The Application of Frieze Groups and Crystallographic Groups in Generating Batak Toba Ornament Motifs Using a Matlab Graphical User Interface

Gorga is a carving or sculpture typically found on the exterior of a Toba Batak traditional house. The Batak people use fractal (geometric) dimensions in Batak gorga carvings. In mathematics, repetitive and symmetrical patterns in planes that result from transformations are included in the plane symmetry groups. Ethnomathematics is a cultural approach to the concept of mathematics. A frieze group can be defined as a symmetrical group which arises from a unidirectional translation and subsequently generates a linear pattern that recurs exclusively in a single direction. There are seven different pattern types in the frieze groups. Meanwhile, crystallographic patterns are flat two-dimensional patterns that form a lattice. There are 17 crystallographic types of patterns with five different types of unit lattices. The purpose of this study is to generate motifs for Batak ornaments based on frieze groups and crystallographic groups using a Matlab Graphical User Interface (GUI). A total of 119 new motifs were generated based on seven types of patterns in the frieze groups, namely, F1,F2,F3,F4,F5,F6, and F7. Meanwhile, in the crystallographic groups, 153 new motifs were generated based on nine types of patterns, namely, p1,p2,pm,pg,cm,pmg,pmm,cmm, and pgg. To keep with trends, the new motifs generated can be used in everyday life as decorations or business symbols that are characteristic of the Toba Batak region

Coxeter's frieze patterns and discretization of the Virasoro orbit

We show that the space of classical Coxeter's frieze patterns can be viewed as a discrete version of a coadjoint orbit of the Virasoro algebra. The canonical (cluster) (pre)symplectic form on the space of frieze patterns is a discretization of the Kirillov symplectic form. We relate a continuous version of frieze patterns to conformal metrics of constant curvature in dimension 2.Comment: typos correcte

Frieze Group in Generating Traditional Cloth Motifs of the East Nusa Tenggara Province

Ethnomathematics studies the relationship between mathematics and culture. Indonesia has many traditional cultures. One of them is traditional cloth. The traditional cloth from East Nusa Tenggara (NTT) province is called tenun ikat. Since the motif of tenun ikat consists of symmetrical and repeated patterns, it can be generated using Frieze groups. The Frieze groups are the plane symmetry groups of patterns whose subgroups of translations are isomorphic to Z. There are seven groups in the Frieze groups, i.e., F_1, F_2, F_3, F_4, F_5, F_6, and F_7. Translation, reflection, rotation, and glide reflection are the transformation used in the Frieze groups. In this paper, Frieze groups are used to generate digital tenun ikat motifs from the basic pattern. Since one piece of original tenun ikat may consist of some basic patterns, the basic patterns are identified first, and then each of them is generated into the desired pattern, according to the suitable Frieze groups. Furthermore, a GUI Matlab program is developed to generate the Frieze groups. Three motifs of tenun ikat are presented to demonstrate the implementation of Frieze groups. With the Frieze group, users can generate other patterns from a basic pattern, so users can generate new motifs of tenun ikat without leaving the cultural characteristics of NTT province

Quantum frieze patterns in quantum cluster algebras of type A

We introduce a quantisation of the Coxeter-Conway frieze patterns and prove that they realise quantum cluster variables in quantum cluster algebras associated with linearly oriented Dynkin quivers of type A. As an application, we obtain the explicit polynomials arising from the lower bound phenomenon in these quantum cluster algebras.Comment: 10 page

The Erechtheion: deciphering the fragments of the Ionic frieze

The Erechtheion, the temple dedicated to Athena Polias on the Athenian Acropolis, was an extraordinary structure. The temple was situated on three different levels and had at least six cults worshipped in the complex. Little is known about the interior of the building or the purpose each room served, but the Ionic frieze that would have adorned the temple is the avenue in which this thesis will explore. The Ionic frieze is believed to be the sole figural decoration on the Erechtheion other than the Porch of the Karyatids, and there is no evidence of pedimental sculpture or statuary akroteria adorning the roof of the building. However, the only extant remains of the frieze are mere fragments of figures and groups of figures. My thesis will explore the possible interpretations of the frieze by first examining the political climate in which the temple and its frieze were created. The myths associated with the gods and heroes included in the sanctuary of the Erechtheion will be considered in my analysis. Lastly, the Erechtheion’s frieze will be regarded in relationship to other fifth century buildings and sculpture in order to determine the frieze’s content and context