There are 17 distinct plane crystallographic groups. These groups are, bydefinition, the symmetry groups of arrangements of discrete geometric figureswhich tile the plane. Thus, each of these arrangements can be represented bya "wallpaper" pattern. An illustrated discussion of these symmetry groups canbe found in the link to the 17 wallpaper groups under Of Particular Note onthe What's New in Mathematics home page.
Here is an illustration of the pattern with the richest symmetry group.