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Explorations in Topology by David Gay

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9

Surface Sums and Euler Numbers

In this chapter, linkages are introduced both as a follow-up to the use of video games in the previous chapter and also as a motivation for the study of surfaces itself. The study of linkages here also leads to the question: Given a polygon (a pattern) with a prescription for gluing sides, can the resulting surface be described? This leads to a careful definition of surface and to a notion of “surface sum,” a way of creating new surfaces out of old. This also leads to the definition of “Euler number” for a surface, a generalization of Euler’s formula for maps on a sphere (and for polyhedra), and a device that can help identify a surface. This chapter sets the stage for the next in which the classification of surfaces ...

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