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Euler's Gem by David S. Richeson

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CHAPTER 20

WHEN TOPOLOGY CONTROLS GEOMETRY

And now remains,That we find out the cause of this effect,Or rather say the cause of this defect,For this effect defective comes by cause.Thus it remains and the remainder thus.—William Shakespeare, Hamlet1

For most of this book we have been moving away from the rigid confines of geometry, working instead in the much more fluid environment of topology. In this chapter and the next we return to geometry. We will examine polygons, polyhedra, curves, and surfaces, made not of rubber, but of the hardest steel. However, these geometric objects can still be viewed with a topological eye—the polygons and curves are homeomorphic to a circle, and the polyhedra and surfaces are homeomorphic to a sphere or a g-holed ...

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