Frieze groups are a mathematical concept used to explain patterns or symmetries arising in various forms of art including wallpaper and architecture. The term frieze refers to the band of decorative designs around a ceiling or another piece of work on a building. Frieze groups are very similar to the frieze form of architecture since both have a finite length (typically wide enough for one pattern sequence) and an 'infinite' width (naturally a frieze on building but the principle is still valid right up to the end).
There are seven types of frieze groups each with their own patterns. Each pattern depends on the symmetry of the group, whether it be translational, mirror, or reflection symmetry. A visual of each group may be viewed here.
Frieze groups are the 'one-dimensional' counterpart to wallpaper groups which have reoccurring patterns in two-dimensions. Frieze groups, as previously mentioned, only have seven different possible patterns. Similarly, the structure of atomic lattices (see lattice groups for more details) have reoccurring patterns which have properties that vary according to the structure of the lattice. Frieze groups are the simple one-dimensional case of repeated designs which appear all throughout history and their larger dimensional counterparts appear everywhere.