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

ABSTRACT

Tenun ikat is a traditional cloth of several regions in the East Nusa Tenggara (NTT) province, such as Flores, Sumba, Timor, Alor, Rote, and Sawu. Tenun ikat is typically used for clothing in everyday life, clothing in traditional dances or ceremonies, dowry in marriage, an indicator of social status, transaction tools, and a gift (Nainupu, 2018). It is made by weaving threads and making the motif by tying it with a rope according to a particular pattern before being dipped in dye. Tenun ikat from 22 regions in NTT province has various motif designs, colors, and sizes, representing their characteristics. The motifs are related to the customs, beliefs, culture, and habits of the local community (Maghiszha, 2019). Tenun ikat from Sumba and Rote regions generally use animal and leaf motifs, respectively. Tenun ikat from Timor region is in the form of silk and has embroidery. Alor region uses natural colors from plants and marine life (Salma et al., 2018). Some motifs of tenun ikat are shown in Figure 1. Weaving the tenun ikat becomes a part of the daily life of people in NTT province, especially women. In this modern era, tenun ikat is not only used by the people of NTT province, but also by people outside of NTT province. Tenun ikat has even appeared at Culture New York Fashion Week in 2017 and Paris Fashion Week in 2018 (Redaksi PI, 2021). Some problems in the weaving process of tenun ikat are the expensive materials and long processing times (Administrator, 2019), which made the price of tenun ikat expensive. Furthermore, some of the younger generations do not want to continue the weaving culture. This can lead to the extinction of the tenun ikat culture. With the advancement of technology, tenun ikat, specifically, its motifs, can be preserved or maintained even developed further. Motifs of tenun ikat can be generated using mathematical formulas and digitized using a computer program.
From Figure 1, it can be seen that the motif of tenun ikat has the characteristic of a geometric pattern. A geometric pattern is a design with a certain pattern, repeating regularly. Ethnomathematics is the study of the relationship between mathematics and culture (Hidayati & Prahmana, 2022). Ethnomathematics helps to build meta-awareness about the role of mathematical knowledge in mathematical society and culture (Rosa et al., 2017). Ethnomathematics can be used to explore mathematics related to culture. Ethnomathematics is a daily mathematical representation or mathematics associated with cultures. Using ethnomathematics, Javanese culture, especially batik, other than consists of philosophy and deep cultural value, it also consists of mathematics concepts, such as geometry transformation concepts (Risdiyanti & Prahmana, 2017). The traditional house and traditional music instrument of Biak can be explored by using ethnomathematics, where the shape of the roof of the traditional house is rectangular, half elliptical and triangular trapezoid, and the traditional music instrument looks like the two most belted cones combined (Sroyer et al., 2018).
Some motifs in the tenun ikat can be generated by applying mathematical formulas. One of the mathematical formulas that can be applied is crystallographic patterns. Some research about finding the crystallographic patterns for traditional clothes can be found in (De Las Peñas et al., 2018;Hobanthad & Prajonsant, 2021;Kartika et al., 2022;Libo-On, 2019;Vasquez et al., 2020). Another mathematical formula that can be used is Frieze groups (Davvaz, 2021). Frieze groups aim to design some repetitions in one dimension, into decorative arts. Some research using Frieze groups concerning culture are shown in Table 1.  (Makur et al., 2020) Most motifs in Towe Songke have group 7 because they can be seen as translation, horizontal reflection, vertical reflection and half turn rotation symmetry. 8 One-color Frieze Patterns in Friendship Bracelets: A Cross-Cultural Comparison (Koss, 2021) Users are creating designs based on symmetry preferences of their local culture.
Indonesian cultures need to be preserved, including the tenun ikat from NTT. Table 1 shows the use of Frieze groups in some regions, but none of them has discussed tenun ikat, even though it is one of Indonesian cultures that is recognized in the world (Redaksi impresinews.com, 2021;Redaksi Kompas, 2019). Most of the tenun ikat weavers are elderly women, so it is feared that there will be no younger generation who understands the culture of tenun ikat (Azizah, 2021). In addition, the high price of tenun ikat (Novemyleo, 2020) makes buyers, especially tourists who just want to buy souvenirs, unable to buy tenun ikat. Therefore, the motifs of tenun ikat will be analyzed, so that the motifs can be preserved and also vary, without leaving their original culture. Besides that, the digitized motifs of tenun ikat allow weavers to use them as stamped motifs on the fabric, so the production cost can be minimized. The goal of this paper is to explore and analyze the motifs of tenun ikat from the mathematical perspective, especially geometry elements in the motifs of tenun ikat using Frieze groups. Furthermore, a computer program is made to generate the motif from a basic pattern, so users can generate new motifs of tenun ikat without leaving the cultural characteristics of NTT province.

B. METHODS
In this paper, Frieze groups are used to identify and generate the basic pattern of tenun ikat. The Frieze groups are the plane symmetry groups of patterns whose subgroups of translations are isomorphic to ℤ. Transformation used in the Frieze group can be seen below. Let ( ) be the origin point and ( ′ ′ ) be the result point.
1. Translation. The matrix transformation for translation is ( ′ ′ ) = ( + ℎ + ). Rotations can be rotations of orders two, three, four, and six-fold, where the angle of rotations is 180 o , 120 o , 90 o , and 60 o , respectively. 4. Glide reflection. Glide reflection is transformation for both translation and reflection. There are seven patterns in the Frieze groups (Gallian, 2021) as shown in Table 2.  Washburn and Crowe (1988) proposed an algorithm that can be used to identify seven Frieze groups by using flowchart (Gallian, 2021), as seen in Figure 2. For example, one basic pattern shown in Figure 3 is used. From this basic pattern, the seven patterns of Frieze groups can be generated as shown in Figures 4, Figures 5, Figures 6, Figures  7, Figures 8, Figures 9 dan Figures 10.  1. Group 2 . Let denotes a glide reflection, then the group 2 is generated as shown in Figure  5. Group 3 . Let denotes a translation to the right of one unit and denotes a vertical reflection, then the group 3 is generated as shown in Figure 6. Group 6 . Let denotes a translation to the right of one unit and denotes a horizontal reflection, then the group 6 is generated as shown in Figure 9. Group 7 . Let denotes a translation to the right of one unit, denotes a horizontal reflection, and denotes a vertical reflection, then the group 7 is generated as shown in Figure  10. The research method in this paper can be shown in Figure 11. The process starts with identifying the basic pattern of a fabric motif, if needed. If a basic pattern that doesn't follow a pattern that already exists is wanted, then the process is continued on generating the basic pattern into a new pattern using Frieze groups. Since some new patterns can be combined into one pattern, then it can generate more into a new pattern using Frieze groups, as shown in Figure 11. 658 | JTAM (Jurnal Teori dan Aplikasi Matematika) | Vol. 6, No. 3, July 2022, pp. 651-664 Figure 11. Research Method

C. RESULT AND DISCUSSION
Several tasks carried out in this article are (1) identifying the basic pattern of tenun ikat, (2) generating a pattern from the basic pattern using Frieze groups, (3) applying a pattern generation using GUI Matlab. In this study, three motifs of tenun ikat from NTT province are given as examples to prove that Frieze groups can be used to generate digitized tenun ikat. The mathematical formulas to generate the basic pattern into the result pattern using the Frieze groups are also explained.

Motif 1
The original motif is shown in Figure 12(a). The motif generation can be seen in Table 3 and the result using the program is shown in Figure 12 The generation using the Frieze group is described in Table 4. The basic pattern 1 is identified from the result pattern 1 using the flowchart in Figure 3. From the flowchart, the result pattern 1 belongs to the group 7 , where the basic pattern 1 is reflected horizontally, and then translated with vertical reflection, and also translated with vertical and horizontal reflections. Furthermore, the basic pattern 2 generates the result pattern 2 with a group of 6 , where the basic pattern 2 is translated and reflected vertically, as shown in Table 3.

Motif 2
The original motif is shown in Figure 13(a). The motif generation can be seen in Table 4 and the result using the program is shown in Figure 13(b). As seen in Table 4, the motif in Figure  13(a) consists of two parts, the first part as shown in the first row and the second part as shown in the second row. The final result is obtained by combining the two parts, as shown in the third row, as shown in Figure 13. For the first part, the result pattern 1 can be obtained from the basic pattern 1 by generating group 7 . The second part is generated from the basic pattern 2 using a group of 7 . Furthermore, the first part and the second part are combined, becoming the basic pattern 3. Using a group of 7 , the result is shown in the result pattern 3. Finally, using a group of 1 , the result pattern 3 is translated to the right one unit, which becomes the result pattern 4, as shown in Table 4.

Motif 3
The original motif is shown in Figure 14(a). The motif generation can be seen in Table 5 and the result using the program is shown in Figure 14(b). Table 5 shows the generation in each step. As shown in Figure 14. The basic pattern 1 is generated using group 7 , becomes the result pattern 1. Then, by using a group of 6 , the result pattern 1 is reflected horizontally and becomes the result pattern 2. Furthermore, the result pattern 2 is combined with another pattern, as can be seen in the basic pattern 3. Using a group of 3 , the basic pattern 3 is reflected vertically, and becomes the result pattern 3. Finally, the result pattern 3 is translated to the right of one unit, using a group of 1 , as shown in the result pattern 4. As shown in Table 5.  Furthermore, Graphical User Interface (GUI) in Matlab is made to generate a basic pattern into the result pattern. Figure 15 is the GUI design for the pattern generation. Users can enter a motif from "Open" button and the motif will be displayed on the axes. One Frieze group is chosen from the "F1" until "F7" buttons, to get the results. The result will be displayed on the new window and users can save the result with "Save" button, as shown in Figure 15. Motif 3 is taken for the example. Let 3 is applied to the 3rd basic pattern from Table 5 (as shown in Figure 16(a)), then the result can be seen in Figure 16(b). The Matlab code for 3 is shown in Code 1 below, where image is the basic pattern, F is the matrix for the basic pattern, G is the matrix for the horizontal reflection, and Z is the result.

Code 1. Matlab Code for 3 Group
F1=double(image); s=size(F1); G1=zeros(s(1),s(2),s(3)); for i=1:s(1) j=1:s(2); k=s(2)-j+1; G1(i,k,:)=F1(i,j,:); end G=uint8(G1); F=uint8(F1); Z=[G F]; As explained in the Introduction, the preservation and innovation of tenun ikat is needed, so that tenun ikat does not become extinct. By identifying the existing of tenun ikat motifs and combining them with the Frieze groups, in this digital era. some new interesting motifs can be created and can be a reference for weavers. From the results of the several motifs identification, it turns out that the Frieze groups can be used to generate tenun ikat, since the motifs of tenun ikat contains of repeated translation, reflection, and rotation.

D. CONCLUSION AND SUGGESTIONS
Ethnomathematics combines mathematics and culture. In this paper, the motifs of traditional cloth from NTT province are generated using mathematical formula, i.e., Frieze groups. Three motifs are presented. Each motif consists of some basic patterns. The basic patterns are identified and then generated into the desired pattern using Frieze groups.