## Group Theory of Wallpaper Patterns

Consider a plane. Move it and put it onto exactly the same plane. (We admit turning it over, but don't admit folding it.) We call such movement a transformation and the set of all such transformations of the plane is called a Euclidean group, which is denoted by E.

Note that NOT moving the plane is a kind of movement and is a special transformation. We therefore give it its own name. We call it an identity transformation.

Theorem

Any element of the Euclidean group E is one of the following:

1. the identity transformation
2. a rotation about a point
3. a translation
4. a reflection on an axis line on the plane
5. a glide reflection on an axis line on the plane

Consider a line on the plane and fix it as the axis. The transformation such that turning over the plane by rotating it through 180 degrees about the axis is called a reflection on this line. We regard the line as a mirror, and it associates any point on the plane with the mirror image of that point. Intuitively, a reflection "flips" the plane across the axis.

The transformation equivalent to performing a reflection on a line and a translation parallel to the same line after the reflection is called a glide reflection on this line.

Now, assume that we have a repeating wallpaper pattern drawn on a plane. Consider the set of all elements in the Euclidean group, E, such that the pattern on the result of the transformation is exactly the same pattern as the original. We denote this subset of E by G. We call G the wallpaper group of this pattern.

(In Japanese this is called "monyo gun." The word "gun" means "group" in English, while "monyo" means "patterns on traditional clothes." In Japan, we have a rich collection of traditional patterns that are used to decorate our clothes, i.e. "ukata patterns." One hundred or more years ago, the Japanese were dressed in much more elaborately designed, patterned clothes. However, we have very few kinds of patterned wallpapers.)

Consider the case where G satisfies the following two conditions:

• (1) G contains translations in two different directions.
• (2) The minimal value of the moving distance of translation belonging to G is not zero. We have a positive number, b, where the moving distance of any translation in G is necessarily greater than b.

Condition (1) means that the pattern is repeated in at least two directions. For the simplest stripes, G has a translation with an arbitrarily short moving distance in the direction of the stripes. However, most repeating patterns satisfy both (1) and (2). From a mathematical point of view, cases satisfying both (1) and (2) are the most interesting. Cases where either (1) or (2) are not satisfied are considered separately, therefore, we assume conditions (1) and (2) are both satisfied below.

Under this assumption, group G coincides with one of seventeen kinds seen on the previous page.

Let us explain how to distinguish the seventeen kinds:

 (A) Without rotations. p1 Without reflections and glide reflections. pm With a reflection. Any axis of a glide reflection is necessarily also an axis of a reflection. pg Without reflections. With a glide reflection. cm With a reflection. There exists an axis of a glide reflection that is not an axis of any reflection. (B) With 180 degrees of rotation. Without rotations of 90 degrees or 60 degrees. p2 Without reflections and glide reflections. pmm With a reflection. Any axis of a glide reflection is necessarily also an axis of a reflection. pgg Without reflections. With a glide reflection. cmm With a reflection. There exists an axis of a glide reflection that is not an axis of any reflection and has a parallel axis of a reflection to it. pmg With a reflection. There is an axis of a glide reflection without parallel axis of a reflection to it. (C) With 90 degrees of rotation. p4 Without reflections and glide reflections. p4m There is a center of 90 degrees rotation lying on an axis of a reflection. p4g With a reflection. There is a center of 90 degrees rotation not lying on any axis of a reflection. (D) With 120 degrees of rotation. Without 60 degrees of rotation. p3 Without reflections. p31m With a reflection. There is a center of 120 degrees of rotation not lying on any axis of a reflection. p3m1 With a reflection. Any center of 120 degrees of rotation lies on an axis of a reflection. (E) With 60 degrees of rotation. p6 Without reflections. p6m With a reflection.

The above table may be difficult to understand, but by using it we can decide the unique type of seventeen symmetry types for any given pattern satisfying the conditions (1) and (2). Please check that the example on the previous page satisfies conditions in the above table.

Another guide might help to decide the type of symmetry. It's helpful to think in terms of a lattice. First, pick an origin point, O, on the plane. (You can pick any point.) The subset of all points on the plane obtained by moving O with translations belonging to G is called the lattice of G. (Note that we are considering only translations in G.) Actual lattices such as window bars or jail bars have many horizontal or vertical bars equidistant from each other. These intersecting points are similar to the mathematical concept called a lattice. Likewise, if we make a pattern by tiling with one kind of tile with all of them oriented in the same direction and putting a mark on every tile at the same point, then the set of all the marks on the completed pattern is the lattice. Picking a characteristic point in the pattern as the origin O, we can easily obtain the lattice.

A general lattice is all intersecting points in the checkered pattern of parallelograms. In some cases it has a characteristic shape. We refer to them by the names below.

 general lattice rectangular lattice square lattice hexagonal lattice centered lattice

For a centered lattice, as in the figure above, we always choose an axis of reflection and consider it the standard direction together with the lattice. We agree that a square is a kind of a rectangle, and a rectangle is a kind of a parallelogram. We have the following theorem:

Theorem

1. In the three cases in the table above (C), the lattice is square.
2. In the five cases in (D) and (E), the lattice is hexagonal.
3. In the case cm of (A) and cmm of (B), the lattice is centered.
4. In the case pm and pg of (A) and pmm, pgg, and pmg of (B), the lattice is rectangular.

By using the concept of lattices and by the above theorem you can easily choose several candidates of seventeen symmetry types. Thereafter, you can decide the unique type by close consideration.

We have explained you the essentials of the mathematical theory of wallpaper patterns. You can improve your understanding further by checking the theory for many concrete examples aroud you. We have prepared an examination for you here.

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