I was interested in the lecture when Paul mentioned the 4th miller index being related to the symmetry of a lattice, so I looked at Chapter 7 of Ashcroft and Mermin, which talks about symmetries of a Bravais lattice.
Bravais lattices can be classified by the symmetries that their lattices exhibit. Since they are all periodic, any translational operation will result in the same lattice. In general, the bravais lattice can be classified by considering all rotations, inversions and reflections that preserve the lattice structure, i.e. it looks the same after the operation. This set of operations is called the symmetry group of the Bravais lattice.
It's also interesting strail to know that in three dimensions Bravais lattices can be classified to 14 symmetry groups. Each group can be applied in one lattice system only. And in three dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.( http://en.wikipedia.org/wiki/Crystal_system)
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