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Explorations in Topology by David Gay

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8

New Worlds

Klein Bottles and Other Surfaces

This chapter develops ideas from Chapters 6 and 7. One idea is to take the rectangular patterns for the surface of a doughnut (torus) and for the Möbius strip as a model and, by varying how the top and bottom of the rectangle are identified, arrive at new and interesting surfaces. Also, the use of a rectangle with identified sides is generalized to the use of other polygons with identified sides. In trying to make sense of these new surfaces, another idea (from Chapter 7) is utilized: cutting up a pattern, labeling cut edges (later to be reglued), reassembling pieces, and gluing edges originally identified. One arrives at a “new” pattern corresponding to the original surface. One can then study this ...

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