# Classifications of wallpaper groups

## Introduction

The purpose of this project will be to demonstrate to you how the wallpaper groups relate to group theory. To illustrate this we will provide a background of definitions and walk through some basic examples. Wallpaper groups have been used throughout history in architecture and art. All 17 groups were used by Egyptian craftsmen, and can also be seen in the Muslim world. It was first proved that there were only 17 possible patterns by Evgraf Fedorov in 1891 and then by George Pólya in 1924 independently of Fedorov. Before we can go into the explanation of wallpaper groups we must first define some terms.

## Definitions

Definition of wallpaper group: (loosely) Simply repeating patterns in the plane $\bb{R}^2$. (P.162, book)

The study of wallpaper groups is the study of the symmetries when a pattern in the shape of a square, parallelogram, rhombus or hexagon is repeated by translations along *two nonparallel vector directions* to **fill the entire plane**.

Fundamental Domain: smallest area of a plane that contains all the information needed such that the entire plane can be filled (generated) using products (relations) of the symmetries of the group.

Lattice – a mathematical equivalent of a wallpaper (shape of the fundamental domain; parallelogram, rhombus, rectangle, square)

(Each lattice has different symmetric properties; hence why there are 17 groups and 4 lattice types.)

Isometry or Rigid motion: Two shapes are isometric if they are congruent;

- Preservation of angles and lengths

- Translation – a shifting of the analog

- Rotation - a turn in the space around a fixed point

- Reflection – an inverted image across a line.

- Glide reflection – a combination of reflection and translation.

(Note that all plane isometries are equal to one of these.)

The classifications of wallpaper groups are based on the different symmetries of the pattern.

Patterns that differ in color, orientation, style, and scale maybe belong same classification (group).

## Constructing a plane symmetry (wallpaper)

To construct a wallpaper you must start with a base unit known as the fundamental domain. From this given information and using the different isometries you can construct the infinite wallpaper. Understanding of this is best displayed by illustration seen in $figure 1.1$. $figure 1.1$ is similar to the wallpaper denoted $p1$. $p1$ is a parallelogram lattice type. The only isometries that $p1$ has are translations. There are two types of translations used in generating the wallpaper of $p1$, vertical and horizontal. Now the colors are only different to illustrate the different types of isometries.

$figure 1.1$

## Key

The blank rhombus is the fundamental domain

A Blue Rhombus indicates a horizontal translation

A Red rhombus indicates a vertical translation

A purple rhombus indicates a combination of the two

## Big yellow rhombus

(This is just the fundamental domain of $P_1$ but remained in our write up as a humorous element due to the fact that it was so big and yellow)

Here is a more complicated example that utilizes the lattice type of a hexagon.

The ideas and diagram was obtained from http://www.scienceu.com/geometry/articles/tiling/wallpaper.html

$figure 1.2$

One way of getting at what it means to have a big symmetry group is to think about moving yourself instead of moving the wallpaper. Pick any tile in the infinite tiling, and imagine you are standing on the tile. Look around at the surrounding tiling, and remember what it looks like. Now pick a direction, and walk in that direction. As you walk along stop at each tile you step on and look around, and ask yourself if surrounding tiling looks exactly as it did on the original tile. (3)

## Link to Illustrations

These animated illustrations help to understand the different classifications of wallpaper groups.

http://www.clowder.net/hop/17walppr/17walppr.html

## Complete table

(coming soon, it's about 1/2 done)

$figure 1.3$ is a complete table with each classification, gives a pictorial example and lists its isometries.

$figure 1.3$

## The Big (yellow rhombus sized) Idea

The wallpaper groups (verb) are a group of actions that are applied to a fundamental domain. The wallpaper (noun) is the result of the actions.

Deciphering some on your own

$Example 1$

$Example 2$

$Example 3$ (The hard one)

Sources:

(1) http://en.wikipedia.org/wiki/Wallpaper_group

(2) Abstract algebra -theory and application by Thomas W. Judson

(3)http://www.scienceu.com/geometry/articles/tiling/images/p4g_2.gif&imgrefurl=http://www.scienceu.com/geometry/articles/tiling/wallpaper.html

(4) Contemporary Abstract Algebra third edition by Joseph A. Gallian

(5) A First Course Abstract Algebra by John B. Fraleigh