70 // Wallpaper Patterns

Wallpaper Patterns A wallpaper pattern repeats the same image in two directions: down the wall and across the wall (or on a slant). The repetition down the wall comes from the paper being printed in a continuous roll, using a revolving cylinder to create the pattern. The repetition across the wall makes it possible to continue the pattern sideways, across adjacent strips of paper, to cover the entire wall. A `drop' from one panel to the next causes no problems, and can actually make it easier to hang the paper.

Wallpaper patterns repeat

in two directions.

The number of possible designs for wallpaper is effectively infinite. But different designs can have the same underlying pattern, it's just that the basic image that gets repeated is different. For instance, the flower in the design above could be replaced by a butterfly, or a bird, or an abstract shape. So mathematicians distinguish essentially different patterns by their symmetries. What are the different ways in which we can slide the basic image, or rotate it, or even flip it over (like reflecting it in a mirror), so that the end result is the same as the start?

For my pattern of flowers, the only symmetries are slides along the two directions in which the basic image repeats, or several such slides performed in turn. This is the simplest type of symmetry, but there are more elaborate ones involving rotations

´ and reflections as well. In 1924 George Polya and Paul Niggli proved that there are exactly 17 different symmetry types of wallpaper pattern ­ surprisingly few.

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