Fair, but two perpendicular Flatlands embedded in the same 3D space won't be able to agree on which is the fourth dimension. It's fine as a shorthand when Flatlanders talk to each other, but "the fourth dimension" still won't be an unambiguous direction, for Flatland A it's actually one of Flatland B's two dimensions, and vice versa. For us any dimensions perpendicular to the entire universe will be "special", but only for us. Native four dimensional critters won't see what's so different about the ana/kata axis, unless our universe happens to be their tabletop RPG.
> Fair, but two perpendicular Flatlands embedded in the same 3D space won't be able to agree on which is the fourth dimension.
A lot of religious/spiritual people will say that God or the metaphysical realm (some people agree, some disagree...just like certain ideas in physics) is where many higher dimensions can be found. Say what you want, but people's incorrect models of reality having more influence than "reality itself" isn't nothing (if it kills people, it's at least something[1]). Besides, almost everyone complains about it, they just don't consider it (thus it is not) dimensional, it's "just reality", kind of like how a lot of phenomena now understood due to science were(!) formerly "just reality".
Well that's where I think the Flatland analogy is helpful, there's nothing inherently mystical about a fourth spacial dimension any more than Flatlanders ought to be worshiping us because we have one more dimension than they do.
> 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.
I guess, but there may be (is, in my opinion) something mystical about higher non-spatial, metaphysical dimensions, and I think the difficulty one experiences conceptualizing a 4th spatial dimension can be useful in attempting to conceptualize the possibility that difficulty also exists in conceptualizing these non-spatial dimensions.
Or in other words: People don't think it be like it is, but it do.
I think you’re making my point. The Fourth Dimension is still uniquely defined for our 3-space (which is the entire universe, as far as we know). I think that’s pretty solid justification for giving it a distinct name.
Hypothetically, if there are other (infinite) 3-spaces embedded in our 4D metaverse, then we either intersect them (which would cause all sorts of problems), or they are parallel and agree with our definition of The Fourth Dimension.
Our universe might be a much more complicated manifold than a plane in the space it's embedded in. Of course, we experience it as a flattish 3D space, but maybe gravity makes it four dimensionally lumpy. Or something else, who knows. Maybe it's actually closed and is the surface of a very, very large 4D sphere.
This would make the ana/kata vectors pointing outwards away from the universe where I am and the ana/kata vectors where you are not line up, and there wouldn't be any way to decide which ana/kata vector is the special one, even between different points inside our universe.
But really, my objection is mostly that "The Fourth Dimension" makes it sound like a "Dimension" is a kind of place, which is confusing.
This is one of the biggest hurdles of learning modern physics. We know, from experiment, that the universe is flat at large distances, and that there is not a fourth dimension into which it is folded or bent.
However, we also talk about "curvature" of space-time, and people's intuitive first understanding of that is that space-time somehow gets bent from the perspective of a higher number of dimensions.
Curvature is very badly explained in physics, starting from the highly misleading rubber sheet analogy.
If you want more pet peeves for your collection, try every explanation of gravity that shows rubber sheets, and (for advanced peeving) rubber sheet diagrams of Schwarzschild black holes that put the event horizon partway down the well, instead of where the coördinate singularity actually places it (which is infinitely far down at a finite radius).
Isn't that just a matter of visualization, though?
You can visualize (a static snapshot of) a 1-dimensional compression wave as the flattening of a 2 dimensional sine wave.
More generally, any n-dimensional space with varying "density" can be viewed an n+1 dimensional space, where the extra dimension is the "density" dimension.
Maybe a "stretch".
The article is extremely clear in explaining the fourth dimension is a direction, not a place. You appear to be projecting your past experiences onto the article.
Quite likely I am! And of course the article was fine. But I will note there was already a comment on it about how "the fourth dimension is time, actually." So this particular phrasing does seem to keep confusing people.
If two flatlands are embedded in 3d space, there's only three dimensions, and one dimension shared between the two flatlands, or one common 3rd dimension if the flatlands are parallel.
What they may disagree on is which the third dimension is. But, so what? subjectivity isn't wrong. You can make the same argument to say that "right" and "left" don't exist because they are subjective.
No, one couldn't; because roywiggins isn't arguing that the directions don't exist. Given that the ideas of there being no privileged frame of reference and the universe being isotropic are fairly basic concepts of modern physics, it is quite surprising how many people don't grasp the quite simple argument that "the" is the wrong article to use.
It was even wrong by 1884 standards, although one could excuse a treatment of it at the novice level from explaining that not all subspaces of 4-dimensional spaces are necessarily parallel.
