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Euler's Gem by David S. Richeson

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CHAPTER 16

RUBBER SHEETS, HOLLOW DOUGHNUTS, AND CRAZY BOTTLES

A mathematician named KleinThought the Möbius band was divine.Said he: “If you glueThe edges of two,You’ll get a weird bottle like mine.”—Anonymous

By the middle of the nineteenth century mathematicians had a much better understanding of how Euler’s formula applied to polyhedra. It was during this time that they began to ask whether it applied to other objects. What if the figure was not a polyhedron made of flat faces, but instead was a curved surface like a sphere or a torus? If so, what must the partitions look like? Recall that in 1794 Legendre used a partition of a sphere by geodesic polygons to prove Euler’s formula, and Cayley showed that when we apply Euler’s formula to graphs, ...

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