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Euler's Gem by David S. Richeson

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CHAPTER 15

NEW PROBLEMS AND NEW PROOFS

The first important notions in topology were acquired in the course of the study of polyhedra.—Henri Lebesgue1

Suppose you were asked: what trees change color and lose their leaves in the autumn? If you replied, “Maple trees do,” then you would have given a correct answer. However, anyone who has driven through the Pennsylvania countryside in October knows that there are also radiantly colored oak, birch, and beech trees standing amid beds of fallen leaves. So, although the answer was correct, it did not give a complete account of all such trees. Could you say that all trees change color in the autumn? No. Pine, fir, spruce, and cedar trees have no leaves to lose. In order to make a general, but true, statement, ...

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