Topological Experiments: The Conical Möbius Band & the Klein Bottle

Stephen Barr's fun book "Experiments in Topology" gives a feeling for homeomorphism (a fundamental concept in topology) with paper models of Möbius Bands and Klein Bottles (and more). Our task in this meetup is to explore The Conical Möbius Strip (Chapter 4 pp. 50-61) and The Klein Bottle (Chapter 2 pp. 34-39 and the first half of Chapter 5 pp. 62-69) in the book. The introductory section of Chapter 1 (pp. 1-9) and the section in Chapter 2 on orientability (pp. 25-28) provide important background information for this session.

On p. 4, Barr defines a homeomorphism ( https://en.wikipedia.org/wiki/Homeomorphism ) with a rule: "distortions are only allowed if one does not disconnect what was connected (like making a cut or a hole), nor connect what was not (like joining the ends of the previously unjoined string, or filling in the hole)". On p. 5 he simplifies this "no-cutting-or-joining rule": "Any distortion is allowed provided the end result is connected in the same way as the original". It is an intuitively conceptual definition instead of a mathematically rigorous one. Experimentation is needed to develop an intuition for its meaning.

The hope and intention for this session is to build our intutions about topology and homeomorphism through exploring paper models of the Möbius band and the Klein bottle.

It is recommended that participants build as many of the paper models discussed in the book as they have time for. Building the models and thinking about them ahead of time will help with our task of exploring the subject during our discussion.

To guide your exploration of the text and to guide our exploration during the meetup, I have organized this list of 20 questions: http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.04.pdf

Here are some selected questions for the discussion from the full list (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.04.pdf):

● Inspired by the book, the questions that follow or your own initiative, what additional experiments did you attempt? What did you learn from these supplemental experiments?

● In considering the variations on a Möbius band in chapter 4 and any additional thought or model-building experiments you might have undertaken, how should we think about the distortion and connectedness in the joined edge of the strip of paper used to make a Möbius band? How is the homeomorphic property preserved in each of these variations?

● According to chapter 1 of Barr's book, what is the rule for homeomorphically distorting one surface into another? What caveats does Barr mention for using his rule? How does Barr's definition compare to Wikipedia's definition of homeomorphism?

● What is the property of orientability? Which of the sphere, disk (the 2D surface bounded by a circle), cylinder, torus, and Möbius band are orientable and which are nonorientable?

● After reading chapter 4, can you explain what Barr's text means on p. 52 where it says "some meaningful restrictions must be placed on the Möbius strip, too, as to how much of the edges ought to be joined" and in wondering "if the amount of edge involved can be increased"? What does it mean to increase or decrease the amount of edge involved? What are the restrictions alluded to?

● On p. 61, Barr concludes chapter 4 by saying "The moral of all this is that when we allow only one kind of distortion (bending), unexpected relationships persist. Suspicion arises that with any distortion allowed, what persists must be invariant indeed, and perhaps overlooked before." How do you interpret this conclusion? What invariants did you infer from the experiments into topology that you undertook in reading chapter 4?

● In considering the variations on a Möbius band in chapter 4 and any additional thought or model-building experiments you might have undertaken, how should we think about the distortion and connectedness in the joined edge of the strip of paper used to make a Möbius band? How is the homeomorphic property preserved in each of these variations?

● In considering the sequence of experiments discussed in chapter 4 where Möbius strips with various extents of connectedness are considered, what did you learn about the nature of homeomorphism, topological invariants, and paper representations of topological surfaces?

● Given the considerations in chapter 4, what subtleties, limitations, and caveats must we heed about the nature of the distortions allowed and the requirements for connectedness and continuity in Stephen Barr's definition of a homeomorphism?

● What happens when the Klein bottle is cut in two?

● In considering the conical Möbius band and the Klein bottle and two of its dissections in the first half of chapter 5, what observations, realizations, or insights have you acquired about the nature of topology, homeomorphism, orientability, and the connectedness of topological surfaces?

This meetup is part of a series exploring the content of Stephen Barr's fun book "Experiments in Topology".

"Experiments in Topology" is available from Dover (http://store.doverpublications.com/0486259331.html)

"Experiments in Topology" at Google Books (https://books.google.com/books?id=9TMx6ABV-98C)

Each event will be made accessible to newcomers. Key concepts will be reviewed and an effort to explain technical terms will be made. If anything is unfamiliar to you, please ask and we will try to clarify.