[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 3.

There are three distinct patterns in this category, and they often display triangular or three-legged motifs.

Classifying an Order-3 Pattern

There are two 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 p3. If so, then we ask question 2:

2. Do all the centres of rotation lie of axes of reflection?

If so, then the pattern is p3m1. If not, then the pattern is p31m.

All three patterns are described in detail below.

Pattern p3

[Embed picture right: p3 pattern. External link]

Pattern p3, an example of which is shown on the right, has the following characteristics:

• Three distinct points of rotation, all of order 3 (as marked)

• A hexagonal lattice structure (shown in white)

• A generating region (shown in red), which in this case is a rhombus occupying one third of the lattice cell. This is rotated three-fold to produce the hexagonal lattice cell.

• No reflections or glide reflections.

You will often see this pattern as three different three-legged rotational motifs, each of which has the same inherent direction of rotation. This is the reason we see no reflections: this would only reverse the apparent direction of rotation for each of those motifs, and so it cannot be a symmetry.

Pattern p31m

[Embed picture right: p31m pattern. External link]

Pattern p31m, an example of which is shown on the right, has the following characteristics:

• Two distinct points of rotation, both of order 3

• A hexagonal lattice structure

• A generating region which is a kite-shaped segment, one sixth of the the lattice cell. This is reflected to fill one third of the lattice cell, then rotated three-fold to fill the entire hexagon.

• There are reflections through the lines which join the midpoints of the opposite sides of the hexagonal lattice.

• There are also glide reflections through lines which join alternate points of the lattice cell.

The two three-legged motifs don't have an inherent 'direction of rotation' as you might see in p3.

Pattern p3m1

[Embed picture right: p3m1 pattern. External link]

Pattern p3m1, an example of which is shown on the right, has the following characteristics:

• Three distinct points of rotation, all of which are of order 3

• A hexagonal lattice structure

• A generating region which is an equilateral triangle, one of six segments of the hexagonal lattice cell formed by joining the centre to two adjacent points. This is reflected to fill an adjacent triangular segment, then rotated three-fold to fill the hexagonal cell.

• There are reflections through each side of the hexagonal lattice cell, and also through the lines which join its opposite points

• There are also glide reflections through lines which join the midpoints of alternate sides of the lattice cell.

None of the three distinct triangular motifs have an inherent direction of rotation, and the centres of all three lie on the lines of reflection.

Real-World Examples

All three order-3 patterns occur widely in the highly geometric designs of Islamic art.

Pattern p3 has a good example at the Sultan's Palace, Tangier, Morocco.

Pattern p31m features in the design of the al-Maridani Mosque in Cairo.

Pattern p3m1 can be seen at the Madrassa of az-Zahiriye in Damascus.

In addition, Dutch graphic artist MC Escher made good use of the order-3 patterns in his work. Here are links to examples using patterns p31m and p3m1.