by **quickfur** » Thu Mar 07, 2013 4:19 pm

One always has to keep in mind the distinction between an (n-1)-hyperplane in n-dimensional space, which is what divides that space, and which serves as boundaries of n-D object, vs. a 2D plane, which only serves the role of dividing space in 3D. So when dealing with 2D planes in 4D, one should always keep in mind that it does not divide space, and so one should treat it more like "edge" (or "ridge") rather than "plane".

In 4D, most 2D planes, if they intersect at all, only intersect in a single point. One crude way of understanding this is to treat them as extrusions of 3D lines in the 4th direction. In 3D, two arbitrary lines usually will not intersect; if they do, it's most likely only in a single point. Only very rarely do you get two lines that intersect in a line (i.e., coincident). Now consider the extrusions of these scenarios into 4D. If you like, visualize this as a 3D cross-section, which would be two random lines drawn in 3D space. If you draw successive cross sections of these two 2D planes, you get a series of 3D cross-sections in which the two lines retain their orientation, but change their position. Thus, they trace out two 2D planes in 4D.

Most of the time, if you take two random lines with two random directions of change in position, will not intersect in any of the cross-sections at all. Which means that in 4D, they don't intersect. Once in a while, you get a series of lines in the 3D cross-sections that move toward each other and eventually cross: this is the case where two planes intersect in a point. Note that this case only has a single intersection point, because after the crossing, the lines in the 3D cross-sections move apart again.

Once in a while, you will get a case where the two lines intersect each other in every cross-section. This is the case where the two planes intersect in a line (collect the intersection points in the cross-sections together to form a line). As you can tell, this is very rare: it requires the two planes to be close enough and oriented similarly enough in the right positions, that every cross-section has an intersection. Another case similar to this one is when the two lines are parallel in the 3D cross-sections, and they move toward each other such that in one cross-section they are coincident. This also is the case of two planes intersecting in a line -- and it's also very rare.

The rarest case is when the two lines are coincident in every cross-section: this is the case where the two planes are coincident and have a 2D intersection.