Using Twist to Understand the Topology of the Projective Plane & its Symmetry

We will explore the nature of twist in a number of paper models of the projective plane with a view toward understanding its symmetry and its topology. Our trusty guide will be pages 82-107 of Chapter 6 in Stephen Barr's "Experiments in Topology". If you are new to the book reading pages 1-9 in Chapter 1 and all of Chapter 2 (pages 20-39) will provide sufficient background.

Here is a set of 11 exercises which will guide our exploration during the event: http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.09.pdf

Here are the main questions for the discussion abstracted from the full list of exercises ( http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.09.pdf ):

● How does the Martin Gardner model of the projective plane work?

● What is the pattern for the number of twists that result after cutting along the axes of various variations on the Martin Gardner model of the projective plane discussed in Stephen Barr's text?

● In considering this rule for the number of twists and the full set of variations on the Martin Gardner model of the projective plane discussed in Stephen Barr's text, can you understand why the projective plane is symmetrical? Can you organize your understanding to be clear enough to explain the projective plane's symmetry to a child?

● How are the topological and geometrical projective planes related? Can you see the symmetry of the topological perspective in each of the geometrical models described at http://blog.cjfearnley.com/2012/07/24/models-of-projective-geometry

"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)

The 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.