Wallpaper group

From Academic Kids

Missing image
Wallpaper_group-p6m-1.jpg
Example of a Persian design with wallpaper group "p6m"

A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. Such patterns occur frequently in architecture and decorative art. The mathematical study of such patterns reveals that exactly seventeen different types of patterns can occur.

Wallpaper groups are related to the simpler frieze groups, and to the more complex three-dimensional crystallographic groups (also called space groups).

Contents

Informal introduction

Wallpaper groups are used to answer the questions:

  • When are two patterns the same?
  • When are two patterns different?

The difficulty with these questions is in the precise definition of "same" and "different". Consider the following examples.

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Wallpaper_group-p4m-2.jpg
Example A: Cloth, Otaheite (Tahiti)
Missing image
Wallpaper_group-p4m-1.jpg
Example B: Ornamental painting, Nineveh, Assyria
Missing image
Wallpaper_group-p4g-2.jpg
Example C: Painted porcelain, China


The wallpaper group associated to such a design is a mathematical object that captures information about the 'symmetries' of the pattern. It turns out that examples A and B have the same wallpaper group; it is called "p4m". Example C has a different wallpaper group, called "p4g". The fact that A and B have the same wallpaper group means that it is impossible to tell them apart on the basis of symmetry alone, whereas C can be so distinguished.

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

Symmetries of patterns

A 'symmetry' of a pattern is, loosely speaking, a way of transforming the pattern so that the resulting pattern looks exactly the same as the one we started with. (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. (To be precise, we need to imagine that the pattern continues indefinitely in all directions. Similarly, we should pretend that the artist executed the design with perfect accuracy, so that small imperfections do not interfere with our concept of exactly the same.) This type of symmetry called a translation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.
  • If we rotate example B clockwise by 90 degrees, around the centre of one of the squares, again we obtain exactly the same pattern. This is called, unsurprisingly, a rotation. Examples A and C also have 90 degree 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 A and B have a different wallpaper group to C.

The wallpaper group does not capture everything about a pattern. For instance, A is arranged 'diagonally', whereas B is arranged 'horizontally and vertically'. From the point of view of wallpaper groups, this is irrevelant, because we have no way of knowing what the correct orientation of A is; perhaps the camera photographing A was tilted by 45 degrees. Similary, we have no way of telling what the correct scale is, so the wallpaper group does not take into account the fact that the repeating 'cells' in A are smaller than those in B.

Formal definition and discussion

A plane crystallographic group is a topologically discrete group of isometries of the Euclidean plane which contains two linearly independent translations.

Isometries of the Euclidean plane

Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).

  • Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane in the direction of v.
  • Rotations, denoted by Rc,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation.
  • Reflections, denoted by Fc,v, where c is a point in the plane and v is a unit vector in R2. (F is for "flip".) This has the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c.
  • Glide reflections, denoted by Gc,v,w, where c is a point in the plane, v is a unit vector in R2, and w is a vector perpendicular to v. This is a combination of a reflection in the line described by c and v, followed by a translation along w.

The independent translations condition

The conditon on linearly independent translations means that there exist linearly independent vectors v and w (in R2) such that the group contains both Tv and Tw.

The purpose of this condition is to distinguish wallpaper groups from frieze groups, which have only a single linearly independent translation, and from two-dimensional crystallographic point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups which only repeat along a single axis.

(It is possible to generalise this situation. We could for example study discrete groups of isometries of Rn with m linearly independent translations, where m is any integer in the range 0 ≤ mn.)

The discreteness condition

The discreteness condition means that there is some positive real number ε, such that for every translation Tv in the group, the vector v has length at least ε (except of course in the case that v is the zero vector).

The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, we might have for example a group containing the translation Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern.

One important and nontrivial consequence of the discreteness condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180 degrees, 120 degrees, 90 degrees, or 60 degrees. This fact is known as the crystallographic restriction theorem (http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/A2.html), and can be generalised quite easily to higher-dimensional cases.

Relationship between the formal definition and the informal concept

Notations for wallpaper groups

Standard "p6m"-style notation

Orbifold notation

Perhaps look at http://www.xahlee.org/Wallpaper_dir/c5_17WallpaperGroups.html.

See also orbifold.

How to prove that there are exactly seventeen groups

Guide to recognising wallpaper groups

To work out which wallpaper group corresponds to a given design, one may use the following table.

Least
rotation
Has reflection?
Yes No
360 / 6 p6m p6
360 / 4
Has mirrors at 45?
Yes: p4m No: p4g
p4
360 / 3
Has rotocenter off mirrors?
Yes: p31m No: p3m1
p3
360 / 2
Has perpendicular reflections?
Yes No
Has rotocenter off mirrors?     pmg    
Yes: cmm No: pmm
Has glide reflection?
Yes: pgg No: p2
none
Has glide axis off mirrors?
Yes: cm No: pm
Has glide reflection?
Yes: pg No: p1


The complete list of wallpaper groups

Each group in the following list has a cell structure diagram, which is interpreted as follows:

  • A pink diamond indicates a centre of rotation of order two (180 degrees).
  • A red triangle indicates a centre of rotation of order three (120 degrees).
  • A green square indicates a centre of rotation of order four (90 degrees).
  • A blue hexagon indicates a centre of rotation of order six (60 degrees).
  • A thick blue line indicates an axis of reflection.
  • A dotted line indicates an axis of a glide reflection.
  • The brown area indicates a fundamental domain, i.e. the smallest part of the pattern which is repeated.

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Software

There exist several software graphic tools that will let you create 2D patterns using wallpaper symmetry groups. Usually, you can edit the original tile and its copies in the entire pattern are updated automatically.

  • Inkscape, a free vector editor, supports all 17 groups plus arbitrary scales, shifts, and rotates per row or per column, optionally randomized to a given degree.
  • Arabeske (http://www.wozzeck.net/arabeske/) is a free standalone tool, supports a subset of wallpaper groups.

See also

External links

References

  • The Grammar of Ornament (1856), by Owen Jones. Many of the images in this article are from this book; it contains many more.
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