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GuySrinivasan comments on Less Wrong Q&A with Eliezer Yudkowsky: Video Answers - Less Wrong
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In 1 or 2 dimensions, random walks return to the origin infinitely often. In 3 dimensions, they have but a 34% chance. There are nontrivial qualitative differences between numbers of spatial dimensions that we don't see when we think "2? 3? 5? 179? It's just a choice of N!"