Experiments in Topology: Dissecting The Klein Bottle

Stephen Barr's fun book "Experiments in Topology" helps us begin to think topologically by building and considering paper models. In this session, we will explore several dissections of the Klein bottle. In Barr's book the Klein bottle is introduced in Chapter 2 pp. 31-39. In Chapter 5 pp. 62-77, the text develops a (possibly incomplete) classification of the dissections of the Klein bottle.

We will try to understand each cut and its effects. We will try to determine the significance of the six dissections of the Klein bottle explored in the text. We will try to understand the role of cutting or dissection in topology and the topology of surfaces. If time permits, we will begin exploring the projective plane in Chapter 6 pp. 78-107 (we will not get past p. 84 and the introduction of Martin Gardner's model for the projective plane).

The introductory section of Chapter 1 pp. 1-9 (especially the account on homeomorphism) and the beginning of Chapter 2 pp. 20-34 (especially the section on Orientability) provide important background information for this session. Since the Möbius strip is integral to an understanding of the Klein bottle, skimming Chapters 3 and 4 (especially the connectivity diagram of the Möbius strip in Fig. 15 on p. 49) might be helpful.

The hope and intention for this session is to build our intutions about topology and thinking topologically about surfaces through exploring paper models of the Klein bottle and its dissections.

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.05.pdf .

Here are some selected questions for the discussion from the full list (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.05.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?

● 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?

● How would you describe a Klein bottle? Where is its hole? Where is its inside and its outside? What is the nature of its self-intersection? Is the self-intersection integral to the idea of a Klein bottle? Why or why not? Does the Klein bottle have a boundary (or an edge)? How would you explain the idea of a Klein bottle to a child?

● How can one see and understand the nonorientability of the Klein bottle? How can you appreciate its nonorientability in a paper model? How would you convince a child that it is nonorientable?

● What happens when the Klein bottle is cut in two? What do you think? How many cases need to be considered?

● In considering the dissections and related experiments with the Klein bottle in Chapter 5, what is the topological significance of the chapter? What did you learn about thinking topologically? What did Barr intend for us to learn?

● Given Stephen Barr's book so far, all of the experimentation and thinking you have done related to the book and our event(s), what observations, realizations, understandings, and insights have you had about the nature of topology, homeomorphism, orientability, connectedness, continuity, paper representations of topological surfaces, the nature of topological surfaces, and topological invariants?

This meetup will be 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 as accessible to newcomers as possible. 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.