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# Wallpaper group

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 Title: Wallpaper group Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Wallpaper group

Example of an Egyptian design with wallpaper group p4mm

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.

Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups).

## Contents

• Introduction 1
• Symmetries of patterns 1.1
• History 2
• Formal definition and discussion 3
• Isometries of the Euclidean plane 3.1
• The independent translations condition 3.2
• The discreteness condition 3.3
• Notations for wallpaper groups 3.4
• Crystallographic notation 3.4.1
• Orbifold notation 3.4.2
• Why there are exactly seventeen groups 3.5
• Guide to recognizing wallpaper groups 4
• The seventeen groups 5
• Group p1 (o) 5.1
• Group p2 (2222) 5.2
• Group pm (**) 5.3
• Group pg (××) 5.4
• Group cm (*×) 5.5
• Group p2mm (*2222) 5.6
• Group p2mg (22*) 5.7
• Group p2gg (22×) 5.8
• Group c2mm (2*22) 5.9
• Group p4 (442) 5.10
• Group p4mm (*442) 5.11
• Group p4mg (4*2) 5.12
• Group p3 (333) 5.13
• Group p3m1 (*333) 5.14
• Group p31m (3*3) 5.15
• Group p6 (632) 5.16
• Group p6mm (*632) 5.17
• Lattice types 6
• Symmetry groups 7
• Dependence of wallpaper groups on transformations 8
• Web demo and software 9
• Notes 11
• References 12

## Introduction

Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.

Consider the following examples:

Examples A and B have the same wallpaper group; it is called p4mm in the IUC notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4mg or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

A complete list of all seventeen possible wallpaper groups can be found below.

### Symmetries of patterns

A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry — when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.

Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "bar code" in the centre of mass of each tile.

The types of transformations that are relevant here are called Euclidean plane isometries. For example:

• If we shift example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the pattern we started with. This type of symmetry is called a translation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.
• If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain exactly the same pattern. This is called a rotation. Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C.
• We can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection. Example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A.

However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a diagonal line, we do not get the same pattern back; what we do get is the original pattern shifted across by a certain distance. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C.

## History

A

• The 17 plane symmetry groups by David E. Joyce
• Introduction to wallpaper patterns by Chaim Goodman-Strauss and Heidi Burgiel
• Description by Silvio Levy
• Example tiling for each group, with dynamic demos of properties
• Overview with example tiling for each group
• Escher Web Sketch, a java applet with interactive tools for drawing in all 17 plane symmetry groups
• Burak, a Java applet for drawing symmetry groups.
• A JavaScript app for drawing wallpaper patterns
• Beobachtungen zum geometrischen Motiv der Pelta
• Seventeen Kinds of Wallpaper Patterns the 17 symmetries found in traditional Japanese patterns.