Also the number of regular polytopes is highest (6) in 4D (https://oeis.org/A060296).
When push comes to shove I'm not sure we have enough experimental physicists...
> Another thing that can happen in 4 dimensional space but not 3 is that you can have two planes which only intersect at the origin (and nowhere else.) In 3 dimensions you'd get at least a line in the intersection.
I tried to envisage how that could be, but failed miserably.
The stackoverflow quote is the intersection of two 2D objects (planes) in a 4D space. If you take it down one dimension, you are looking at the intersection of two 1D objects (lines) in a 3D space. That intersection is always 0D object (a point). It's also a point if the lines intersect in 2D space. I would humbly conjecture two intersecting lines in any space with more than 1 dimensions would be a point.
My naive theory would be that the intersection of two N dimension objects in a space with more dimensions than N would always be a N-1 dimension object. Apparently not. The intersection of two infinite planes in 4D can be a point. Not any point mind you, only the origin. They must be very special planes.