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Euler's Gem by David S. Richeson

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CHAPTER 23

HENRI POINCARÉ AND THE ASCENDANCE OF TOPOLOGY

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.—G. H. Hardy1

If Euler’s theorems on the bridges of Königsberg and polyhedra mark the birth of topology, and the contributions of Listing, Möbius, Riemann, Klein, and other nineteenth-century mathematicians signify topology’s adolescent years, then its coming of age was signaled by the work of Henri Poincaré. Before this there were theorems that we now categorize as topological, but it was not until the waning years of the nineteenth century ...

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