# Wallpaper Patterns

Wallpaper catalog of Remondini - Bassano - Italy, 18th century

• Visions of Symmetry, pg. 31-44, 77-78.

Relevant examples from Escher's work:

• Escher's sketches of Polya's 17 wallpaper patterns. Visions of Symmetry, pg. 24-26.

Begin learning about Wallpaper Patterns with the Wallpaper Exploration.

## Wallpaper Patterns

A wallpaper pattern is a plane figure which has more than one direction of translation symmetry. Most actual wallpaper fits the bill, because it must repeat left to right in order to hang in strips, and it must repeat vertically so that multiple strips can be cut from the same roll.

Having one line of translation symmetry forced frieze patterns to be infinitely long strips. For wallpaper patterns, the multiple translation directions force the pattern to cover the entire infinite plane. A finite portion of a wallpaper pattern is enough to establish the translation symmetry which is used to extend to the entire plane. Generally, when drawing wallpaper patterns, show enough of the pattern so that the translation symmetries are obvious. Practically, it takes at least 9 repetitions of the pattern (in a 3x3 array) to clearly display the translation symmetry.

In a wallpaper pattern, the lattice of translations is the collection of all translated images of a point. To mark the lattice of translations, choose a point in the figure and then mark all translations of that point. Be careful not to mark reflected or rotated versions of the point.

The lattice is so named because connecting nearby dots with edges results in a grid, or lattice, structure. These lattice lines depend on which dots one chooses to connect, so they are not usually shown.

The example below is adapted from a pre-Columbian Peruvian fabric pattern. In the left portion, the basic pattern is shown. The center portion has the lattice marked with red dots, and the right portion connects nearby lattice points with lines to show that points lie on a rectangular grid.

## The Seventeen Wallpaper Groups

As with frieze groups, the classification of wallpaper symmetry groups is done by a process of elimination. The first crucial step is known as the crystallographic restriction:

A rotation symmetry of a wallpaper pattern must be a rotation of order 2, 3, 4, or 6.

Because two reflection axes which meet at an angle $\theta$ produce a rotation symmetry whose angle is $2\theta$, the crystallographic restriction also puts a strong restriction on the possible reflection and glide reflection symmetries. For example, if a wallpaper pattern has rotations of order 2 but no higher order rotations, then it can have at most two sets of reflection axes, meeting at right angles.

A painstaking and lengthy argument eventually reduces the possibilities down to 17 wallpaper symmetry groups. This remarkable classification was first published by the Russian mathematician Fedorov in 1891, but not widely recognized. George Pólya's rediscovery of the 17 groups in 1924 [1] brought the subject into the mainstream, and Escher read Pólya's work with great interest, hand copying Pólya's seventeen diagrams into his own notes. The adventurous reader will find a rigorous treatment of the classification in the Related Sites on this page.

The remainder of this section is a list of the 17 wallpaper symmetry groups, with their Crystallographic names, descriptions of their symmetries, and a sample of each.

 p1 This is the simplest symmetry group. It consists only of translations. There are neither reflections, glide-reflections, nor rotations. The two translation axes may be inclined at any angle to each other. pm This group contains reflections. The axes of reflection are parallel to one axis of translation and perpendicular to the other axis of translation. The lattice is rectangular. There are neither rotations nor glide reflections. pg This group contains glide reflections. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation. There are neither rotations nor reflections. cm This group contains reflections and glide reflections with parallel axes. There are no rotations in this group. The translations may be inclined at any angle to each other, but the axes of the reflections bisect the angle formed by the translations, so the lattice is rhombic. p2 This group differs only from p1 in that it contains 180° rotations, that is, rotations of order 2. As in all symmetry groups there are translations, but there neither reflections nor glide reflections. The two translations axes may be inclined at any angle to each other. pgg This group contains no reflections, but it has glide-reflections and 180° rotations. There are perpendicular axes for the glide reflections, and the rotation centers do not lie on the axes. pmg This group contains reflections, and glide reflections which are perpendicular to the reflection axes. It has rotations of order 2 on the glide axes, halfway between the reflection axes. pmm This symmetry group contains perpendicular axes of reflection, with 180° rotations where the axes intersect. cmm This group has perpendicular reflection axes, as does group pmm, but it also has rotations of order 2. The centers of the rotations do not lie on the reflection axes. p3 This is the simplest group that contains a 120°-rotation, that is, a rotation of order 3. It has no reflections or glide reflections. p31m This group contains reflections (whose axes are inclined at 60° to one another) and rotations of order 3. Some of the centers of rotation lie on the reflection axes, and some do not. There are some glide-reflections. p3m1 This group is similar to the last in that it contains reflections and order-3 rotations. The axes of the reflections are again inclined at 60° to one another, but for this group all of the centers of rotation do lie on the reflection axes. There are some glide-reflections. p4 This group has a 90° rotation, that is, a rotation of order 4. It also has rotations of order 2. The centers of the order-2 rotations are midway between the centers of the order-4 rotations. There are no reflections. p4g Like p4, this group contains reflections and rotations of orders 2 and 4. There are two perpendicular reflections passing through each order 2 rotation. However, the order 4 rotation centers do not lie on any reflection axis. There are four directions of glide reflection. p4m This group also has both order 2 and order 4 rotations. This group has four axes of reflection. The axes of reflection are inclined to each other by 45° so that four axes of reflection pass through each order 4 rotation center. Every rotation center lies on some reflection axes. There are also two glide reflections passing through each order 2 rotation, with axes at 45° to the reflection axes. p4m is very common and is easy to recognize because of its square lattice. p6 This group contains 60° rotations, that is, rotations of order 6. It also contains rotations of orders 2 and 3, but no reflections or glide-reflections. p6m This complex group has rotations of order 2, 3, and 6 as well as reflections. The axes of reflection meet at all the centers of rotation. At the centers of the order 6 rotations, six reflection axes meet and are inclined at 30° to one another. There are some glide-reflections.

## Wallpaper Flow Chart

It can be quite difficult to find and mark all symmetries of a wallpaper pattern. However, it is possible to identify the symmetry group of a wallpaper pattern without finding all of its symmetries by focusing on the most important features. Following this flow chart is a quick way to identify symmetry groups of wallpaper patterns.

To use the flow chart, begin with the rotation order question on the left, and follow arrows towards the right until the symmetry group is determined. Practice with the Wallpaper Symmetry Exploration.

### Examples

This pattern is a common brick layout. What is its symmetry group?

Rows of bricks make horizontal strips that will remain horizontal after rotation, which means that the pattern has at most order 2 rotation. It does have order 2 rotation symmetry, for instance at the center of each brick. In the flow chart, starting at "Largest rotation order?", follow the arrow marked 2.

The pattern does have reflection symmetry, both horizontally and vertically, so in the flow chart the next two questions are answered yes.

The final question will determine if this pattern has symmetry pmm or cmm: Are all rotation centers on mirror lines?

This question may be hard to answer, because you need to find all the rotation centers. Looking at samples of pmm and cmm can help. In pmm, strips are aligned, while in cmm patterns, strips are staggered. That suggests that this brick layout has cmm symmetry group.

To confirm that the pattern has cmm symmetry, we need to find a rotation center not on a mirror line. There should be a piece of this pattern displaying C2 symmetry considered as a finite rosette, and we learned to recognize those by looking for details in the shape of the letters S or Z. Here, two bricks together make a sort of 'S':

The rotation center is marked, and is not on a mirror line.

This brick layout pattern has symmetry group cmm.

## Escher's Use of Symmetry

Escher's visit to the Alhambra. Recent effort to find all 17 wallpaper symmetry groups in the Alhambra.

Escher's Regular Division of the Plane Drawings served as source material for his finished printed works. The vast majority of these fall into one of seven symmetry groups: p1, p2, p3, p4, p6, pg, and pgg. These are exactly the symmetry groups which have no reflection symmetry - only translation, rotation, and glide reflection. If two creatures meet on a line of mirror symmetry, they must have a flat edge, and recognizable figures from life rarely have perfectly straight edges. Because of this, Escher mostly avoided mirror symmetry, although he did create a few drawings where bilateral symmetry of the motif leads to overall mirror symmetry of the pattern.

Some animal motifs, generally larger animals, are usually seen from the front or side, and so look silly when viewed upside down or at an angle. Escher's was careful, at least in his later work, to avoid symmetries containing rotation when working with such animals. On the other hand, Escher writes [2]:

When a rotation does take place, then the only animal motifs which are logically acceptable
are those which show their most characteristic image when seen from above.

For example, insects and lizards occur frequently in Escher's work when rotation symmetry is present.

## The Classification of Plane Symmetry Groups

All discrete symmetry groups of plane figures are completely classified. Figures with no translation symmetry, the rosette patterns, have either cyclic or dihedral symmetry. Figures with one line of translation symmetry have one of the seven frieze groups for their symmetry, and figures with more than one line of translation symmetry fall into one of 17 symmetry classes.

## Notes

1. Pólya,G. 1924. Über die Analogie der Kristallsymmetrie in der Ebene. Zeitschrift für Kristallographie 60, p278-282
2. Visions of Symmetry, pg. 78