Wallpaper Patterns of Order 6

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This is one of a series of entries on the seventeen wallpaper patterns. The main entry describes what we mean by a wallpaper pattern 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 6.

There are two distinct patterns in this category, and they are two of the most interesting and intricate of all the wallpaper patterns.

Classifying an Order-6 Pattern

As there are only two patterns in this category, there's only one question we need to ask to distinguish between them:

Does the pattern have any reflections?

If so, then the pattern is one which we call p6m. If not, then the pattern is p6. Both patterns are described in detail below.

Pattern p6

[Embed picture right: p6 pattern. External link]

Pattern p6, an example of which is shown on the right, has the following characteristics:

  • Three distinct points of rotation, of orders 6, 3 and 2 (as marked)

  • A hexagonal lattice structure (shown in white)

  • A generating region (shown in red) which in this case is an equilateral triangle. This is rotated six-fold to fill the hexagonal lattice cell. Any 60-degree sector of a regular hexagon will also serve as a generating region for this pattern.

  • No reflections or glide reflections.

You will often recognise this pattern as a series of identical six-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 in these motifs, and so cannot be a symmetry.

Pattern p6m

[Embed picture right: p6m pattern. External link]

Pattern p6m, an example of which is shown on the right, has the following characteristics:

  • Three distinct points of rotation, of orders 6, 3 and 2 - the same as with pattern p6.

  • A hexagonal lattice structure.

  • A generating region which is half of an equilateral triangle. This is reflected then rotated six-fold to fill the hexagonal lattice cell.

  • There are reflections through the lines joining opposite corners of the hexagonal lattice, as well as the lines which join the midpoints of opposite sides of the hexagon.

  • There are glide reflections through the lines which join the midpoints of non-opposing sides of the hexagon.

This pattern is very noticeably hexagonal, yet its hexagonal motifs have no inherent 'direction of rotation' as you might see in p6. One of the most interesting features is the small generating region, which is repeated 12 times in each lattice cell. This produces an overall effect which is often very beautiful and highly symmetric.

Real-World Examples

Pattern p6 can be found in WJ Odell's Festival of Britain wallpaper (1951). This is almost a p6m pattern, but the small white circles give the motifs a direction of rotation.

Pattern p6m features in many of the intricate patterns found in Islamic art. Examples include carved masonry from the Red Fort in Delhi, and the Maharajah's Palace, in Jaipur.

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