A frieze is an infinite horizontal strip with discrete translational symmetry and a frieze group is the symmetry group of some frieze. Although it may seem that there can be infinitely many frieze groups, in fact there are only 7 different types. I will refer to these different types with names that relate to footsteps for each of the frieze groups, as mathematician John Conway did.
Symmetries: glide reflections and translations
A glide reflection is the composite of a translation in one direction and a reflection in the direction vertical to the translational direction.
Symmetries: vertical reflections and translations
Symmetries: translations and 180° rotations
Symmetries: vertical reflections, glide reflections, translations and 180° rotations
Symmetries: translations, horizontal reflections, glide reflections
Symmetries: horizontal and vertical reflections, translations and 180° rotations
Frieze Groups in Real Life
Ceramic Design from the San Ildefonso Pueblo, New Mexico
This pattern has translational symmetry, vertical and horizontal reflection symmetry as well as order 2 rotational symmetry where the axes of reflection cross. It has symmetry group FH.
Tile Pattern from the Basilique Saint-Denis, Paris
The translation symmetry of this pattern is vertical. The only additional symmetry of the pattern is a reflection axis, which is along the direction of translation. This pattern has symmetry group FM.
Stylised Pattern from a Navajo Rug
All the triangular arrow shapes point to the left, so there is no vertical reflection symmetry or rotational symmetry. The top triangles and bottom triangles are not lined up, so there is no horizontal reflection symmetry, but there is a glide reflection so the symmetry group is FLΓ.
Portion of Pavement in Rhodes, Greece
This pattern is known as the Greek Key or Meander pattern. Firstly, there is no reflection symmetry. This is because the red meander line, which looks like the letter ‘S’, has a mirror image that does not appear in the pattern.
Also, the order 2 rotation centre at the centre of the ‘S’ is also rotation symmetry of the entire strip pattern.
So, this pattern has symmetry group FZ.