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The Shape of Space

Jeffrey R. Weeks
Chapman and Hall/CRC
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Benjamin Linowitz
, on
See Colin Adams's review of the second edition.
First published in 1985, The Shape of Space is an introduction to the low-dimensional geometry and topology necessary to discuss what is currently known about the topological shape of the universe. What makes this book so remarkable is that it is able to do all of this while remaining accesible to an extremely broad audience: the book can be read and enjoyed by any student of mathematics (regardless of background), and even by high school students who have seen pictures of Mobius strips and want to learn more.
Weeks begins The Shape of Space by discussing Edwin Abbott's Flatland and creating a narrative about A. Square's explorations of his two dimensional universe. (Spoiler: A. Square's universe turns out to be the connected sum of a torus and a projective plane!) These explorations motivate surfaces, their topologies, and a discussion of how one can (topologically) distinguish one surface from another. Once a few basic concepts are in place, Weeks goes over orientability, the Classification of Surfaces, and flat manifolds before moving on to geometries on surfaces and 3-manifolds. The book ends by applying all of this to the problem of determining our universe's global topology.
The book is extraordinarily readable. It comes across not as a textbook, but as Weeks' attempt to tell you about the aspects of geometry that he finds fascinating without bogging you down with technicalities. For example, the topologies of a flat torus and a Klein bottle are described, not by technical proofs about metrics on quotient spaces, but rather by imagining what it would be like to play chess or tic-tac-toe on such a surface. Moreover, there are hundreds of beautiful figures (now in color!; see below) illustrating what life would be like were you to live in various manifolds, and engaging exercises on every page. This book does a wonderful job of giving its readers topological and geometric intuition. This intuition comes at a cost, however. Although the reader will come away with a thorough understanding of the proof of the Classification of (closed, connected) Surfaces, he's unlikely to be able to rigorously define any of these terms. Similarly, the same reader that gains a thorough understanding of what it would be like to live in Poincare dodecahedral space is unlikely to finish the book able to define a topological space, much less a 3-manifold. This isn't necessarily a bad thing. It just means that the book is unlikely to fit anywhere in the standard undergraduate math curriculum (aside from, perhaps, a "Topics in Geometry" course). On the other hand, the book would make for great supplementary reading for a topology course, or for a student that has finished a first course in topology and wants to learn more.
The two biggest changes made in the third edition are:
  • The book is now in color! Both of the previous editions of this book had hundreds of figures and diagrams, all of which were black and white. These figures convey considerably more information now that they are in color. As an example, Weeks's discussion of what life would be like in a 3-torus (the three dimensional version of a flat torus; here opposite faces of a cube are glued together) makes use of many drawings of cubes with colored faces. Seeing drawings of cubes whose faces are actually colored is considerably more intuitive than looking at a black and white drawing of a cube whose faces are labelled "Y" for yellow, "B" for blue, etc.
  • The discussion of the proof of the Classification of Surfaces no longer makes use of connected sums. Instead, what were formerly algebraic proofs have been replaced by simple pictorial proofs using handles and crosscaps.
This extraordinary book provides its readers with a very friendly introduction to low dimensional geometry and topology and the problem of determining the shape of our universe. Readers will come away with strong intuition for the topology of 3-manifolds that are not usually discussed until late in graduate school. That Weeks was able to accomplish all of this in a manner that can be understood by non-mathematicians is simply incredible.


Benjamin Linowitz ([email protected]) is an Assistant Professor of Mathematics at Oberlin College. His website can be found at