Chapter VIII

Frieze and Wallpaper Groups

In the previous chapter we classified all the finite isometry groups. They
were the cyclic and dihedral groups, all of which arise as the symmetries for
bounded figures in the plane. These groups are also known as the Rosette
groups. There are very few artists who are not familiar with patterns
arising from the Rosette groups.

In this chapter we expand our investigation to all of the ornamental
groups. In addition to the Rosette groups, the ornamental groups include
the frieze groups --- groups arising as the symmetry groups for repeti-
tive linear design --- and the wallpaper groups --- groups arising as the
symmetry groups for repetitive planar designs. As the word ornamental sug-
gests, the patterns described by these groups are of central importance in
artistic decoratoin. This should not, however, overshadow the importance of
these groups in science nor the importance of the theoretical mathematical
structure their analysis illustrates.