# Maths in Art 7: Wallpaper Patterns

William Morris is possibly the only person famed for textile design. The author, artist and textile designer created patterns that still influence home décor.

Since it would be extremely time consuming to create wallpaper to cover the whole wall one segment is created and then copied to create a pattern. These copies can be reflections, rotations or translations (simply moving the original).

The image on the left shows one block. The rest of the wallpaper can be created by reflecting on one of the longer sides and then copying, translating, above and to the sides.

In mathematics groups study the symmetries of objects. A wallpaper group is the group of all patterns made using the same rules. The group for Morris’ pattern above and left is called ‘pm’.

Another group ‘pmm’ involves a reflection on both edges. The colourful design is used by Weave.

Sections don’t just have to be square though. ‘p4m’ is like ‘pmm’ but there is an added reflection along the diagonal. The two other patterns in the gallery below are examples of a p4m wallpaper pattern as is a chess board.

Elizabeth Olwen creates a number of wallpapers. I think these are more interesting as they contain some rotations.

In 1891 mathematician Evgraf Fedorov managed to prove that there are exactly 17 wallpaper groups. Any repeated pattern on a flat surface must be in one of these groups. (Note this does not hold for higher dimensions or for different geometries, i.e. the patterns on a football.)

Many of Escher‘s tessellations are wallpaper patterns.

The gallery below shows some more interesting wallpapers. You can follow me here on WordPress or on Facebook. Checkout the rest of my posts using the categories on the left.