# 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.

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\bullet n D_n 4\bullet n n\bullet n \bullet \mathbf 1 D_n C_n n\geq 1 \ast6\bullet \ast3\bullet \ast2\bullet \ast632 2,3,6 \ast333 60^\circ 3\ast3 C_3 D_3 2\ast22 444 \ast \ast\ast \times \ast\ŧimes \times\times \circ \circ \circ\cdots\circ ab\ldots c \ast ab\ldots c\ast de\ldots f\ldots \times\cdots\times 1/4 \circ 2 \ast \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 2 \ast 632 1+\frac{5}{12}+\frac{1}{3}+\frac{1}{4}=2 3\ast 3 \frac{2}{3}+1+\frac{1}{3}=2 2\ast 22 \frac{1}{2}+1+\frac{1}{4}+\frac{1}{4}=2 \ast\times 1+1=2\$.