A Proof That Some Spaces Can’t Be Cut
The question is deceptively simple: Given a geometric space — a sphere, perhaps, or a doughnut-like torus — is it possible to divide it into smaller pieces? In the case of the two-dimensional surface of a sphere, the answer is clearly yes. Anyone can tile a mosaic of triangles over any two-dimensional surface. Likewise, any three-dimensional space can be cut up into an arbitrary number of pyramids.
But what about spaces in higher dimensions? Mathematicians have long been interested in the general properties of abstract spaces, or manifolds, which exist in every dimension. Could every four-dimensional manifold survive being sliced into smaller units? What about a five-dimensional manifold, or one with an arbitrarily large number of dimensions?
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