Actually, configuration space have an infinite number of dimensions. Current physics, (or at least QM) doesn't describe our universe as having 3 spatial dimensions. That's a projection, the classical illusion —which is a logical consequences of the underlying physics of course.

Imagine 2 pearls on a thread. You have 2 ways to represent them: the obvious one is 2 points on a line. A less obvious (but just as valid) is a single point on a plane: the X axis would represent the first pearl, and the Y axis would represent the second pearl. Similarly, 2 billiard balls on a billiard can be represented by a single point in a 4 dimension configuration space. And the entire universe require many many more dimensions than that.

There are many more subtleties. I suggest you read the Quantum Physics Sequence for a comprehensible explanation of all this mess. http://lesswrong.com/lw/r5/the_quantum_physics_sequence/