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.
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.