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Euler's Gem by David S. Richeson

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CHAPTER 22

NAVIGATING IN n DIMENSIONS

LISA: Where is my dad?PROFESSOR FRINK: Well, it should be obvious to even the most dimwitted individual who holds an advanced degree in hyperbolic topology that Homer Simpson has stumbled into the third dimension . . . [drawing on a blackboard] Here is an ordinary square—CHIEF WIGGUM: Whoa, whoa. Slow down, egghead!PROFESSOR FRINK: —but suppose we extend the square beyond the two dimensions of our universe, along the hypothetical z-axis, there [everyone gasps]. This forms a 3-dimensional object known as a “cube,” or “Frinkahedron” in honor of its discoverer.—The Simpsons, “Treehouse of Horror, VI”

Thus far, all of our topological objects have been curves or surfaces—objects that are locally 1- or 2-dimensional, ...

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