In 1985, at the same time he was finishing his Ph.D. under the direction of William Thurston at Princeton, Jeffrey Weeks published the book The Shape of Space. Since then, Weeks has gone on to become one of the preeminent experts in hyperbolic 3-manifold theory. His computer program SNAPPEA, begun while still in graduate school, has evolved into the most fundamental tool for understanding hyperbolic 3-manifolds. In addition to his mathematical research, Weeks has worked on the development of software to teach topology to children of all ages. In more recent years, he has joined with cosmologists to attempt to determine the topological shape of the spatial universe. In 1999, Jeffrey Weeks received a MacArthur grant for all of his contributions to mathematics, to science and to education.
In the meantime, The Shape of Space has become acknowledged as the best intuitive introduction to the ideas behind low dimensional topology and its connections with geometry. That geometry plays a key role for surfaces has been known for a long time. But it wasn't until the late 70's and early 80's that Thurston demonstrated that geometry also plays a critical role in dimension three.
But where can one go oneself or send a student to read about these connections? How can one get a sense of what it means for a surface or a 3-manifold to have a certain geometry without spending years learning background material? All the other books and papers on this subject assume a relatively high level of expertise. Where does one go to get started?
To quote from Weeks' preface to the first edition, "The Shape of Space fills the gap between the simplest examples, such as the Mobius strip and the Klein bottle, and the sophisticated mathematics found in upper-level college courses. It is intended for a wide audience: I wrote it mainly for the interested non-mathematician (perhaps a high school student who has heard of Mobius strips and wants to learn more), but it also provides the intuitive examples that are currently missing from the college and graduate school curriculum."
Both editions of the book are in large friendly type with plenty of inviting figures throughout. The book does not read like a text. It is more like a friendly discussion.
The first three quarters of the new edition follows the original. Part I of the book is about surfaces and 3-manifolds. Starting with a discussion of Edwin Abbott's Flatland, the book quickly gets into surfaces and questions about how one might determine what surface one lives on. Representations of tori as squares with opposite edges identified are presented via torus tic-tac-toe and chess. These games are best played on the web using software written by Weeks and available at www.northnet.org/weeks/SoS.
These ideas form the basis for understanding the 3-torus, the analog of the torus one dimension up, which is the first 3-manifold considered. The next chapter sets up vocabulary, explaining the differences between topology and geometry, between intrinsic and extrinsic properties (depending on the particular embedding of the space), between local and global properties, between homogeneous and nonhomogeneous geometries and between closed and open manifolds. There are distinctions here that many students do not grasp even after a course in topology. The distinctions are made tangible by considering how Flatlanders on surfaces experience them. Then comes a chapter on orientability, including a discussion of the Klein bottle, tic-tac-toe and chess on the Klein bottle (again, best played on the web) and the projective plane.
Chapters on connected sums and products are followed by a chapter on flat (Euclidean)manifolds. First comes a discussion of what it looks like if you are in a flat 2-dimensional torus. Then comes the view for various flat 3-manifolds constructed by gluing together opposite faces of a cube and of a hexagonal prism. This covers seven of the ten compact flat manifolds that exist. All of these manifolds can be explored utilizing more amazing software at www.northnet.org/weeks/SoS. The last chapter of Part I is a short discussion of orientability and 2-sidedness.
Part II is on geometries on surfaces. It begins with discussions of the geometries of the sphere and the hyperbolic plane(including construction of hyperbolic paper). Although intuition is developed, readers looking for derivations of formulas and mechanics should look to more technical sources. This material is followed by chapters on geometries on surfaces, and the Euler number and Gauss-Bonnet formula.
In Part III, we turn to geometries on 3-manifolds, with chapters on 4-space, the hypersphere, hyperbolic space, geometries on 3-manifolds I (including the Seifert-Weber dodecahedral space and the Poincaré Dodecahedral space), bundles, and geometries on 3-manifolds II, including examples of manifolds with the eight homogeneous geometries that a closed 3-manifold can possess.
Much of Part III of the book will be new even to students who have had a course or two in topology. This is not material that is normally covered in topology courses.
The last part of the book contains most of the new material to this edition. Part IV is called "the Universe". The first two chapter gives background about the universe and attempts to understand it. What are the facts that we know?
The next chapter gets into the first method for determining the shape of the universe, cosmic crystallography. The idea is that by looking at the statistics of the distances between galactic superclusters, one might be able to identify the 3-manifold that is the spatial universe. The final chapter of the book concerns a second method for determining the shape the universe using matching circles in the cosmic background radiation. It is this method with which Jeffrey Weeks has been directly involved and on which he is one of the world's experts. The book ends with answers to all the exercises, a useful bibliography and an appendix new to this edition of the book that contains a copy of a paper written by Jeffrey Weeks and George Francis explaining John Conway's "ZIP" proof of the classification of surfaces.
This book provides excellent supplementary material for any course in topology. I have found that students are particularly fascinated by the applications to cosmology. The book is also the optimal choice to give to that student who is intrigued by topology, and wants to learn more.
Can The Shape of Space be used as a text? Yes. But the problem solutions at the end create some difficulties for a professor who wants their students to figure things out on their own. Also, the course itself would not fit well with the standard curriculum. Students would not come out with a background in point-set topology or any of the usual topics. But they would come out with an excellent intuition of what topology is about and how geometry relates to that topology.
The original version of this book was a must read for anyone with curiosity about geometry in low dimensional topology. The new edition takes us even further, including perhaps the most exciting application of topology one could hope for, the plans to determine the topological shape of the universe. What more could you ask?
Colin Adams is the Francis C. Oakley Third Century Professor of Mathematics at Williams College. Author of The Knot Book and co-author of How to Ace Calculus and How to Ace the Rest of Calculus, his research interests include knot theory and hyperbolic 3-manifold theory.
SURFACES AND THREE-MANIFOLDS
Orientability vs. Two-Sidedness
GEOMETRIES ON SURFACES
The Hyperbolic Plane
Geometries on Surfaces
The Gauss-Bonnet Formula and the
GEOMETRIES ON THREE-MANIFOLDS
Geometries on Three-Manifolds I
Geometries on Three-Manifolds II
The History of Space
Circles in the Sky
Appendix A Answers
Appendix B Bibliography
Appendix C Conway's ZIP Proof