# Difference between revisions of "Wallpaper Patterns"

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The vast majority of these fall into one of seven symmetry groups: | The vast majority of these fall into one of seven symmetry groups: | ||

{{group|p1}}, {{group|p2}}, {{group|p3}}, {{group|p4}}, {{group|p6}}, {{group|pg}}, and {{group|pgg}}. | {{group|p1}}, {{group|p2}}, {{group|p3}}, {{group|p4}}, {{group|p6}}, {{group|pg}}, and {{group|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 symmetyr, although he did create a few drawings | ||

+ | where bilateral symmetry of the motif leads to overall mirror symmetry of the pattern. | ||

==Exercises== | ==Exercises== |

## Revision as of 02:35, 5 February 2007

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

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.

The **lattice of translations**.

## The Seventeen Wallpaper Groups

File:Test1.svg File:Test2.svg File:Test3.svg

## 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 symmetyr, although he did create a few drawings
where bilateral symmetry of the motif leads to overall mirror symmetry of the pattern.

## 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.