TITLE: How does intuition fail for higher dimensions?
QUESTION [22 upvotes]: From this answer:

Now, Algebraic Geometry is one of the oldest, deepest, broadest and
  most active subjects in Mathematics with connections to almost all
  other branches in either a very direct or subtle way. The main
  motivation started with Pierre de Fermat and René Descartes who
  realized that to study geometry one could work with algebraic
  equations instead of drawings and pictures (which is now fundamental
  to work with higher dimensional objects, since intuition fails there).

What are these failings? Can you give me some examples?

REPLY [11 votes]: Intuition already breaks down at the fact that two planes can meet in a single point. The moral is that visualizing intersections in higher dimensions is hard.
Another example: the volume of the $n$-sphere. It starts at 2, then increases until dimension 4, then it decreases again, and in fact, goes to zero. (there is in fact a very long discussion of this on mathoverflow)