> there's nothing inherently mystical about a fourth spacial dimension
Weirdly enough, there are big distinctions between three and four dimensions when it comes to geometry and topology. For example: “Four is the only dimension n for which R^n can have an exotic smooth structure. R^4 has an uncountable number of exotic smooth structures; see exotic R^4.” [0]
It turns out that having four dimensions really is magically different from having any other finite number.
Does this mean anything physical for 4-D spacetime? I'd guess yes because "manifold" is a general turn that includes the asymmetric curvature of spacetime.
Weirdly enough, there are big distinctions between three and four dimensions when it comes to geometry and topology. For example: “Four is the only dimension n for which R^n can have an exotic smooth structure. R^4 has an uncountable number of exotic smooth structures; see exotic R^4.” [0]
It turns out that having four dimensions really is magically different from having any other finite number.
[0] https://en.wikipedia.org/wiki/4-manifold#Special_phenomena_i...