Ornament

And now … the mathematics:

But where do the given symmetries of the exhibit come from? Now this has to do with group theory again. The distance-preserving self-mappings of Euclidean space form a group: in fact, one can perform two such isometries in succession and obtain an isometry again, and for each such mapping there is a unique inverse mapping, which is an isometry again. There are basically two different types of images: rotations and reflections. The reflections reverse the orientation, i.e. if you imagine a sheet of paper, a reflection corresponds to the process of turning the sheet along the mirror axis. With a rotation, on the other hand, the sheet is only rotated around one point (but you are still looking at its front side afterwards). If you do two reflections in a row, you get a rotation — then you look at the front of the sheet again.

In this exhibit, however, we are not interested in the group of all possible rotations and reflections of the plane (that every isometry is of such a type is the subject of the exhibit “Dust Circles“), but only discrete subgroups of these. We will not define precisely what this is here; however, it can be briefly summarised as follows: Each point is either fixed under a symmetry operation of the group or mapped onto another point that has a certain minimum distance from it. For each such group, there are many symmetrical patterns obtained from that group. Now, in order to illustrate the group well, it therefore makes sense to study such patterns (although this is of course not a $1-1$ correspondence). This is what we want to do in the following.

So what are the repeating patterns in the plane? In the following we describe the four fundamental properties of repeating patterns in the plane and introduce a signature for each such pattern.

Local symmetries

What local symmetries (i.e. symmetries around a point) can a plane figure have? We will describe this below and assign a signature for each species.

The star $\ast$ denotes a mirror or kaleidoscopic symmetry. A single star also means that the given figure has no other symmetries. The next signature we introduce is $\ast2\bullet$ . The associated symmetry is called star-two-point symmetry. This means that the figure has exactly two mirror lines (which are perpendicular to each other) at one point. The star $\ast$ also stands for the mirrors of a kaleidoscope (here two) meeting at one point. The point $\bullet$ means that all symmetries fix a point (as is the case here, but not with a mere $\ast$ -symmetry, since a whole straight line is fixed).

This signature can now be easily extended: Thus $\ast n\bullet$ means that at a point exactly $n$ mirror lines meet. Behind this is the Dihedral group $D_n$ . See also figure 1 below.

Gyrations are another type of local symmetry where there is no mirror symmetry on a straight line, only rotational symmetry. For example, the unpopular swastika has a fourfold rotational symmetry, but no mirror symmetries. This form of symmetry is given the signature $4\bullet$ . An $n$ -fold rotational symmetry is thus abbreviated to $n\bullet$ , see Figure 2. Behind it here is the cyclic group (realised as a rotational group) of order $n$ .

The case where a figure has no symmetry at a point is abbreviated by the signature $\bullet$ . Apart from the symmetries around a point just listed, there are no other discrete symmetry groups (there are only the groups $\mathbf 1$ , $D_n$ and $C_n$ for $n\geq 1$ ). A circle, for example, has infinitely many point symmetries (namely all rotations around this point and all reflections through an axis containing this point). But this is inadmissible, because there are an infinite number of symmetries.

Frieze pattern

Frieze patterns are patterns in which a figure not only has certain symmetries in one point, but in which this point can also be shifted in one direction within the pattern. They appear frequently in ancient architecture (see Figure 3 below).

Repeating patterns in the plane

Frieze patterns have only one direction in which they can be moved by a certain amount. In this exhibit, however, we are interested in patterns that can be continued periodically in two different directions and thus fill the whole plane (see figure 4 below).

Here, as we will see, the local symmetries around a point play a major role.

Kaleidoscopic patterns

Patterns whose symmetries are defined by reflections are called kaleidoscopic. The name is based on the fact that a pattern seen in a kaleidoscope is such a kaleidoscopic pattern (see also the exhibits “Kaleidoscope Mirror” and “Kaleidoscope“).

But how can such patterns be described? The answer is: by how their mirror lines intersect. This can be seen clearly in Figure 5 below. There are three interesting types of points here: In the first type six mirrors meet (local symmetry $\ast6\bullet$ ), in the second type three mirrors meet (local symmetry $\ast3\bullet$ ) and in the third only two mirrors meet (local symmetry $\ast2\bullet$ ). The signature of the entire kaleidoscopic pattern is thus $\ast632$ (here the point is missing, since not all reflections fix a point). The order in which the numbers $2,3,6$ are noted here is irrelevant, because this only reflects at which corner of the excellent triangle you start or whether you are looking at a mirrored triangle.

Gyrations

The pattern shown in figure 6 also has many mirror symmetries. If it had only these, its signature would be $\ast333$ . However, a closer look reveals that there is another symmetry: If you look at the red triangle, you will notice that this triangle — an equilateral triangle — can be rotated by $60^\circ$ at its centre, so that the pattern underneath remains intact. On this point, however, the pattern has no mirror symmetries. The signature of the pattern is set to $3\ast3$ because there is one kind of points with threefold rotational symmetry (local symmetry group $C_3$ ) and one kind of points with threefold mirror symmetry (local symmetry group $D_3$ ).

This is how one proceeds now: For a given pattern, one always counts the number of types of points with non-trivial local symmetries. For example, the following pattern (Figure 7) has the signature $2\ast22$ .

However, it can also occur that there are no mirror symmetries at all. For example, the following pattern (Figure 8) has the signature $444$ because there are four different point varieties with fourfold rotational symmetry, but no mirror symmetries appear at all.

An obvious question now is which patterns or which signatures are possible at all. For example, are there patterns with mirror lines in only one direction. Figure 9 below shows two of these; one with one type of mirror line and another with two different types. Accordingly, the first has the signature $\ast$ , while the second has $\ast\ast$ .

Miracles and Miracle Rings

In patterns like these (i.e. when the mirror lines delimit an infinitely large region) there can then be small miracles that can occur: Namely, connecting lines of oppositely directed spirals that do not cross a mirror line. For each such line, make a $\times$ in the signature. The following illustration shows a pattern with one and two such lines (here you only take so many such lines that each further one can be composed of them). In the first pattern there is still one type of mirror line. Therefore, it is given the signature $\ast\times$ . The second pattern, on the other hand, has no mirror symmetries at all and is therefore only given the signature $\times\times$ .

A miracle is a combination of translation and mirroring of a fundamental area that cannot be explained by a mirroring or rotation of the overall pattern alone. But there is even the possibility of a repetition of the fundamental area, which cannot be explained by a rotation or a reflection, nor by a miracle. Such repetitions always occur in pairs (since the pattern is supposed to extend infinitely in two directions). We call such a phenomenon wonder-ring (from English: wonder-ring) and abbreviate it with $\circ$ . For example, the following sample has the signature $\circ$ .

Summary

We have described four different phenomena so far: mirror lines (kaleidoscopic patterns), gyrations, miracles and wonder rings. The content of the exhibit “Ornament” is now that these phenomena are sufficient to describe any periodic pattern that extends infinitely in two directions. Wonder rings and gyrations maintain orientation, whereas mirror lines and miracles reverse it. We summarise this again in the following table:

	Wonder-Ring	Gyration	Mirror lines	Wonder
Symbol	$\circ\cdots\circ$	$ab\ldots c$	$\ast ab\ldots c\ast de\ldots f\ldots$	$\times\cdots\times$

Table 1: The four fundamental phenomena

With the help of this notation, we can now determine all possible types of periodic patterns in the plane that propagate infinitely in two directions. To do this, we introduce costs for each of the above symbols:

Symbol (orientation receiving)	Cost	Symbol (orientationreversing)	Cost
$\circ$	$2$	$\ast$ oder $\times$	$1$
$2$	$1/2$	$2$	$1/4$
$3$	$2/3$	$3$	$1/3$
$4$	$3/4$	$4$	$3/8$
$n$	$\frac{n-1}{n}$	$n$	$\frac{n-1}{2n}$
$\infty$	$1$	$\infty$	$1/2$

Table 2: Costs of each symbol

There is now the following mathematical theorem, which is the centre of this exhibit: This states that the possible signatures of plane periodic patterns are exactly those with a total cost of exactly $2$ . However, we do not want to prove this theorem at this point, but only apply it: For example, the first pattern shown had the signature $\ast 632$ . This corresponds to the cost $1+\frac{5}{12}+\frac{1}{3}+\frac{1}{4}=2$ as desired. The sample with the signature $3\ast 3$ also has a total cost of $\frac{2}{3}+1+\frac{1}{3}=2$ . The pattern with the two kaleidoscopic symmetries and the one rotational symmetry with signature $2\ast 22$ has accordingly also the total cost of $\frac{1}{2}+1+\frac{1}{4}+\frac{1}{4}=2$ . Finally, the pattern with the one mirror symmetry and the one miracle whose signature is accordingly $\ast\times$ also has a total cost of $1+1=2$ .

Now it’s your turn. First determine the signatures of the seventeen different samples shown at the exhibit and check their total cost. Can you show that there can only be exactly these seventeen types by analysing the possible signatures?

Literature

[1] Conway, J.H., Burgiel, H. und Goodman-Strauss, C.: The Symmetries of Things, 2008.

[2] https://en.wikipedia.org/wiki/Wallpaper_group

[3] https://en.wikipedia.org/wiki/Dihedral_group

[4] https://en.wikipedia.org/wiki/Cyclic_group