The Seven Frieze Groups

Around the frieze of an older building there is often a pattern formed by the repetition of some figure or motif over and over again. The essential property of an ornamental frieze pattern is that it is left fixed by some “smallest translation.” We can call AB the length of translation τ_A,B and say τ_A,B is shorter than τ_C,D if AB < CD. Other symmetries in addition to translations are often apparent in a frieze as well. Of course, there is infinite variety in the subject for such patterns. However, by discounting the scale and subject matter and by considering only the symmetries under which such patterns are left invariant, we shall see that there are only seven possible types of ornamental frieze patterns.