Nice thing about tunnels is that unlike unbroken spatial metaphors, 3 is all you need. Any graph can be embedded in 3d iirc (follows from Nash embedding theorem I think?). Even if idea space you're exploring is 10d, so long as you explore small connected patches, 3d is enough
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It’s much easier than this. You can use a fun fact about a twisted cubic: no four distinct points on a twisted cubic are coplanar. So you take n points, draw a complete graph and use induction.
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In fact, this gives you not just an embedding, but one with all straight lines - so Fary’s Theorem is trivial for 3D graphs.
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Oh, neato, though I do like squiggly lines. I think if you add a sort of small epsilon tunnel diameter and a slightly larger delta "cave" radius for junctions you will generally need to violate the straight line geometry? At least if you want to minimize the embedding volume

