Frieze group
Fries or ribbon ornament groups are special groups that are in mathematics, more precisely, the discrete geometry studied.
The basic idea
Ribbon ornaments (or friezes ) are patterns that are formed by a certain smallest unit (eg, a pattern or figure) along a fixed direction (the so-called Fries direction ) to each other repeatedly sets. A band ornament group in turn is seen clearly the symmetry group of a given band ornament, while one imagines the band ornament but extended in both directions to infinity before. Band ornaments can have many different shapes and figures that are mathematically difficult to handle, so that band ornament groups only bad can be defined mathematically in this way.
However, all band ornament groups one thing in common: Due to its design, a belt ornament is mapped definitely on yourself when you move it to a unit or to a multiple of this unit along the frieze direction. Such shifts or translations ( parallel shifts ) therefore belong to the symmetry group of a band ornament. But they are also the only possible shifts
Band ornament maps to itself, in particular occur no shifts in another direction as the Frisian direction (or in the opposite direction). The only shifts that may arise within a symmetry group of a frieze and therefore must be changed by a multiple of a vector whose length is the distance between the two smallest units of the same and pointing in the direction of Fries. Denoting such a vector with, then for any displacement from the symmetry group of the frieze. In other words, all occurring in the symmetry group of the ribbon ornament shifts belong to the group generated, especially the translations form a band ornament group is a cyclic group. This property of frieze symmetry groups is used for the mathematical definition of a band ornament group.
Mathematical definition
The belonging to a generating translation translation vector is also called frieze vector, it is uniquely determined up to the sign. The straight line is called Fries direction. Since the amount of translations of a group consisting of isometric itself represents a group, called T ( G) and the translation group G. It should be noted that the n-fold series connection of the Figure and indicated that:
Classification of frieze groups
On closer examination, it is found that there are " many " frieze groups do not exist, ie many frieze groups are similar in a sense. Here one thinks of the similarity term that is defined in the next section.
Classification
The similarity of the groups has, so that it divides the properties of a equivalence relation with the definition of the similarity, the set of all subsets of disjoint.
It turns out that there are only 7 types of ribbon ornament groups with respect to the similarity. So you select from any band ornament group G, then there exists a subgroup H of one of the types described below 7 and a bijective affine map with. All types are systematically numbered and listed in the list below and described ( in the following is a generating element of the translation group of G).
Types of tape ornament Groups:
Proof of the classification ( sketch)
The proof of the classification is a frieze group via the so-called point group S ( G ) G. The point group of G consisting of all the linear parts of the illustrations from G:. In the proof one considers what point groups are possible and reconstructs possible frieze groups.
Since the translation group is cyclic, it contains a translation with the shortest possible translation vector. This shows it is easy, because all related to translations vectors are of the form, if a vector belonging to a generating element of the translation group. Apparently, the shortest possible belonging to a translation vectors. We now consider an arbitrary element of a frieze group with ( ie, with displacement vector v and linear component L ). Due to the group properties of G is with and also in G, but is expected right after that, that is a translation, so must their displacement vector of the form. Since L is an isometry and receives lengths is as long as, so it must be one of the shortest possible translation vectors, ie. It is now expanding to an orthogonal basis and considered due to the Isometrieeigenschaften of L, that must apply. About the existence and uniqueness theorem all possible linear Shares of any picture of a band ornament group comes up. Thus it is found that there are a total of 4 possible pictures, so 5 possible point groups (namely the set of all mappings and their subgroups ). For the reconstruction of the frieze groups of G then you think about it at any point group, which are displacement vectors for a picture of G in question. Here, one finds: Contains a point group, a reflection in a line in the direction frieze, two cases must be distinguished:
- The frieze group itself contains a reflection
- The frieze group itself contains only thrust reflections
This consideration leads to the splitting of the cases 4 and 5, in two " sub- events " 4.1 and 4.2 or 5.1 and 5.2. After it has been found by the above considerations, any types of groups, it shows the construction of a suitable imaging explicitly that groups that belong to the same case, are similar to each other. Moreover, we prove that groups belonging to different cases, may not be similar.
Examples
The following are some examples of each type of frieze group. The frieze depicted here has the corresponding frieze group as a symmetry group. The gray elements belonging respectively to the pattern in black symmetry elements are shown: dots indicate inversion centers, mirror axes dashed lines, solid or dotted lines real thrust axes of reflection, the arrow points to one of two possible Fries vectors. The following figure shows a frieze of type F5.1 ( from a carpet ):
Variables and Glossary of Terms
Directory of key variables used in the article: