International Society of communication and Development among universities www.europeansp.org Spectrum (Educational Research Service),ISSN:0740-7874 Mathematical Transformations and Songket Patterns Nor Maizan Abdul Aziza, Rokiah Embong b* a,b Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia Abstract Songket the beautiful traditional textile of the Malay in Malaysia is generally appreciated for its aesthetics motifs and patterns, without any thought on the mathematical concepts underlying those patterns. The symmetries in the motifs and the transformations concepts embedded in the pattern of two pieces of sampin songket were studied by utilizing the mathematical software GeoGebra. Based on the symmetries and transformations identified, the songket patterns could be categorized under the symmetry groups standardized by the Union of Crystallography namely the Frieze Pattern and the Wallpaper Pattern. The identified symmetries and transformations concepts were then used to generate new songket patterns, patterns with better symmetry, precise and accurate thread counts. For majority of students, mathematics is considered as a difficult and uninteresting subject, hence the idea of integrating mathematics, arts and one’s culture such as the songket patterns creation activity could act as a catalyst in students’ understanding of some mathematical concepts. It could help teachers to integrate concrete, abstracts ideas and creative elements in the classroom. This might make learning mathematics more interesting. © 2016 The Authors. Published by European Science publishing Ltd. Selection and peer-review under responsibility of the Organizing Committee of European Science publishing Ltd. Keywords: Songket; Mathematical Transformations; Symmetry Groups 1. Songket Songket is one of the beautiful traditional crafts produced by the Malays in Malaysia. Songket is a luxurious hand-woven fabric that belongs to the brocade family of textiles. Songket comes from the Malay word menyongket (or menyungkit) which means to ‘lever up’ or ‘to pick’ and insert metallic threads, traditionally gold or silver threads during the weaving process to form motifs. Songket portrays the sophistication of the Malays in term of knowledge and aesthetic. The motifs and designs in the songket fabric not only portray the weaver’s special skill but it also Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021 captures the Malay’s cultural identity, civilization and aesthetics. Using a technique that is hundreds of years old, songket is woven on the kek (weaving loom), a foot operated wooden frame (Fig.1) using what is commonly referred to as the supplementary weft method, a decorative weaving technique in which extra (supplementary) threads ‘float’ across a colourful woven ground to create an ornamental effect (Yayasan Tuanku Nur Zahirah, 2008). Traditionally the supplementary threads are gold and silver but nowadays varieties of other metallic colours such as lime green, © 2016 The Authors. Published by European Science publishing Ltd. Selection and peer-review under responsibility of the Organizing Committee of European Science publishing Ltd. 2 Abdul Aziz; Embong/ Spectrum (Educational Research Service),ISSN:0740-7874(2017)101_111 turquoise blue, pink or purple are also used. The threads called tekat (weft ‘floats’) is inserted in between the benang pakan (the latitudinal or crosswise weft threads) and the benang loseng (the longitudinal or upright warp threads). The metallic weft ‘floats’ are passed over and under a certain number of carefully counted warp threads on the loom to produce patterns that contrasts in colour and texture with the ground cloth. The use of metallic threads gives songket its exquisite beauty that distinguishes it from other type of hand-woven textile. The weaving is done carefully and there is harmonious balance between the choice of colour, design, structure and motifs. Fig. 1. A Weaver weaving on the Kek (taken from Bibah Songket) Traditionally, songket are worn by the royal families and dignitaries during official ceremonies and functions, and by the bridal couples on their wedding days. Songket is gaining its popularity due to the recognition it receives at both domestic and international levels. The hand-woven songket has been getting a makeover in terms of the design, material and usage. Nowadays, songket is not only used as the traditional attire in special functions and weddings, but it is also used as deco items, accessories, handbags, home gifts, etc as portrayed in Fig. 2. Fig. 2. Contemporary Songket Products (taken from Yayasan Tuanku Nur Zahirah and Bibah Songket) Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021 2. Mathematics and Arts The traditional philosophers believed that harmony and beauty surrounding us can be perceived from mathematics. Geometry patterns can be found everywhere in nature such as in the honeycomb, pineapple, flowers, trees and snowflakes. Often humans copied and adapted these patterns to enhance their world. Mat Rofa (2007) states that geometry is an instrument of investigation into the beauty of objects existing in the universe, and geometry offers the intermediary between the material and spiritual world and the glimpse of perfection. There is a strong connection between mathematics and art. The idea of gaining entrance to mathematics via art is very appealing. M.C. Escher, the graphic artist, illustrates how he uses inspiration from mathematical ideas that he read about to develop exquisite work of art. On the other hand, A. Fomenko, a Russian mathematician, uses art as a mean to express abstracts mathematical concepts (Stylianou & Grzegorczyk, 2005). The concept of symmetry and repetition was known to ancient artisans long before its study was formalized, but it was geometrical inquiry that Abdul Aziz; Embong/ Spectrum (Educational Research Service),ISSN:0740-7874(2017)101_111 3 opened up its mathematical nature (Keeney, n.d.). Mathematicians treat symmetry as ideal, but in nature, symmetry is approximate. Objects in nature are symmetrical; human, animals, trees, flowers, leaves and so on. In art, as in nature, beauty can be found in the approximation of symmetry, rather than in its precision (Bier, 2000). In the Malay culture, the design of a house, handicrafts such as tudung saji (food cover), wau (kite), gasing (top spin), pots and so many others depict rich sources of geometry and its application in the Malay daily life. The Malay songket weavers managed to capture the beauty in their surrounding and translated them into beautiful songket motifs, and if one were to take a closer look at the patterns in songket, he or she will notice that the beautiful patterns and creative designs produced by the songket weavers are also rich with geometrical shapes (e.g. square, rhomboids, triangle and hexagon) and symmetrical patterns. However, the public rarely noticed it, let alone appreciate the mathematics underlying the patterns formations. The weavers themselves, especially the older generations, are not aware of their use of mathematical concepts, geometrical properties, and calculation and measurement in their work (Nor Maizan, 2016). Transverse symmetry is one of the techniques used in the weaving process (Norwani, 2002). Most songket motifs have the tendency to become geometrical in design because of this transverse symmetry weaving techniques. According to Ascher (1994) weaving involves geometric visualization. The weavers express the visualization through actions and materials. It requires the creation and conception of pattern and knowing what techniques and colours to use in the weaving process so that the patterns will emerge. Zakiah (1987) discovers that in the weaving process, the traditional Malay songket weavers have to memorize all the patterns or use samples from the old songket and copy them directly, without any patterns drawn on paper as guidance. The expert weavers rarely draft the songket patterns before they weave (Norwani, 2002). They refer to the existing songket patterns and creatively change certain motifs in certain sections with their own creation. All these knowledge, the patterns and the skills are handed down secretly from generation to generation. 2.1. Group Theory Group theory is essentially the study of symmetry. Symmetry applies to anything that stays invariant under some transformations. In general, a definition of a group is a set (let label it as G) with an operation (let use the symbol *) that obey these four rules: 1. Closure: If a and b are in G, then a*b is also in the group. 2. Associative: If a, b, c are G, then (a*b)*c = a*(b*c). 3. Identity: There is an element e in G such that a*e = e*a = a for every element a in G. 4. Inverse: For every element a in G, there is an element a-1 such that a*a-1 = e and a-1*a = e. (Pinter, 1990) The study of systems that obey these four rules is the basis of group theory. The emphasis is on the basic quality of groupness in which the groups have in common. The example of groups can be seen in areas of chemistry, such as the structure and behavior of molecules and crystals, in the area of physics such as the conservation of energy under the “translation” of time, the conservation of momentum under the “translation” of space and the conservation Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021 of angular momentum under rotation (Conrad, n.d.). In mathematics, group theory can found in the various geometric transformations in the Euclidean plane R2. The repeated patterns in a plane of Euclidean 2-space can be described by the four basic linear transformations; translations, reflections, rotations and glide reflection. A figure is said to be symmetrical if it has at least one of the four isometries. Symmetrical figures are called designs. Designs with translation symmetry are called patterns (Washburn & Crowe, 1988). Both symmetry and repetition can be accomplished by transformations. A transformation is a process where each point on a shape is moved in a formulaic way (Keeney.n.d).