Topology as Fluid Geometry

(This review covers all three volumes of Two-Dimensional Spaces.)

Cannon’s trio of slim volumes form a compendium of mathematical marvels that will doubtless inspire both budding mathematician and seasoned practitioner. Taken together, they may be viewed as Cannon’s personal update of or variation on Hilbert & Cohn-Vossen’s classic Geometry and the Imagination. The books overflow with mathematical charm. Many readers will be hooked by Cannon’s aesthetics and proof exposition, where geometric intuition and topological arguments play leading roles.

The first volume, whose main focus is to “explore classical attempts to measure distances and areas in the plane and give natural applications of those attempts to classical problems in geometry and to algebra, number theory, and measure theory”, requires the fewest prerequisites of the three volumes, e.g. a calculus sequence and a first linear algebra course at a university in the USA. In its preface Cannon confesses: “It explains an entire string of results that teased me as an undergraduate because they were stated without proof. I sorely wanted to understand why they were true. This book is written for the “me” who was a young college student”.

The book begins with a trio of proofs of the Pythagorean Theorem and discusses some of its avatars and consequences. Next, Cannon is on to lengths, areas and Archimedes’ discovery of the volume of the sphere using weights and balance arms. There is a neat derivation of the volume of spheres and balls in all dimensions that every calculus instructor should master (and be inspired to present)! The chapter on Areas by Cut and Paste contains a proof of the Bolyai-Gerwien theorem and that on Areas by Counting exposes Pick’s theorem. The penultimate chapter, Unsolvable Problems in Euclidean Geometry. proves the classical trinity of Greek impossibilities, and also explains Hilbert’s proof that \(\pi\) and \(e\) are transcendental. The last chapter presents a proof of the wonderful Hausdorff-Banach-Tarski Paradox, which “shows that the \(3\)-dimensional sphere can be broken into finitely many pieces that are too complicated and fuzzy to be assigned a well-defined area since they can be rigidly reassembled to form two copies of the original sphere”.

The second volume, Topology as Fluid Geometry, assumes more sophistication on the part of the reader. Cannon assumes that the reader has had “a first course in topology and is comfortable with open and closed sets, connected sets, compact sets, limits, and continuity”. The first chapter presents two proofs of the Fundamental Theorem of Algebra. It begins with the following endearing anecdote:

Georg Pólya wrote to me when I was a young Mormon missionary in Austria. He said that I should solve a hard mathematical problem every week so that I wouldn’t rust (“Wer rastet, der rostet”). He also gave me a list of books that I might find in a used bookstore.

The second chapter proves Brouwer’s Fixed Point Theorem in the plane via an engaging series of examples and techniques — tiling a checkerboard, the \(15\)-puzzle, the polygonal Jordan Curve Theorem and the one-ended arc trick — that lead to a proof of the No Retraction Theorem, which is then used to establish Brouwer.

The later portions of this second volume continue with more sophistication and its concomitant beauty. Here the reader will meet

finite curves of infinite length, finite curves of positive area, space filling curves, disks whose interiors have smaller areas than their closures, \(0\)-dimensional sets through which no light rays can penetrate, continuous functions that are nowhere differentiable, sets of fractional dimension.

Some guidance might help beginners get a better handle on the subtle topological concepts and arguments that dominate this segment. For instance, the statements and proofs of Wilder’s topological characterization of the \(2\)-sphere and that of Moore’s Decomposition Theorem may challenge undergraduates who have only seen a semester of topology. I recommend that such readers skip parts they find difficult and encourage them to return to such in the future. The volume closes with the analysis of \(2\)-dimensional manifolds. Both structure and classification theorems are proved. The reader is introduced to the notions of triangulation, genus, and Euler characteristic on the way to proving the Riemann-Hurwitz theorem.

The heart of the third and final volume of Cannon’s triptych is a reprint of the incomparable introduction (written jointly with Floyd, Kenyon, and Parry) to Hyperbolic Geometry (Flavors of Geometry, MSRI Pub. Vol. 31, 59–115). I strongly urge readers to read this piece to get a flavor of the quality of exposition that Cannon commands. Cannon then introduces a natural metric-invariant definition of curvature in dimension \(2\) via measuring the difficulty of flattening a surface into the plane without distorting lengths and areas. It closes with an introduction to the Gauss map and a proof of Gauss’s Theorema Egregium.

My minor and major complaints must begin with bemoaning the lack of indices! Certain chapters contain endnotes and end-of-chapter hints to the lists of exercises, but these are lacking in a number of places where most beginners would desire more help and guidance. Alarmingly scant attention is paid to historical sources. Finally, there are instances of poor copyediting that are distractions. They do not muddy the content but leave minor blemishes.

The third chapter (Gravity As Curvature) of the third volume (Non-Euclidean Geometry and Curvature) puzzled me, since it appeared to be incomplete! The entire chapter consists of less than a page, and comprises the single subsection (3.1. Einstein Identified Gravity with the Curvature of Space-Time). Cannon writes in his preface:

This is the last of a three volume set describing a very personal arc of thought that begins with earth measurement (that is, geo-metry), passes through the topology of \(2\)-dimensional surfaces, and ends with space-time measurement (that is, geo- chrono-metry, where Einstein identifies gravity with the curvature of space-time)

However, there seems to be no real discussion of space-time measurement or of Einstein’s ideas regarding gravity. Given the surprising brevity of Chapter 3, I was suspicious that there might have been a printing error.

Such quibbles apart, Cannon’s books are worth every cent. I have in the past gifted Hilbert & Cohn-Vossen, and Rademacher & Toeplitz to my students. Now I have Cannon’s trio to add to my list of giftables.

I end this review by leaving the reader with Cannon’s invitation to this wonderful work!

These three volumes form a very personal excursion through those parts of the mathematics of \(1\)- and \(2\)-dimensional geometry that I have found magical. In all cases, this point of view is the one most meaningful to me. Every section is designed around results that, as a student, I found interesting in themselves and not just as preparation for something to come later. Where is the magic? Why are these things true?

Where is the tension? Every good theorem should have tension between hypothesis and conclusion. — Dennis Sullivan

Where is the Sullivan tension in the statement and proofs of the theorems? What are the key ideas? Why is the given proof natural? Are the theorems almost false? Is there a nice picture? I am not interested in quoting results without proof. I am not afraid of a little algebra, or calculus, or linear algebra. I do not care about complete rigor. I want to understand. If every formula in a book cuts the readership in half, my audience is a small, elite audience. This book is for the student who likes the magic and wants to understand.

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Tushar Das is an Associate Professor of Mathematics at the University of Wisconsin–La Crosse.