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Euler's Gem by David S. Richeson

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CHAPTER 17

ARE THEY THE SAME, OR ARE THEY DIFFERENT?

It was very often repeated that Geometry is the art of reasoning well on badly made figures; still these figures, not to mislead us, must satisfy certain conditions; the proportions can be grossly distorted, but the relative positions of the various parts should not be disrupted.—Henri Poincaré in the introduction to Analysis Situs1

One of the most important recurring questions in mathematics is: are the two mathematical objects X and Y the same? In different contexts we have different criteria for what “the same” means. Often, when we say the same we mean equal, such as the expression 5 · 4 + 6 − 23 and the number 18, or the polynomials x2 + 3x + 2 and (x + 2)(x + 1). In other circumstances ...

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