Field
What a lattice forbids
Only two-, three-, four- and six-fold rotations are compatible with periodicity. The proof takes one line and the exception took a Nobel Prize.
The crystallographic restriction
A repeating pattern may have rotations of order two, three, four or six, and nothing else whatever. The proof is one line of arithmetic, and everything finite in the subject descends from it.
Why five-fold is impossible
A second proof, geometric rather than algebraic: assume a five-fold centre, and out of it construct a lattice vector shorter than the shortest one there is.