Wallpaper Patterns

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

Suggested reading:

• 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.

Wallpaper Patterns

This section is unfinished.

Start learning about Wallpaper Patterns with the Wallpaper Exploration.

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.

Practice finding the lattice with Lattice Exploration.

The Seventeen Wallpaper Groups

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.
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.

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?

Escher's Use of Symmetry

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 :

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.

Exercises

Wallpaper Exercises

Related Sites

Theory

• Proof of the 17 wallpaper group classification by George Baloglou.
• The Discontinuous Groups of Rotation and Translation in the Plane by Xah Lee.

Examples

• Tilings by Dror Bar-Natan.
• The 17 wallpaper groups (animated) by Hop.
• The 17 wallpaper patterns in traditional Japanese art at Mathematics Museum (Japan).
• The 17 wallpaper groups shown with symmetries marked by Xah Lee.
• The 17 wallpaper groups by David Joyce.