if we lived in 4 spatial dimensions our cables would never get tangled. really makes you think
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wouldn’t they just get tangled worse...?
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they have more space to pass "through" each other. you can sort of visualize it by taking color to be a fourth dimension. you can move stuff in that dimension by changing its color and that lets you pass parts of cables "through" each other
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yea i get that, just as being able to move things through the 3rd dimension allows you to tidy up 2d messes by picking up objects "out" of the field and reinserting them elsewhere – but it also introduces a whole new kind of potential mess, no? in that same sense?
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like, a knot is a 3d phenomenon, right – you can't have a 2d knot. I imagine you could resolve a 3d knot by going through the 4th dimension, but what about 4d knots?
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so i'm saying things very tersely but the "passing through" move i've been describing basically proves that 1d shapes can't be tangled / knotted in 4d; basically in 4d you can undo all of the crossings that appear in a tangle / knot diagram
this is cool and not obvious!
it's actually a cool metaphor to combine with your "walking through cultural walls" metaphor; when you have an extra degree of freedom you can move in ways other people can't, do things that seem impossible to them, kinda thing
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yea I get that re: 4d lets you untangle lower dimensional things in "magical" ways
and to continue with the cultural walls metaphor: having an extra degree of freedom also introduces you to new problems that nobody else is capable of seeing or even comprehending
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Wait, what's the broader rule here?
1d shapes can't tangle in 1d (obv) or 2d (they can circle but not cross). I believe flat 2d sheets can't knot in 3d (again, can't cross), but can in 4d.
Do sheets not knot in 5d? Does adding more dimensions ever restore "knotiness"?
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Actually, that makes some kind of intuitive sense? You need +1d to loop, +1d more to cross, but beyond that your object doesn't "fill" any more dimensions so it just gets easier to straighten out.