Positional Relationship in 4D I don't know if the title is right... I'm not an English speaker...
Let's think about planes. In third dimension, only 3 kinds of positional relationship exists : parallel, share a straight line, exactly the same. However, in fourth some relationships can be added. For example, $xy$ plane and $zw$ plane share only one point $(0, 0, 0, 0)$. What are the other relationships in fourth dimension?
 A: Let $V$ and $W$ be two vector subspaces of dimension $n_V$ and $n_W$ respectively in a $\mathbb{R}^n$ space. Using that $$dim(U \cap V)=dim(U)+dim(V)-dim(U+V)$$ we can conclude the that $$max\{dim(U)+dim(V)-n,0\} \le dim(U \cap V) \le min\{dim(U),dim(V)\}.$$
Hyperplanes can be seen as a point+a vector subspace. Let's consider that both planes cut the origin, then they can essentially be seen as two subspaces. In your case, $n_V=n_W=2$, since we have two planes; and n=4. Therefore, the dimension of the vector subspaces intersection is $0 \le dim(U \cap V) \le 2$. The planes can be the same plane, cut in a line, or cut in a single point. We must also consider the case where one of them does not cut the origin (we can always make a translation so that one of them cuts the origin). In this case, the planes can be parallel, cut in a line, cut in a point or cross in space without cutting. However, I don't see a straightforward way to prove it.
An intuitive way of "seeing" in 4D is using color as the fourth dimension. Imagine that the red color equals to w=1 and blue to w=2. Imagine a red z=0 plane, and a blue y=0 plane. Both planes cross without cutting, because they have different w (color).
