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.

Dieses Bild hat ein leeres Alt-Attribut. Der Dateiname ist Ornament-1024x771.jpg
Figure 1: Image of the exhibit

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 1-1correspondence). This is what we will do in the following. <!-- /wp:paragraph -->  <!-- wp:paragraph --> So what are the repeating patterns in the plane? In what follows, we describe the four fundamental properties of repeating patterns in the plane and introduce a signature for each such pattern. <!-- /wp:paragraph -->  <!-- wp:separator --> <hr class="wp-block-separator"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>Local symmetries</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> Which <em>local symmetries</em> (i.e. symmetries around a point) can a plane figure have? We will describe this below and assign a signature for each type. <!-- /wp:paragraph -->  <!-- wp:paragraph --> The star\astdenotes a <em>mirror or kaleidoscopic symmetry</em>. 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 <em>star-two-point symmetry</em>. This means that the figure has exactly two mirror lines (which are perpendicular to each other) at one point. The star\astalso stands for the mirrors of a kaleidoscope (here two) meeting in one point. The point\bulletmeans that all symmetries fix a point (as is the case here, but not for a mere\astsymmetry, since a whole straight line is fixed). <!-- /wp:paragraph -->  <!-- wp:paragraph --> This signature can now be easily extended: Thus\ast n\bulletmeans that exactlynmirror lines meet in a point. Behind this is the <a href="https://en.wikipedia.org/wiki/Dihedral_group">dihedral group</a>D_n. See also the following figure 1. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:paragraph --> <em>Gyrations</em> are another type of local symmetry where there is no mirror symmetry on a straight line, only <em>rotational symmetry</em>. For example, the unpopular <em>swastika</em> has a fourfold rotational symmetry, but no mirror symmetries. This form of symmetry is given the signature4\bullet. Thus, ann-fold rotational symmetry is abbreviated asn\bullet, see Figure 2. Behind it here is the <a href="https://en.wikipedia.org/wiki/Cyclic_group">cyclic group</a> (realized as a rotational group) of ordern. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:paragraph --> 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_nandC_nforn\geq 1). For example, a circle has infinitely many point symmetries (namely, all rotations about that point and all reflections through an axis containing that point). But this is inadmissible, because there are infinitely many symmetries. <!-- /wp:paragraph -->  <!-- wp:separator --> <hr class="wp-block-separator"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>Frieze pattern</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> Frieze patterns are patterns where a figure not only has certain symmetries in one point, but, where this point can also be shifted in one direction within the pattern. They often appear in ancient architecture (see Figure 3 below). <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:separator --> <hr class="wp-block-separator"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>Repeating patterns in the plane</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> Frieze patterns have only one direction in which they can be moved by a certain amount. However, in this exhibit we are interested in patterns that can be periodically continued in two different directions, thus filling the entire plane (see Figure 4 below). <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:paragraph --> Here, as we will see, the local symmetries around a point, play a major role. <!-- /wp:paragraph -->  <!-- wp:separator --> <hr class="wp-block-separator"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>Kaleidoscopic patterns</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> Patterns whose symmetries are defined by reflections are called <em>kaleidoscopic</em>. The name is based on the fact that a pattern seen in a kaleidoscope is such a <a href="https://en.wikipedia.org/wiki/Kaleidoscope">kaleidoscopic</a> pattern (see also the exhibits <a href="http://test.erlebnisland-mathematik.de/portfolio/kaleidoskopspiegel/">"Kaleidoscope Mirror"</a> and <a href="http://test.erlebnisland-mathematik.de/portfolio/kaleidoskop/">"Kaleidoscope"</a>). <!-- /wp:paragraph -->  <!-- wp:paragraph --> But how can such patterns be described? The answer is: by how their mirror lines intersect. You can see this well 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). Thus the signature of the whole kaleidoscopic pattern is\ast632(here the point is missing, because not all reflections fix a point). In which row order the numbers2,3,6are noted here does not matter, because this only reflects at which corner of the distinguished triangle one starts or whether one considers a mirrored triangle. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:separator --> <hr class="wp-block-separator"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>Gyrations</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> Also the pattern shown in figure 6 has many mirror symmetries. If it had only these, its signature would be\ast333. However, a closer look reveals that there is another symmetry: Namely, if we look at the red triangle, we notice that this triangle -- an equilateral triangle -- can be rotated by60^\circat its center, so that the pattern below it is preserved. At this point, however, the pattern has no mirror symmetries. The signature of the pattern is set to3\ast3because there is one kind of point with threefold rotational symmetry (local symmetry groupC_3) and one kind of point with threefold mirror symmetry (local symmetry groupD_3). <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:paragraph --> This is how one proceeds now: For a given pattern, one always counts the number of types of points with nontrivial local symmetries. For example, the following pattern (Figure 7) has the signature2\ast22. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:paragraph --> However, it can also occur that there are no mirror symmetries at all. For example, the following pattern (Figure 8) has the signature444, since there are four different point varieties with fourfold rotational symmetry, but no mirror symmetries appear at all. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:paragraph --> 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. In the following figure 9 two of them are shown; one with one type of mirror lines and another with two different types. Accordingly, the first has the signature\ast, while the second has\ast\ast. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:separator --> <hr class="wp-block-separator"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>Miracles and miracle rings</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> In such patterns as these (i.e., when the mirror lines bound an infinitely large region), there may then be small <em>miracles </em>that can occur: Namely, connecting lines of oppositely directed spirals that do not cross a mirror line. For each such line, one makes a\timesin the signature. The following figure shows a pattern with one and two such lines (here one takes only so many such lines that each further one can be composed of them). The first pattern still has one type of mirror lines. Therefore it gets the signature\ast\ŧimes. The second pattern, on the other hand, has no mirror symmetries at all and therefore receives only the signature\times\times. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:paragraph --> A miracle is a combination of translation and mirroring of a <a href="https://en.wikipedia.org/wiki/Fundamental_domain">fundamental domain</a>, which cannot be explained by a mirroring or a rotation of the total pattern alone. But there is even the possibility of a repetition of the fundamental area, which can be explained neither by a rotation or a mirroring, nor by a miracle. Such repetitions always occur in pairs (since the pattern is said to extend infinitely in two directions). Such a phenomenon we call <em>wonder-ring</em> and abbreviate it with\circ. For example, the following pattern has the signature\circ. <!-- /wp:paragraph -->  <!-- wp:image --> <figure class="wp-block-image"><img alt=""/></figure> <!-- /wp:image -->  <!-- wp:separator --> <hr class="wp-block-separator"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>Summary</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> We have so far described four different phenomena: mirror lines (kaleidoscopic patterns), gyrations, miracles and miracle rings. The content of the exhibit "Ornament" is now that these phenomena are sufficient to describe any periodic pattern which extends infinitely in two directions. Miracle rings and gyrations get the orientation here, whereas mirror lines and miracles reverse it. We summarize this again in the following table: <!-- /wp:paragraph -->  <!-- wp:table {"className":"is-style-stripes"} --> <figure class="wp-block-table is-style-stripes"><table><tbody><tr><td></td><td>Miracle ring</td><td>Gyration</td><td>Mirror lines</td><td>Miracle</td></tr><tr><td>Symbol</td><td>\circ\cdots\circ</td><td>ab\ldots c</td><td>\ast ab\ldots c\ast de\ldots f\ldots</td><td>\times\cdots\times</td></tr></tbody></table><figcaption>Table 1: The four fundamental phenomena</figcaption></figure> <!-- /wp:table -->  <!-- wp:paragraph --> 1/4With 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: <!-- /wp:paragraph -->  <!-- wp:table {"className":"is-style-stripes"} --> <figure class="wp-block-table is-style-stripes"><table><tbody><tr><td>Symbol (orientation preserving)</td><td>Costs</td><td>Symbol (orientation reversing)</td><td>Costs</td></tr><tr><td>\circ</td><td>2</td><td>\astoder\times</td><td>1</td></tr><tr><td>2</td><td>1/2</td><td>2</td><td>1/4</td></tr><tr><td>3</td><td>2/3</td><td>3</td><td>1/3</td></tr><tr><td>4</td><td>3/4</td><td>4</td><td>3/8</td></tr><tr><td>n</td><td>\frac{n-1}{n}</td><td>n</td><td>\frac{n-1}{2n}</td></tr><tr><td>\infty</td><td>1</td><td>\infty</td><td>1/2</td></tr></tbody></table><figcaption>Table 2: Cost of each symbol</figcaption></figure> <!-- /wp:table -->  <!-- wp:paragraph --> There is now the following mathematical theorem which is the center of this exhibit: This states that the possible signatures of plane periodic patterns are exactly those with total cost of exactly2. However, we do not intend to prove this theorem at this point, but only to apply it: For example, the first pattern shown had the signature\ast 632. This corresponds to a cost1+\frac{5}{12}+\frac{1}{3}+\frac{1}{4}=2as desired. The pattern with signature3\ast 3also has total cost\frac{2}{3}+1+\frac{1}{3}=2. Accordingly, the pattern with the two kaleidoscopic symmetries and the one rotational symmetry with signature2\ast 22also has 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\timesalso has total cost1+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.

[2] https://de.wikipedia.org/wiki/Ebene_kristallographische_Gruppe

[3] https://de.wikipedia.org/wiki/Diedergruppe

[4] https://de.wikipedia.org/wiki/Zyklische_Gruppe