[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 lattices, fundamental domains and associated symmetries, and how we begin to classify them by identifying points of rotation. This entry completes the classification process for patterns which have no rotational symmentry. We will call these patterns of order 1.

There are four distinct patterns in this category.

Classifying an Order-1 Pattern

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

1. Does the pattern have any reflections?

If it does, then skip to question 3, otherwise we ask question 2:

2. Does the pattern have glide reflections?

If it doesn't, then the pattern is p1. If it does have glide reflections, then the pattern is pg

3. Given the pattern has reflections, are these rectangular or rhombic?

If rectangular, then the pattern is pm. If rhombic, then the pattern is cm.

All four patterns are described in detail below.

Pattern p1

[Embed picture right: p1 pattern. External link]

Pattern p1, an example of which is shown on the right, has the following characteristics:

• No points of rotation.

• A parallelogram lattice structure (shown in white)

• A generating region (shown in red), which in this case is the whole of the lattice cell.

• No reflections or glide reflections.

This is the simplest of all the wallpaper pattern types, with no symmetry whatsoever.

Pattern pg

[Embed picture right: pg pattern. External link]

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

• No points of rotation.

• A rectangular 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. This is glide-reflected to fill the remaining half of the rectangle.

• No reflections, but there are two glide reflections, both in the same direction.

The definitive features of this pattern are the rectangular lattice and the glide reflections in one direction.

Pattern pm

[Embed picture right: pm pattern. External link]

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

• No points of rotation.

• A rectangular lattice structure (shown in white)

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

• There are two reflections, both in the same direction, one through through the lines of the lattice cell and the other through its midpoint. There are no glide reflections.

This pattern is rectangular, having two parallel reflections. It often appears as two stripes of pattern.

Pattern cm

[Embed picture right: cm pattern. External link]

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

• No points of rotation.

• A rhombic lattice structure (shown in white)

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

• There is a reflection and a glide reflection, both in the same direction.

This pattern is distinctly rhombic, with parallel reflections.

Real-World Examples

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

Pattern p1 can be seen in wallpapers with large floral patterns, such as the Anaglypta High Leaf

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