Three non-coplanar translations were used to define a unit cell, a parallelepiped that contains all the symmetry elements of the pattern. We considered three-dimensional lattices and forms in the context of the triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and isometric systems. Two-dimensional patterns can contain glide planes and many popular wallpaper patterns repeat motifs by glide reflection. Three dimensions are needed for the screw rotation that leads, for example, to the helix popular among organic molecules. Typically, the magnitude of the translation part of the glide reflection is one half the lattice translation parallel to the glide reflection.
Diagonal glides are also possible.
Glide reflection is an enantiomorphous operation. Screw rotation is a congruent operation. Because all symmetry elements of a pattern repeat one another, only certain collections of symmetry elements are possible for patterns . These collections are the 17 plane groups in two dimensions and the 230 space groups in three dimensions. All two-dimensional patterns belong to one of the 17 plane groups described in the handout.