Topological Surfaces from a Strip of Paper (feat. minimal length Möbius strip)

What can we learn about the basic concepts of topology from paper representations of topological surfaces? Guided by chapters 1-3 of Stephen Barr's fun book Experiments in Topology, we will explore topological surfaces (https://en.wikipedia.org/wiki/Surface_(topology)) through the device of building paper models of them. Through this process we will explore several basic notions of topology at an elementary level including homeomorphisms (https://en.wikipedia.org/wiki/Homeomorphism), simply connected surfaces (https://en.wikipedia.org/wiki/Simply_connected_space), Euler's formula (https://en.wikipedia.org/wiki/Euler_characteristic), and orientability (https://en.wikipedia.org/wiki/Orientability).

To help guide the discussion and to help focus your efforts reading the book and building the paper models it describes, I have prepared a list of 26 questions and problems (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.01.pdf).

Selected questions for the discussion from the full list (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.01.pdf):

• According to Barr's book, what is the rule for homeomorphically distorting one surface into another?

• Compared to a topologically ideal surface, what properties does a sheet of paper have that make it inherently non-topological?

• What artistry can we employ in the building of models of topological surfaces from strips of paper to rectify these deficiencies, at least to some extent?

• What topological properties are impossible to represent with a physical sheet of paper as a model of a topological surface?

• How can you represent a topological plane, cylinder, torus, and Möbius strip with a strip or sheet of paper?

• Which of the following surfaces are simply connected: the sphere, a plane, a cylinder, a torus, an annulus, a Möbius strip, a mug with a handle, and a disk (the 2D surface bounded by a circle)? Why?

• What is the topological property of orientability? Which of the topological sphere, plane, cylinder, torus, and Möbius strip are orientable and which are nonorientable? Do your paper models of these topological surfaces exhibit the orientable property? How?

• Which of the following topological surfaces are homeomorphic to another surface in the list: a sphere (as a 2D surface), a plane, a cylinder, an annulus, a torus, a Möbius strip, a mug with a handle, and a disk? Why?

• In a paper model of a Möbius strip, what happens when you cut through the middle of the strip? What are the resulting surface(s)? How many sides, edges, and separate pieces are there? What logic explains this behavior?

• Can one build a paper model of a Möbius strip whose width is greater than its length? That is, can the width to length ratio (width/length) ever exceed 1?

• What is the maximum width to length ratio for a paper model of a Möbius strip?

• How can one build such a "minimum length" Möbius strip from paper?

• What is the width to length ratio in your "minimal length" paper Möbius strip model(s)?

• What is width to length ratio of the minimal length paper Möbius strip which can still be cut through the middle of the strip? Why does this property of cutting a Möbius strip through the middle fail when the width to length ratio exceeds that amount?

To prepare for the discussion, it is recommended that you read chapters 1-3 in Stephen Barr's book "Experiments in Topology" and think about the full list of 26 questions and problems prepared for the event (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.01.pdf).

More information about Stephen Barr's 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/about/Experiments_in_Topology.html?id=KFeGa7Ok954C)