[Work in progress]

This is one of a series of entries on the seventeen wallpaper pattern groups. The main entry describes what we mean by a wallpaper pattern, their symmetries and associated lattices, and how we begin to classify them by identifying points of rotation. This entry completes the classification process for patterns which have a highest order of rotation of 2.

There are five distinct patterns in this category, and they often display features in parallel lines.

Classifying an Order-2 Pattern

There are up to four questions we need to ask to distinguish between the patterns in this category:

1. Does the pattern have any reflections?

If not, then the pattern is one known as p2. If so, then we ask question 2:

2. Does the pattern have glide reflections?

If so, then the pattern is pgg. If not, then we ask question 3:

3. Are the reflections in only one direction?

If so, then the pattern is pmg. If not, then we ask question 4:

4. Are the reflections rectangular or rhombic?

If rectangular, then the pattern is pmm. If rhombic, then the pattern is cmm.

All five patterns are described in detail below.

Pattern p2

[Embed picture right: p2 pattern. External link]

Pattern p2, an example of which is shown on the right, has the following characteristics:

• Four distinct points of rotation, all of order 2 (as marked)

• A parallelo-hexagonal lattice structure (shown in white)

• A generating region (shown in red), which in this case is an area occupying one half of the lattice cell, formed by joining two opposite points. This is rotated by 180 degrees to fill the parallelo-hexagonal lattice cell.

• No reflections or glide reflections.

There is no symmetry in this pattern other than a 180 degree rotation.

Pattern pgg

[Embed picture right: pgg pattern. External link]

Pattern pgg, an example of which is shown on the right, has the following characteristics:

• Two distinct points of rotation, both of order 2 (as marked)

• A rectangular lattice structure (shown in white)

• A generating region (shown in red), which in this case is an area occupying one quadrant of the lattice cell. This is glide-reflected to fill the remaining quadrants of the rectangle.

• No reflections, but there are glide reflections in two directions.

The definitive features of this pattern are the perpendicular glide reflections.

Pattern pmg

[Embed picture right: pmg pattern. External link]

Pattern pmg, an example of which is shown on the right, has the following characteristics:

• Two distinct points of rotation, both of order 2 (as marked)

• A rectangular lattice structure (shown in white)

• A generating region (shown in red), which in this case is an area occupying one quadrant of the lattice cell. This is rotated to fill the remaining quadrants of the rectangle.

• There is a single reflection through a line joining the midpoints of opposite sides of the lattice in one direction, but a glide reflection through the line joining the midpoints in the other direction.

The definitive feature of this pattern is the reflection in one direction coupled with a perpendicular glide reflection.

Pattern pmm

[Embed picture right: pmm pattern. External link]

Pattern pmm, an example of which is shown on the right, has the following characteristics:

• Four distinct points of rotation, all of order 2 (as marked)

• A rectangular lattice structure (shown in white)

• A generating region (shown in red), which in this case is an area occupying one quadrant of the lattice cell. This is reflected to fill the remaining quadrants of the rectangle.

• There are reflections in both directions through the lines of the lattice cell and the lines joining their midpoints.

This pattern is rectangular and highly symmetric, having perpendicular reflections.

Pattern cmm

[Embed picture right: cmm pattern. External link]

Pattern cmm, an example of which is shown on the right, has the following characteristics:

• Three distinct points of rotation, all of order 2 (as marked)

• A rhombic lattice structure (shown in white)

• A generating region (shown in red), which in this case is the right-angled triangle forming a quadrant of the lattice cell. This is reflected to fill the remaining quadrants of the rhombus.

• There are reflections in both directions through the lines joining opposite points of the lattice cell and glide reflections through the lines joining the midpoints of adjacent sides.

This pattern is distinctly rhombic, with perpendicular reflections.

Real-World Examples

Many of the order-2 patterns occur regularly in the design of modern fabrics and printed wallpapers.

Pattern cmm has good examples in London's Victoria & Albert Museum, including the Art Deco print of Minnie McLeish and P Shuttleworth's Toccata wallpaper.

These patterns also feature in many of the common brickwork patterns, including running or strtecher bond (cmm), herringbone (pgg) and raking stretcher bond (p2).