[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 4.

There are three distinct patterns in this category, all of which exhibit a characteristic 'squareness'.

Classifying an Order-4 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 p4. If so, then we ask question 2:

2. Are there reflections in four directions?

If so, then the pattern is p4m. If not, then the pattern is p4g.

All three patterns are described in detail below.

Pattern p4

[Embed picture right: p4 pattern. External link]

Pattern p4, an example of which is shown on the right, has the following characteristics:

• Three distinct points of rotation, two of order 4 and one of order 2 (as marked)

• A square lattice structure (shown in white)

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

• No reflections or glide reflections.

You will often see this pattern as two different four-pointed rotational motifs, each of which is pointing in the same direction of rotation. This is the reason we see no reflections: this would only reverse the apparent direction of rotation, and so it cannot be a symmetry.

Pattern p4m

[Embed picture right: p4m pattern. External link]

Pattern p4m, an example of which is shown on the right, has the following characteristics:

• Three distinct points of rotation, two of order 4 and one of order 2 - the same as with pattern p4

• A square lattice structure

• A generating region which is half of one of the quadrants of the lattice cell , divided by one of its diagonals. This is reflected to fill the quadrant, then rotated four-fold to fill the square lattice cell.

• There are reflections through both the diagonals of the lattice as well as through the lines which join the midpoints of its opposite sides. There are also reflections through the edges of the lattice.

• There are glide reflections through the lines which join the midpoints of adjacent sides of the lattice.

This pattern is very noticeably square, yet its two square motifs don't have an inherent 'direction of rotation' as you might see in p4. The generating region, being repeated 8 times in each lattice cell, produces an overall effect which is highly symmetric.

Pattern p4g

[Embed picture right: p4g pattern. External link]

Pattern p4g, an example of which is shown on the right, has the following characteristics:

• Two distinct points of rotation, one of order 4 and one of order 2

• A square lattice structure

• A generating region which is an isosceles triangle. It's half of one of the quadrants of the lattice cell, divided by the line joining the midpoints of two sides of the lattice. This is reflected to fill the quadrant, then rotated four-fold to fill the square lattice cell.

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

• There are glide reflections through the diagonals of the lattice.

This pattern exhibits different behaviour around its centres of rotation. You may see sub-patterns rotating in either direction around the 4-centres, but these behave differently at the 2-centres, where two of the sub-patterns appear to point towards the centre, while two point away from it.

Real-World Examples

Pattern p4 can be found in a popular brick and paving pattern known as the pinwheel.

Pattern p4m is regularly seen in decorative panels in Islamic art. Here's an example from the Alhambra in Granada, Spain.

Pattern p4g is another which features in brick and tiling patterns, for example this basket weave pattern.