Classification of wallpaper groups

Statement
This article completely classifies all the space groups in two dimensions, also called the wallpaper groups. The classification up to affine space type is the same as the classification up to crystallographic space type, and there are a total of seventeen types.

The seventeen types are as follows:

Facts used

 * 1) uses::Crystallographic restriction: This is the main lemma used in the classification, and it states that if a lattice possesses a nontrivial rotational symmetry, then it is spanned by two shortest vectors of equal length at an angle of $$\pi/2, \pi/3, 2\pi/3$$. Note that the case of $$\pi/3$$ and $$2\pi/3$$ gives equivalent lattices.