# Wallpaper Patterns

Suggested reading:

- Visions of Symmetry, pg. 31-44, 77-78.

Relevant examples from Escher's work:

- Escher's sketches of Polya's 17 wallpaper patterns. Visions of Symmetry, pg. 24-26.

## Contents

## Wallpaper Patterns

Start learning about Wallpaper Patterns with the Wallpaper Exploration.

A wallpaper pattern is a plane figure which has more than one direction of translation symmetry. Most actual wallpaper fits the bill, because it must repeat left to right in order to hang in strips, and it must repeat vertically so that multiple strips can be cut from the same roll.

In a wallpaper pattern, the **lattice of translations** is the collection of all translated images of a point.
To mark the lattice of translations, choose a point in the figure and then mark all translations of that point. Be careful not
to mark reflected or rotated versions of the point.

The lattice is so named because connecting nearby dots with edges results in a grid, or lattice, structure. These lattice lines depend on which dots one chooses to connect, so they are not usually shown.

Practice finding the lattice with Lattice Exploration.

## The Seventeen Wallpaper Groups

## Wallpaper Flow Chart

It can be quite difficult to find and mark all symmetries of a wallpaper pattern. However, it is possible to identify the symmetry group of a wallpaper pattern without finding all of its symmetries by focusing on the most important features. Following this flow chart is a quick way to identify symmetry groups of wallpaper patterns.

## Escher's Use of Symmetry

Escher's Regular Division of the Plane Drawings served as source material for his finished printed works.
The vast majority of these fall into one of seven symmetry groups:
*p1*, *p2*, *p3*, *p4*, *p6*, *pg*, and *pgg*.
These are exactly the symmetry groups which have no reflection symmetry - only translation, rotation,
and glide reflection. If two creatures meet on a line of mirror symmetry, they must have a flat edge, and
recognizable figures from life rarely have perfectly straight edges. Because of this, Escher mostly avoided
mirror symmetry, although he did create a few drawings
where bilateral symmetry of the motif leads to overall mirror symmetry of the pattern.

Some animal motifs, generally larger animals, are usually seen from the front or side, and so look silly when
viewed upside down or at an angle. Escher's was careful, at least in his later work, to avoid symmetries containing
rotation when working with such animals. On the other hand, Escher writes ^{[1]}:

- When a rotation
*does*take place, then the only animal motifs which are logically acceptable - are those which show their most characteristic image when seen from above.

For example, insects and lizards occur frequently in Escher's work when rotation symmetry is present.

## The Classification of Plane Symmetry Groups

## Exercises

## Related Sites

- Proof of the 17 wallpaper group classification by George Baloglou.
- Tilings by Dror Bar-Natan.
- The 17 wallpaper groups (animated) by Hop.
- The 17 wallpaper patterns in traditional Japanese art at Mathematics Museum (Japan).
- Wallpaper groups by David Joyce.

## Notes

- ↑ Visions of Symmetry, pg. 78