The crystallography restriction states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of $rot_\theta(x)-x$ is less than the magnitude of $x$ if $\theta$ is less than $\pi / 3$. (So if there's a rotation by less than $\pi/3$, it will fail to be a lattice, since we can always get smaller and smaller elements.)
That sounds fine to me, and I get why real crystals would have the requirement that all atoms be some epsilon away from each other. I don't understand why nature requires the lattice to be closed though - obviously real crystals aren't completely closed under addition, or else they would be infinitely large.
So why does the crystallography restriction limit the symmetries of real crystals? (Or does it not, and this application of the theorem is made up by math teachers to convince their students that math is useful?)