Your last paragraph had me intrigued. I'm now wondering what other observations we have that are simply special cases due to low dimensionality.
However, in four dimensions, you have six regular polytopes: video [1] webpage [2] I only found this out relatively recently; while this is no secret it only seems to be recently getting around in the math general interest video channels and such.
After that, the regular case takes over and you only have 3 forever more; the tetrahedron analogue, the cube analog, and the octahedron analog. So here's a case where 0 through 4 dimensions is a special case, and 5 is the first that fits the general pattern.
In 3d space, we have inverse-square laws. Think of a light source, the area that is covers at a given distance is the area of a sphere, so it makes sense that it would decrease with that area, or 4piR*R. Gravity is also inverse-square in 3D space (though relativity makes this much more complicated, I expect). Implication is that in a 4D space, it would be an inverse-cubed relation.
It turns out that stable orbits can only happen with inverse squared laws.
In 4 dimensions, this is not true. A rotation in XY cannot compose with one in ZW so at least two rotations are required.
Here's an example that's easiest to sort-of-visualize:
1. Rotate the y-z plane of 4-d space by half a turn (180 degrees). That is, transform every point in 4-space according to w->w, x->x, y->-y, z->-z.
2. Rotate the w-x plane, also a half-turn. So w->-w, x->-x, y->y, z->z.
Composing these two transformations yields w->-w, x->-x, y->-y, z->-z. That composed transformation is an inversion, not a rotation. Is it clearer now? (It was late when I wrote last night, sorry. Let me try and rephrase how I defined a rotation: a linear transformation that carries one unit vector to a second unit vector, and leaves unchanged any vectors that are orthogonal to both. In my previous comment the plane I was talking about was the plane that contains both given vectors. Of course, the null rotation doesn't pick out any particular plane, though it's not otherwise special in these terms. IANAM, but would appreciate correction if I'm goofing.)