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,

Abstract

Songket the beautiful traditional 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 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 groups standardized by the Union of Crystallography namely the 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 in Malaysia. Songket is a luxurious hand- that belongs to the family of . Songket comes from the Malay word menyongket (or menyungkit) which means to ‘lever up’ or ‘to pick’ and insert metallic threads, traditionally or threads during the 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 . 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. 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 “” 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 . A figure is said to be symmetrical if it has at least one of the four . 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). The original figure is known as the pre-image, and the resultant figure is called the image. For isomorphic linear transformation, not only are all the vertex and arc angles in the image have the same measure as the corresponding angles in the pre- image but the measures of segment and arc lengths are also identical. If a figure has been enlarged or reduced (dilation), then it has undergone a non-isomorphic linear transformation. Limitless number of patterns can be made from one or combination of the five geometric transformations. The songket pattern is an example where various geometric transformations could be observed. Based on the combination of symmetries and transformations, the 4 Abdul Aziz; Embong/ Spectrum (Educational Research Service),ISSN:0740-7874(2017)101_111

songket pattern could be categorized under the frieze pattern or wallpaper pattern symmetry groups.

2.2. Frieze Pattern

An infinite stripe of repeating pattern is called a frieze pattern or border pattern or an infinite strip pattern or one- dimensional pattern (Washburn & Crowe, 1988, 2004). A frieze pattern has only one direction of translation symmetry. The direction could be horizontal, vertical or even set at angle. The translation length is the distance between repeats of the pattern. Lengthwise, frieze patterns may have reflection symmetry or just translation symmetry. Crosswise, frieze patterns may have reflection, glide reflection or 180 rotations or no symmetry. Each possible combination of symmetries is called a frieze group and altogether there are seven possible frieze groups. The names of the were standardized by the International Union of Crystallography (http://www.iucr.org), known as IUC notation. These names all begin with “p” followed by three characters. The first is “m” if there is a vertical reflection, and “1” if not. The second is “m” if there is a horizontal reflection; “a” if there is a glide reflection, otherwise “1”. The third is “2” if there is a 180 rotations, and “1” if not (rotation centres must be spaced at half of the translation length). Hence, the seven symmetry groups are p111, p1m1, pm11, p112, pmm2, pma2, p1a1. Table 1 shows the seven symmetry groups and the associated translation and reflection symmetries.

Table 1. Frieze Groups and Symmetries Frieze Group Symmetries p111 Translation p1m1 Translation, horizontal reflection pm11 Translation, vertical reflection p112 Translation, 180o rotations pmm2 Translation, vertical and horizontal reflection, 180o rotations pma2 Translation, vertical reflection, glide reflection, 180o rotation p1a1 Translation, glide reflection

Frieze patterns is commonly found in wallpaper borders, decorative designs on buildings, ironwork railings, needlepoint stitches and many others. Study on 210 beadwork specimens of Wisconsin woodland Indian tribes (Nishimoto & Berken, 1996) concludes that 148 are strip beadwork that utilizes all the seven different types of frieze pattern. Barkley (1998) also managed to classify 96 of the 119 of Ute (indigenous peoples in the Rocky Mountain USA) beadwork and designs into the seven frieze’s pattern group.

2.3. Wallpaper Pattern Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021

If a figure discloses translation in two or more directions, it is called wallpaper pattern or two-dimensional pattern (Washburn & Crowe, 1988, 2004). The wallpaper pattern is produced by combining a number of transformations: a translation, a reflection, a glide reflection and rotation. Rotation symmetry of a wallpaper pattern must be a rotation of order 2 (180 rotation), order 3 (120 rotation), order 4 (90 rotation) or order 6 (60 rotation). Each wallpaper pattern is based on a pair of translation in different directions. The translation is repeated many times to give a repeating image and to cover the entire infinite plane. Based on the combinations of transformations, there are 17 possible wallpaper groups. The most widely used is the IUC notation. The names are p1, p2, pm or p1m, pg or p1g, pmm or p2mm, pmg or p2mg, pgg or p2gg, cm or c1m, cmm or c2mm, p4, p4m or p4mm, p4g or p4gm, p3, p3ml, p3lm, p6, p6m or p6mm. The interpretation of symbol (read left to right) is as follows: 1st symbol p or c denotes primitive or centreed cell; 2nd symbol, integer n denotes highest order of rotation; 3rd symbol denotes a symmetry axis normal to the x-axis: m indicates a reflection Abdul Aziz; Embong/ Spectrum (Educational Research Service),ISSN:0740-7874(2017)101_111 5

axis, g indicates no reflection, but a glide-reflection axis, 1 indicates no symmetry axis; 4th symbol denotes a symmetry axis at angle  to x-axis, with  dependent on n, the highest order of rotation;  = 180 if n (the second symbol) is 1 or 2,  = 45 if n = 4, and = 60 if n = 3 or 6. The symbol m, g, 1 are interpreted as in 3rd. No symbol in the third and fourth position indicates that the group contains no reflections or glide-reflections. The short forms of this four-symbol notation given above are internationally accepted (Lee, n.d.). Table 2 shows the wallpaper groups and their respective transformations.

Table 2. Wallpaper Groups and Transformation Transformations p1 No rotation, no reflection, no glide reflection p2 180o rotations, no reflection, no glide reflection pm or p1m No rotation, reflection axis at angle 90o to x-axis pg or p1g No rotation, no reflection, glide reflection pmm or p2mm 180o rotations, reflections in two directions, reflection axis at angle 90o and 180o to x- axis, all rotations centres are on the reflection axes pmg or p2mg 180o rotations, reflection axis at angle 90o to x-axis, glide reflection pgg or p2gg 180o rotations, no reflection, glide reflections cm or c1m No rotation, reflection axis at angle 90o to x-axis, glide reflection in an axis which is not a reflection axis cmm or c2mm 180o rotations, reflections axis at angle 90o and 180o to x-axis, not all rotations centres are on the reflection axes p4 90o rotations, no reflection p4m or p4mm 90o rotations, reflections, reflections in lines that intersect at 45o p4g or p4gm 90o rotations, reflections, no reflections in lines that intersect at 45o p3 120o rotations, no reflection p3m1 120o rotations, reflections, all rotations centres are on the reflection axes p31m 120o rotations, reflections, not all rotations centres are on the reflection axes p6 60o rotations, no reflection p6m or p6mm 60o rotations, reflections

Examples of all the 17 wallpaper groups can be seen in Alhambra, a Moorish palace in Granada, Spain (Montesinos Amilibia, 1987) and in the traditional Japanese patterns (Urabe, n.d.). A study done by Gerdes (2001) on the basket weaving by the Tonga weavers in Mozambique managed to identify eight of the seventeen wallpaper groups in the -plaited baskets.

Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021 3. Songket Pattern

The basic structure of songket and sampin songket (Fig.3) consists of  the ‘body’, the main part of the cloth  the ‘head’, the panel of the cloth  the ‘foot’ border and the ‘head’ border of the cloth  the ‘ground’ of the cloth, the background colour of the cloth.

body foot border head

head border body

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Fig. 3. Basic structure of sarong songket and sampin songket A piece of songket cloth reflects the life of the people; parts of the culture are transformed into songket motifs and patterns. Most of the traditional weavers live in the rural villages. They are close to nature and hence they can capture the beauty in their environment and incorporate them as motifs and patterns in their work as shown in Fig. 4 and Fig. 5. There are hundreds of motifs. Most of the traditional songket motifs are designed and named after the flora and fauna of their environment, the animals kingdom, the activities around the village, the moods and temperaments of nature, the world of beliefs and the cosmos and the Malay cakes and sweets.

Fig. 4. Shoots and the Corresponding Motif (Taken from Halimaton, et al., 2009)

Fig. 5. Corolla of the Persimmom Fruit and the Corresponding Motif

3.1. Symmetries and Mathematical Transformations in Songket Pattern

Songket patterns seem to be rich with symmetries, hence GeoGebra software was utilized to identify the symmetries embedded in the patterns of two sampin songket, named it as Songket 1 (Fig. 6) and Songket 2 (Fig. 7).

Fig. 6. Songket 1 Fig. 7. Songket 2

Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021 Most of the songket motifs in the two sampin songket are symmetrical figures; hence each of those motifs is mathematically identified as a design. To verify that claim, the thread count of each motif in the ‘border’, the ‘body’ and the ‘head’ of the two selected sampin songket is initially transformed into grid form on a graph paper. The grid form of the motif was then drawn in GeoGebra. An example of a design, the tampuk manggis motif in the ‘body’ pattern of Songket 1 and the corresponding grid form is shown in Fig. 8.

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Fig. 8. Tampuk Manggis Motif Transformed into Grid Form in GeoGebra

GeoGebra was used to identify the reflection line(s), the rotation angle(s) and the centre of rotation(s). This was done for all the motifs in the patterns of Songket 1. GeoGebra was then used to construct the repeating patterns by doing the translation in two directions on the basic designs (Fig. 9) for the ‘body’ patterns and translation in one direction for the ‘border’ and the ‘head’ patterns (Fig. 10 – Fig. 14). Once the songket patterns have been duplicated into grid form in GeoGebra, symmetries in the patterns were studied and the classification of the patterns according to the frieze pattern symmetry groups and the wallpaper symmetry groups was done.

Fig. 9. Songket 1 ‘Body’ Pattern Fig. 10. Songket 1 ‘Head’ Pattern

Fig. 11. Songket 1 ‘Foot Border” Pattern 1 Fig. 12. Songket 1 ‘Foot Border” Pattern 2

Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021 Fig. 13. Songket 1 ‘Foot Border” Pattern 3 Fig. 14. Songket 1 ‘Head Border” Pattern

For patterns with two directional translations in Fig. 9, the smallest rotations is 180 , there are reflections in two directions and all rotation centres are on reflection axes, therefore the ‘body’ patterns of Songket 1 falls under the pmm or p2mm wallpaper group. All the ‘border’ patterns and the ‘head’ pattern for Songket 1 could be reproduced in GeoGebra by doing one directional translation; therefore the patterns could be categorized under the frieze symmetry group. There exist both vertical reflection and horizontal reflection in the ‘foot border’ Pattern 1 (Fig. 11) and the ‘head’ patterns (Fig. 10), hence both patterns are of the pmm2 frieze group. For the ‘foot border’ Pattern 2 (Fig. 12), only translation could be identified, therefore it is of the p111 frieze group. In the ‘foot border’ Pattern 3 (Fig. 13) and the ‘head border’ (Fig. 14) only vertical reflection could be identified hence the songket patterns are categorized as the pm11 frieze group. Fig. 15 is the GeoGebra reproduction of the ‘body’ patterns of Songket 2 with the vertical, horizontal and glide reflection lines and the centres of 180 rotations shown. Since not all rotations centres are on reflection axes, the 8 Abdul Aziz; Embong/ Spectrum (Educational Research Service),ISSN:0740-7874(2017)101_111

pattern is of the cmm wallpaper group. The repetitive patterns in Fig. 16 – Fig. 19, the ‘border’ and ‘head’ patterns of Songket 2 have only one directional translation. The symmetries in the ‘head’ (Fig. 19) and the ‘foot border’ Pattern 1 (Fig. 16) are consistent with the properties of the pmm2 frieze pattern group. Only vertical reflection could be detected in the ‘foot border’ Pattern 2 (Fig. 17) and the “head’ border pattern (Fig. 18), therefore both patterns fall under the pm11 frieze pattern group.

---- reflection axes ----- glide reflection axes 180◦ rotations centre on reflection axes 180◦ rotations centre not on reflection axes

Fig. 15. Songket 2 ‘Body’ Pattern

Fig. 16. Songket 2 ‘Foot Border’ Pattern 1 Fig. 17. Songket 2 ‘Foot Border’ Pattern 2

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Fig. 18. Songket 2 ‘Head Border’ Pattern Fig. 19. Songket 2 ‘Head’ Pattern

Table 3 shows the classification of the symmetries groups for Songket 1 and Songket 2.

Table 3. Symmetry Group Wallpaper Group Frieze Group Songket 1 ‘body’ patterns pmm/p2mm ‘head’ patterns pmm2 ‘border’ patterns pmm2 pm11 p111 Abdul Aziz; Embong/ Spectrum (Educational Research Service),ISSN:0740-7874(2017)101_111 9

Songket 2 ‘body’ patterns cmm ‘head’ patterns pmm2 ‘border’ patterns pmm2 pm11

This classification verifies that the songket patterns produced by the non-mathematician Malay weavers could indeed be categorized under the universally accepted Euclidean 2-space mathematical patterns categorization.

4. Songket Pattern Creation

This section illustrates on how those identified mathematical transformation concepts namely symmetries, reflections, glide reflection, rotations, and translations can be used to create new songket patterns. To preserve the Malay philosophy and heritage and to cater the need of the traditional songket weavers who prefer the transverse symmetry concept of weaving, symmetrical traditional motifs are used as the basic designs. As an illustration, the traditional bunga lawang (star anise spice) motif is chosen. Fig. 20 shows the bunga lawang motif with all its reflection axes and its 900 rotations centre.

Fig. 20. Bunga Lawang Motif

Two different ‘body’ patterns were created using different combinations of mathematical transformations on the bunga lawang motif. In creating Pattern 1 (Fig. 21), the smaller motif (pink) was obtained by doing the dilation of the original bunga lawang motif (red) by 0.5 units, and then translation in two directions of the motifs was repeatedly done. For Pattern 2 in Fig. 22, the bunga lawang motif (brown) is still used as the basic design, the smaller motif (yellow) is 0.7 unit dilations and 450 rotations of the original motif, the two motifs was then translated in two directions repeatedly.

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Fig. 21. Pattern 1 Fig. 22. Pattern 2

The patterns created have shown that by using just one basic motif, two different and beautiful patterns has been created. Hence, by having some creativity and utilizing other different combinations of transformations on just the bunga lawang motif many more beautiful songket patterns with this traditional motif could be generated. Pattern 3 in Fig. 24 was designed using combination of three traditional motifs; the bunga kenanga motif, the 10 Abdul Aziz; Embong/ Spectrum (Educational Research Service),ISSN:0740-7874(2017)101_111

daun inai motif and the cabit motif (Fig. 23) and the mathematical transformation concepts applied were rotations, glide reflections and translations.

Bunga Kenaga Motif Daun inai motif Cabit Motif

Fig. 23. Three Traditional Motif

Fig. 24. Pattern 3

Patterns generated using mathematics will have more precise and accurate thread count. Hence, error and wastage can be reduced. This idea of using mathematics in creating songket patterns could lead to collaboration among weavers, mathematicians and programmers to revive and enhance the songket industry. If the songket designers, weavers and mathematicians were to work together in creating new patterns, the weavers will be able to give idea and assist the mathematicians in incorporating the cultural and the philosophical elements into the patterns. Thus, limitless number of beautiful songket patterns that still preserve the traditional elements could be created by utilizing and combining the skill, creative and artistic value of the weavers, with the mathematical ideas from the mathematicians. Those mathematicians could then work together with the computer programmers to produce algorithms, computer programs or software that could be used to generate new songket patterns or to develop machine that could automate the songket weaving process. This will help reduce the production cost and the songket price for mass production market sector of the songket industry.

5. Songket and school curriculum

For majority of students, mathematics is considered as a difficult and uninteresting subject (Rokiah, 1998). The idea of integrating mathematics, art, philosophy and one’s culture, using the Universal Integrated Approach as recommended by Nik Azis (2009), Mat Rofa & Habsah (2007) and Rokiah (1998) could act as a catalyst in students’ understanding of some mathematics concepts. Adding something that they can relate to, components of their culture, Downloaded from erss.europeansp.org at 0:24 +0330 on Monday October 4th 2021 with the mathematics concepts that they have to learn might make learning mathematics more interesting (Rokiah, 2011). For example, instead of using the regular geometrical shapes such as circle, triangle and rectangle, the traditional songket motifs, which are rich with symmetrical concepts, could be used to explain the concepts of symmetry, reflection, rotations and dilation. The application of the isomorphic and non-isomorphic transformation and translation concepts could be explained by doing the songket patterns creation activity. As mentioned by Vogel (2005), activities on patterns formations also contain components of mathematical learning. This combination of mathematics, art and culture could help teachers to integrate concrete, abstract ideas and creative elements in the classroom. It would help in making lessons more interesting, enhance students’ understanding of the subject matter, increase their creativity and make them to be more appreciative of their culture.

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References

Ascher, M., & D'Ambrosio, U. (1994, June). Ethnomathematics: a Dialogue. For the Learning of Mathematics, 14(2), 36-43. Barkley, C. A. (1998, May). Symmetry Patterns of Ute Beadwork. ISGEm Newsletter, 13(2). Bier, C. (2000). Choices and Constraints: Pattern Formation in Oriental Carpet. Forma, 15, 127-132. Conrad, K. (n.d.). Why is group theory important? Retrieved January 27, 2016, from http://www.math.uconn.edu/~kconrad/math216/whygroups.html Gerdes, P. (2001). Exploiring Plaited Plane Patterns among the Tonga in Inhambane (Mozambique). [Special Issue of Symmetry: Culture and Science]. Symmetry in Ethnomathematics, 12(1-2), 115-126. Keeney, R. (n.d.). Linear Transformations in Works of Art. Retrieved january 28, 2010, from http://www.unm.edu/~abqteach/math2002/02-02- 05.htm Lee, X. (n.d.). The 17 Wallpaper Groups. Retrieved October 27, 2014, from http://xahlee.info/Wallpaper_dir/c5_17WallpaperGroups.html Mat Rofa, I., & Habsah, I. (2007). Philosophy of Mathematics: A Case Study in Malaysia. International Conference on Mathematical Sciences 2007. Universiti Kebangsaan Malaysia. Montesinos Amilibia, J. M. (1987). Classical tessellations and three-manifolds. Berlin: Springer-Verlag. Nik Azis, N. P. (2009). Nilai dan Etika dalam Pendidikan Matematik. Kuala Lumpur: Universiti Malaya. Nishimoto, K., & Berken, B. (1996, November). Symmetry Patterns of the Wisconsin Woodland Indians. International Study Group on Ethnomathematics (ISGEm) Newsletter, 12(1). Retrieved January 1, 2010, from http://www.ethnomath.org Nor Maizan, A. A. (2016). An ethnomathematical case study on the mathematical concepts, mathematical practices and beliefs of the Malay songket weavers. Unpublished doctoral dissertation. Universiti Teknologi MARA, Malaysia. Norwani, M. N. (2002). Songket Malaysia. Kuala Lumpur: Dewan Bahasa dan Pustaka. Pinter, C. C. (1990). A Book of Abstract Algebra second Edition. New York: Dover Publication. Rokiah, E. (1998). Pengajaran Matematik Pensyarah ITM. Unpublished doctoral dissertation. Kuala Lumpur: Universiti Malaya. Rokiah, E. (2011). Membudayakan Etnomatematik dalam Pembelajaran & Pembelajaran. Simposium Kebangsaan Sains Matematik ke-19 (SKSM 19). UiTM Cawangan Pulau Pinang. Stylianou, D. A., & Grzegorczyk, I. (2005, Mar). Symmetry in Mathematics and Art: An Exploration of an Art Venue for Mathematics Learning. Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 15(1), 30-44. Urabe, T. (n.d.). Geometry of Wallpaper Patterns. Retrieved October 27, 2014, from Tohsuke Urabe Mathematics Laboratory, Ibaraki University, Japan: http://faculty.ms.u-tokyo.ac.jp/users/urabe/pattrn/PatternE.html Vogel, R. (2005). Patterns- a fundamental idea of mathematical thinking and learning. ZDM, 37(5), 445-449. Washburn, D. K., & Crowe, D. W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Seattle and London: University of Washington Press. Washburn, D. K., & Crowe, D. W. (2004). Symmetry Comes of Age: The Role of Pattern in Culture. Seattle & London: University of Washington Press. Yayasan Tuanku Nur Zahirah. (2008). Songket Revolution. Kuala Lumpur: Yayasan Tuanku Nur Zahirah. Zakiah, H. (1987). Malaysia Kita: Tradisi dan Budaya. , Kuala Lumpur: Times Book International.

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