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This isn't a question about how to visualize higher dimensions, or how intuit them, or how unintuitive they are.

Rather, it's a hypothetical question about the kinds of questions that might be easier to answer (not necessarily prove, but to suggest) if we could visualize $n$ dimensions as easily as we could 2 or 3. As an example, it's not hard to imagine we would probably have a better idea about kissing numbers if we could picture higher dimensions as easily as the euclidian plane.

What other kinds of (unsolved) problems might lend themselves to analysis more easily if we could intuit $n$ dimensions?

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When I asked this question, multiple commenters were surprised to learn that there don't exist three orthogonal lines with latices coordinates and length $2$ in $3$D space. Multidimension geometry in general tends to be unintuitive because of how used to $2$D we are. In the same vein of thought, manifolds would be a lot easier to study.

A bucket-load of problems in graph theory would likely be easier, as it would grant us the ability to display extremely complex graphs without overlapping any edges. You really only need $3$D to do this, but many people find that hard to conceptualize.

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