# Frieze Patterns

Frieze of timber-frame house in Einbeck, Germany

Relevant examples from Escher's work:

## Frieze Patterns

An infinite strip with a repeating pattern is called a frieze pattern, or sometimes a border pattern or an infinite strip pattern. The term "frieze" is from architecture, where a frieze refers to a decorative carving or pattern that runs horizontally just below a roofline or ceiling. Some examples of frieze patterns:

Since the patterns repeat, we show only a finite portion, but you should keep in mind that these pictures should extend infinitely far in both directions. Although most of the frieze patterns in this book will be horizontal, there's no reason a frieze pattern cannot be vertical, or even set at an angle.

## Explorations

Begin learning about frieze patterns with:

## Translation Symmetry

All frieze patterns have translation symmetry. A horizontal frieze pattern looks the same when slid to the left or right, a vertical frieze pattern looks the same when slid up or down, and in general any frieze pattern looks the same when slid along the line it is layed out upon. Again it is important that the pattern extend infinitely far in both directions, so that there are no "ends" that appear different.

The translation length is the distance between repeats of the pattern. A simple way to describe translation symmetry is with an arrow, which gives both the length and direction of the symmetry.

The position of the arrow is not important:

Longer arrows may also give translation symmetries:

However, the short arrow describes a translation which generates all the longer translations. These long translations are simply the short translation repeated multiple times. We will only label the shortest translation in a given direction.

Clearly if a strip has translation symmetry to the left, then it has it to the right. That is, the reverse arrow also describes a translation symmetry of the figure:

Again, the reverse arrow provides no extra information, so we will not label it. In summary, displaying one minimum length arrow indicates an entire family of translation symmetries, all of which point in one direction or its reverse. We say that this collection of symmetries is a direction of translation symmetry.

We can now give a more precise description of a frieze pattern:

A frieze pattern is a figure with one direction of translation symmetry.

Frieze patterns can have other symmetries as well. For example, the figure below has infinitely many reflection symmetries as well as a horizontal translation symmetry, both marked in red:

Practice looking for symmetry in frieze patterns with the Frieze Marking Exploration before continuing with the next section.

## Glide Reflection Symmetry

The pattern below displays a new kind of symmetry:

It has the translation symmetry common to all frieze patterns. However, the symmetry between the left footprints and the right footprints does not come from translation, nor is it due to a reflection since the prints are not next to each other. Instead, a combination of a translation and a reflection is needed. This combination is called glide reflection symmetry.

In figures, we label glide reflection symmetry with a dashed arrow. The length of the arrow shows the translation length and direction, and the body of the arrow lies on the axis of reflection. In the figure below, the translation symmetry is also shown by a solid arrow:

As with translations, there can be longer glide reflections and glide reflections that point in the opposite direction. We only show the shortest possible glide reflection in one direction.

Some figures have both reflection and translation symmetry along the same line. This is not glide reflection symmetry, because both the glide and the reflection are already symmetries of the figure:

Glide reflection symmetry can be hard to spot. Keep an eye out for zig-zag or wavy lines, as these often have glide reflection symmetry down their centers.

## Classification of Frieze Symmetry Groups

Frieze groups are symmetry groups of frieze patterns. They have one direction of translation symmetry. In this section, we assume the strip is turned so its translation is horizontal.

Try the Frieze Group Exploration before continuing.

The first observation is that frieze patterns can contain only a very limited selection of symmetries besides the single direction of translation symmetry. The only possibilities are 180° rotations, reflections with vertical axes, a reflection with horizontal axis, and a glide reflection with horizontal axis. By a process of elimination, these can only occur in seven possible combinations. Thus there are seven possible frieze groups.

In this book, we use the names of symmetry groups that were standardized by the International Union of Crystallography, known as the IUC notation. These names all begin with "p" followed by three characters. The first is "m" if there is a vertical reflection, "1" if not. The second is "m" if there is a horizontal reflection, "a" if there is a glide reflection, and otherwise "1". The third is "2" if there is a 180° rotation, "1" if not.

### The Seven Frieze Groups

 The seven frieze groups: File:Frieze-printable.pdf.

For each group, the IUC name, a description, and a marked sample pattern are shown. The symmetries of each pattern are marked in red. The translation symmetry is shown with a solid arrow, mirror lines are dashed, glide reflection is shown with a dashed arrow, and red diamonds indicate order 2 rotation centers.

 p111 Has only the translation symmetry. p1m1 Has a horizontal mirror symmetry. pm11 Has vertical mirror symmetries spaced half the translation length. pmm2 Has vertical mirror symmetry, horizontal mirror symmetry, and 180° rotations where the mirrors intersect. p112 Has only 180° rotations, spaced at half the translation distance. p1a1 Has a glide reflection, half the length of the translation. pma2 Has a glide reflection, with alternating vertical mirrors and 180° rotations.

There are many naming systems for Frieze groups besides the IUC notation. Learn some with the Frieze Names Exploration.

### Examples

The pattern below is a ceramic design from the San Ildefonso Pueblo.[1].

 This pattern does not have symmetry group D2, it is not a rosette pattern. This pattern has infinitely many symmetries, including a translation symmetry.

The pattern the horizontal and vertical reflection axes as marked, and order 2 rotation centers wherever these axes cross. It has symmetry group pmm2.

The strip on the right is a tile pattern from the Basilique Saint-Denis. [2] 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 p1m1.

The picture below shows a stylized pattern from a Navajo rug. What is its symmetry group?

Because all the triangular arrow shapes "point" to the left, this pattern has no vertical reflection symmetry, as it would reverse the arrow shapes to point to the right. The same reasoning shows the pattern has no rotational symmetry. The top triangles and bottom triangles are not lined up, so there is no horizonal reflection symmetry, but there is a glide reflection. The pattern has symmetry group p1a1.

The picture below shows a portion of pavement from the Greek island of Rhodes. It is known as a "Greek Key" or "Meander" pattern. What is its symmetry group?

 'S' detail with rotation center Mirror image of 'S'

First, notice the figure has no reflection symmetry. To be sure of this, we look for a detail in the figure whose mirror image does not appear. For example, part of the red meander line looks like the letter 'S'. Since the mirror image of the 'S' does not appear anywhere in the pattern, the pattern has no reflection symmetry, not even glide reflection symmetry.

The order 2 rotation center at the center of the 'S' detail is also a rotation symmetry of the entire strip pattern. Knowing that the rotation centers must be spaced at half the translation length helps to spot the other order 2 rotation symmetries as shown in the marked image below:

The symmetry group of this Greek key pattern is p112.

## Notes

1. Washburn and Crowe, Symmetries of Culture, Figure 4.4
2. Eugène Viollet-le-Duc, Dictionnaire raisonné de l’architecture française du XIe au XVIe siècle, 1856