# Ornament

In this exhibit, you can draw colorful lines with your finger, which are then multiplied according to a selected symmetry rule. But where do these symmetries come from? And what is the exhibit all about? This is the question we want to explore in this in-depth text.

You encounter symmetrical objects and patterns every day. A parquet floor is usually laid in a periodic pattern. The tiles in your bathroom on the wall as well. But what are the fundamentally different ways to fill a plane with a periodic pattern. This is the question that this exhibit, “Ornament” is designed to answer vividly. The seventeen different choices you have are, in a certain sense, “all” there are.

#### And now … the mathematics of it:

But where do the given symmetries of the exhibit come from? Now this has to do again with group theory. The distance-preserving self-mappings of the Euclidean space form a group: In fact, one can execute 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. Here, there are basically two different types of mappings: Rotations and Reflections. The reflections turn the orientation around, i.e. if you imagine a sheet of paper, a reflection corresponds to the process that you turn the sheet along the mirror axis. With a rotation, on the other hand, the sheet is turned around only one point (but you still look at its front side afterwards). If you do two reflections in a row, you get one rotation — then you look at the front side 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 type isr the subject of the exhibit “Dust Circles”), but only discrete subgroups of these. What this is exactly, we will not define precisely here; but it can be briefly summarized as follows: Each point under a symmetry operation of the group is either fixed or mapped to another point which has a certain minimum distance from it. For any such group, there are many symmetric patterns obtained from that group. Therefore, in order to illustrate the group well now, it is convenient to study such patterns (although, of course, this is not a 1\ast\ast2\bullet\ast\bullet\ast\ast n\bulletnD_n4\bulletnn\bulletn\bullet\mathbf 1D_nC_nn\geq 1\ast6\bullet\ast3\bullet\ast2\bullet\ast6322,3,6\ast33360^\circ3\ast3C_3D_32\ast22444\ast\ast\ast\times\ast\ŧimes\times\times\circ\circ\circ\cdots\circab\ldots c\ast ab\ldots c\ast de\ldots f\ldots\times\cdots\times1/4\circ2\ast\times121/221/432/331/343/443/8n\frac{n-1}{n}n\frac{n-1}{2n}\infty1\infty1/22\ast 6321+\frac{5}{12}+\frac{1}{3}+\frac{1}{4}=23\ast 3\frac{2}{3}+1+\frac{1}{3}=22\ast 22\frac{1}{2}+1+\frac{1}{4}+\frac{1}{4}=2\ast\times1+1=2\$.

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

#### Literature

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