Sorting a collection of shapes is child’s play. Circles here, squares there, triangles in their own pile.
But if you take the task seriously, there’s a lot more to it. In fact, one of the largest subdisciplines in mathematics — topology — is devoted exactly to this kind of endeavor, and after centuries of concerted effort, mathematicians aren’t even close to finishing.
Topologists study the properties of general versions of shapes, called manifolds. Their animating goal is to classify them. In that effort, there are a few key distinctions. What exactly are manifolds, and what notion of sameness do we have in mind when we compare them?
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