Try shifting it down: Another thing that can happen in 3 dimensional space but not 2 is that you can have two infinitely long rectangles which only intersect as a line at the origin (and nowhere else.) In 2 dimensions you'd get at least an overlapping quadrilateral in the intersection. (That doesn't imply I grasp it in 4.. ha)
I don't understand your comparison. It's true the intersection of two overlapping rectangles in a 2D space form another rectangle. But that's always true for all dimensions, isn't it? For example the intersection two overlapping lines in 1D space is another line, the intersection of two overlapping boxes in 3D space is another box.
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.
