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Counterintuitive Properties of High Dimensional Space (2018)

(people.eecs.berkeley.edu)
247 points nabla9 | 1 comments | 13 Oct 24 21:09 UTC | HN request time: 0.204s | source
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bmitc ◴[14 Oct 24 00:42 UTC] No.41833078[source]▶
Actually, the most counterintuitive is 4-dimensional space. It is rather mathematically unique, often exhibiting properties no other dimension does.
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elcritch ◴[14 Oct 24 00:56 UTC] No.41833147[source]▶
How so?
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1. hansvm ◴[14 Oct 24 01:32 UTC] No.41833370[source]▶
The intuitive way to think about it is that with very few dimensions you have very few degrees of freedom, so it's easy to prove things possible or impossible. With lots of dimensions, you have enough wiggle room to prove most things possible. Somewhere in between, you have enough complexity to not trivialize the problems but not enough wiggle room to be able to easily circumvent the issue.

Often in practice, that boundary is around 3-4 dimensions. See the poincaré conjecture, various sphere packing shenanigans, graph embeddings, ....