Posted on 17 September 2018 by **cjf**

Stephen Barr’s “Experiments in Topology” (originally published in 1964, reprinted in 1989 by Dover) is extraordinary because it treats a sophisticated mathematical subject with accessible language that can be understood by motivated junior high students. It is extraordinary because its wonderful and copious figures are remarkably clear and elucidating. It is extraordinary because it captures the flavor, depth, and breadth of a very subtle subject with carefully written passages that boil down significant complications into understandable overviews. It is extraordinary because he gives a concrete enough treatment that the attentive reader can learn something substantial of the subject while the mysteries that are both alluded to and implicit may drive the curious reader to explore its nooks and crannies. It is extraordinary because its learning-by-doing (experimental / exploring) style is infectious and the reader may be emboldened to ask their own questions and attempt their own experiments. It is extraordinary because its effective survey of key ideas in the major branches of topology make it a useful reference. It is extraordinary because it can reward the casual reader with some basic guideposts for apprehending an advanced subject while the serious student who builds all its models and tries to understand their integrated significance can extract many deeper insights from the text.

I prefer this latter, in-depth, approach and organized, through Math Counts, a mathematics group in Philadelphia, 11 deep explorations on topics from the book each with a group of 6-10 mathematically-oriented colleagues. We thoroughly enjoyed the book and our explorations, although after 11 months we wanted to move on to other ideas.

The book uses homeomorphism as its first principle for exploring topology. Barr gives several definitions, but I found the characterization on page 5 to be the most helpful: “Any distortion is allowed provided the end result is connected in the same way as the original.” This exemplifies the kind of careful but informal style of the book.

In chapters 2-6 we are introduced to the topological surfaces that can be built with a sheet of paper, namely, the plane, cylinder, torus, Möbius band, Klein bottle, and projective plane. This might be called the piecewise linear topology of 2 dimensions and it provides an introduction to ideas from geometric topology and differential topology. Barr introduces algebraic topology with Euler’s formula in chapter 1 and Betti numbers in chapter 8. The treatment of graph theory in chapters 7-8 includes an effective overview of the four color theorem which one of our participants, Kurt Tekolste, observed follows some of the ideas of the actual proof (which was first announced in 1976, 12 years after the publication of the book). Barr includes a little bit of knot theory in chapters 6, 8, and 9. Chapter 9 on the eversion, or inside-outing, of the torus is a wonderful further introduction to differential topology. The basic ideas of point-set topology are introduced in chapters 10-11.

In short, the book gives a competent introduction to the material and the flavor of a broad swath of the main subfields of topology though Barr does not use the academic disciplinary names preferring an informal treatment. I would like to thank George Zipperlen for pointing out the breadth of coverage in the book and for providing us a lot of useful context about the various branches of topology.

For each of our monthly meetings, I prepared a set of exercises on a bite-sized part of the book (my questions were sometimes a bit off because I had never studied topology in any depth before). Working through those questions, both in preparation before a meeting and during each group session, took more time than I expected. We came to realize that beneath the surface of Barr’s elementary presentation live many subtleties which neither the book nor our mathematically experienced participants were able to easily explain to each other. We had doubts about the ideas and the language in the book. It took a lot of effort to integrate each of our partial insights into apprehension. In fact, it took so much time and effort that we usually failed to complete the prepared exercise set, so the next month’s frequently overlapped significantly. But in every instance we came to understand the posed questions and the book in a deeper way. And we got to enjoy the rewards of determined exploration: realization and discovery over and over again! This in-depth process convinced us that Barr’s writing is very good: once we understood what he was saying and what the often unstated caveats were, we were in awe at how effectively he spoke mathematical truth in language accessible to middle school students!

Although I agree with Bruce Trumbo’s assessment that much of Barr’s language will cause difficulties for the reader, I think that applies to all mathematics writing: unfamiliar language about unfamiliar subjects is always difficult to interpret especially when the precision of mathematical thinking is desired. It was the general assessment of our group that overall Barr’s writing is precise and nuanced despite its informal style. It is topology itself that is almost too complicated for such an elementary treatment! Barr should be commended for his brilliant success in introducing such a subtle subject to a broad population of readers while maintaining the integrity of mathematical precision even though it may take substantial effort for readers to understand many of its passages in any depth, let alone the integrated significance of them into the broad enterprise that is modern topology. However, it should be emphasized that the book rewards less intense study by giving a feeling for the subject even if you choose not to fully untangle the nuances in the guideposts provided by the text.

One delight in the book, which George Zipperlen observed, is how the chapter on the conical Möbius strip subtly motivates the cross-cap. Indeed, the conical Möbius strip, in the limit, is the cross-cap. At our June 2018 event, George said “Be patient with Barr, what seems like a side excursion will come back.” Indeed, I suspect that most of the juxtapositions and suggestions of the text are meaningful even though we did not have the wherewithal to explore very many of them: it is a rich book and a rich subject!

Not only that, but your own ideas can lead you to fascinating tangents that are not in the book. One of our participants, David Sternman, imagined cutting a torus along a Möbius strip which led Michael Reichner to post a link that led us to George Hart’s great demonstration video of how to cut a bagel for a better breakfast. And Jeannie Moberly impressed us with her wonderful construction of Chris Hilder’s Boy’s Surface model.

We noticed very few errors in the Dover edition. One of our participants, Bob Miller, helped us realize there is a typo on pages 47-48 where it says “bringing B and B’ together and joining BC to B’C’ ” it should say “bringing B and B’ together and joining B’C to BC’ ”. On page 91, the phrase “is not is no worse” should simply be “is no worse”. On page 138, “Laocoön statute” should be “Laocoön statue”. In Fig. 25 on page 144, the dashed line “y” should be external and not internal to the inside-out torus.

Here is a list of links to the event descriptions and associated exercise sets for each of the 11 events I organized on Barr’s book (several of the comments posted to these events are also interesting):

- Topological Surfaces from a Strip of Paper (feat. minimal length Möbius strip) (Oct 2017)
- Variations of the Möbius Band to Explore the Nature of Homeomorphism (Nov 2017)
- Exploring Homeomorphism through Experiments on the Möbius Band (Dec 2017)”
- Topological Experiments: The Conical Möbius Band & the Klein Bottle (Jan 2018)
- Experiments in Topology: Dissecting The Klein Bottle (Feb 2018)
- Exploring the Topology of the Projective Plane (Mar 2018)
- Map Coloring; Martin Gardner’s Projective Plane & variations (Apr 2018)
- The Symmetry of the Projective Plane (and the curious property of twist) (May 2018)
- Using Twist to Understand the Topology of the Projective Plane & its Symmetry (Jun 2018)
- Betti Numbers and the Symmetry of the Projective Plane (Jul 2018)
- 10 exercises in support of reading and discussing chapter 8 pages 123-128 (on Betti numbers) and chapter 6 pages 96-107 (on the projective plane)

- Deliberations in The Trial of the Punctured Torus (Aug 2018)
- Note: 13 exercises on chapter 9 are included in the event description instead of a separate PDF

In my research on Barr’s book, I found these additional resources:

- Dover’s product page for the book. Dover also provides a PDF of chapter 1 of the book.
- H. M. Cundy’s review in The Mathematical Gazette Vol. 50, No. 373 (Oct., 1966), pp. 323-324.
- Bruce Trumbo’s review in Mathematics Magazine Vol. 38, No. 1 (Jan., 1965), p. 50.
- Karen M. Daniels of the University of Massachusetts Lowell has 14 slides based on Barr’s book to introduce topology to her Graduate Geometric Modeling students.
- Maria E. Salcedo’s paper “Knotted Ribbons and Ribbon Length” includes a section on Barr’s treatment of the shortest Möbius strip.
- Matthew R. Francis has three blog posts on the book: Hearing Around Corners, Simply Connect (Further Adventures in Topology), and Time to Make the Paper Donuts.
- Alexander Bogomolny’s Cut-The-Knot site has a landing page with a brief review of the book and a copy of its table of contents and back cover.

I have found little biographical information about Stephen Barr. If you know anything about him or if you know of another good resource on the book or have some experience or question about it, please post a comment as I’d love to learn more.

Whoever Stephen Barr is or was, the book “Experiments in Topology” is an able tour guide to lead both casual and serious readers to apprehend some key ideas in topology, one of the most storied and important branches of modern mathematics.