces and the quotient space descriptions of the torus and klein bottle. Later chapters describe 3-manifolds, fibre bundles(!), and the 8 geometries relevant to Thurston's geometrization conjecture. The focus of the book is on applying these concepts to investigating the shape of our spatial universe. This is a particularly apt goal, given that many times in the book the reader is asked to imagine living in various kinds of spaces.

He has a very good set of exercises designed to increase one's visualization powers. For example, in the chapter on 3-manifolds, he has the reader color various covering space pictures of 3-manifolds like the 3-torus, according to some specifications; this really helps one understand how covering maps work.

As someone who was familiar with topology before reading the book, I can say that the book has definitely increased my understand of 3-manifolds, which is more than I can say for most topology books. In particular, I found the material on fibre bundles very enlightening.

Jeffrey wrote this book with the high school student in mind, but even as a second year student in Mathematics I found this book quite informative. Most textbooks in Analysis or Topology do not give you an intuitive feel for the subject. I recommend this book for anyone taking a course in Topology, even Graduate students.
This book is well written with many illustrations and exercises to help you get an intuitive understanding of 3 Dimensional manifolds. This helped me a lot in my second year Analysis class as I had an intuitive notion of manifolds taught in class.
At the same time the book is easy enough for high school students who always wondered what a Mobius strip or a Klein bottle was but did not find any books on it. This book would make Topology interesting for everyone. I give it a five star rating.

Is space finite or infinite? Does it have borders? What shape does it have? These are among the most pressing and interesting questions in astrophysics and cosmology today. To answer (or at least understand) these questions, one must possess an understanding of topology, a branch of mathematics dealing with properties of shapes that are not changed upon deformation.

This book is an ideal introduction to topology for beginners with little or no mathematical background. It introduces topological manifolds (especially 2- and 3-manifolds) and their applications to cosmology and the shape of space. It is filled with diagrams, examples and exercises with full solutions at the end of the book.

The book assumes almost no knowledge of mathematics or physics, and is thus suitable for high-school and beginning college students. It is a must read for students contemplating a career in pure mathematics or theoretical physics, and who want to get a taste of the applications of pure mathematics to the physical world.

For those wishing to go a step further on the subject of the shape of space, the author published a paper (Nature 425, 593 - 595, 09 October 2003) claiming that the universe is a dodecahedral 3-manifold, based on cosmic microwave background measurements. This book may be a nice introduction for this paper and for subsequent papers that will surely ensue, trying to describe the shape of space.

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.

Actually this book deserves 10 stars. It gives the average person access to the mysterious worlds of topology and nonEuclidean geomotries. It is not a text book but rather a carefully considered exploration of these ideas. It would be an excellent summer read for someone who was going to take topology or nonEuclidean geometry in school in the fall.

One warning though, not all the ideas can be made easy. One still needs to work through the exercises and do a lot of pondering, but this book makes it possible.

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

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.

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 sprinkled throughout.

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.