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Knots, Molecules, and the Universe: An Introduction to Topology

Erica Flapan
American Mathematical Society
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Tom Schulte
, on

Flatland: A Romance of Many Dimensions, the 1884 satirical novella by the English schoolmaster Edwin Abbott written at least in part to comment on the hierarchy of Victorian culture, has been well known for a long time. Many authors find it a worthy starting point for a discussion of two-dimensional universes, higher dimensions, tesseracts, and more. This includes some books reviewed in MAA Reviews, such as: Voltaire's Riddle: Micromégas and the Measure of All Things (MAA, 2010), Surfing Through Hyperspace: Understanding Higher Universes in Six Easy Lessons (Oxford University Press, 1999), Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems and Musings (John Wiley, 1999), and The Shape of Space (CRC/Chapman & Hall, 2001). This book also begins with Abbott’s lucid descriptions, and introduces a new explorer: A. 3D-Girl. Using these elements, the first chapter helps the reader become comfortable with gluing diagrams that describe unexpected 2-D universes and with varieties of 3-D surfaces by way of geodesics and triangles on spheres and tori.

This book introduces the elementary mathematics of topology with applications to chemistry and cosmology. Basically self-contained, it does not assume or require any mathematical background beyond the most basic algebra and set theoretical concepts and notation. Clearly meant to be fun, the book can be an enjoyable read for the interested and nontechnical, or used as an undergraduate textbook.

There are some ambitious undertakings, especially considering the audience, as the second chapter “Visualizing Four Dimensions” and the exploration of the Klein Bottle later in “Part 1: Universes”. Sticking with the material, the reader will gain a good understanding of orientability, manifolds, the Euler characteristic, and the genus of a surface. Assuming no real prior mathematical sophistication on the reader, proving the one-sidedness of a non-orientable 3-manifold is patiently walked through, beginning with defining “if and only if”. This Part concludes with a thorough and enlightening dive info Euclid’s Axioms, given that non-Euclidean surfaces must be confronted.

Part 2’s introduction to knot theory emphasizes invariants, the taxonomy of knots, and their analogy to surfaces through genus and Euler characteristic. Part 3 presents unified applications of knots, topology, and geometry to DNA and protein molecules. Part 3’s two chapters covering chemical, geometric, intrinsic, and topological chirality are rich and engaging. The exploration of DNA covers packing, replication, and recombination described with prior definitions. The Möbius strip makes a surprise reappearance in the closing sections on Möbius ladders in proteins.

The lively style and at times informal theorems support developing intuition and comprehension about topology without an exclusionary argot or technical details. Profuse illustrations are constant aids to clarity in this work. This book is indexed and includes very good chapter exercises, but not solutions.

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Tom Schulte untangles algebra for students at Oakland Community College in Michigan.