A relation between tudung saji weaving patterns and group theory

Abstract

Tudung saji is a traditional utensil used by the Malays to cover their food to be served. Tudung saji is woven with strands of dried leaves using a specific technique called triaxial weave, where the strands are plaited in three directions. Previously, a tool known as triaxial template had been created to represent the patterns of tudung saji weaving in a planar pattern. Based on this template, many beautiful and symmetrical patterns were successfully generated, creating some of the original patterns of tudung saji. These patterns are categorized according to the number of colours of the strands, from the basic 2-strand up to 6-strand template. The purpose of study is to find several finite groups to represent the triaxial weaving patterns on two dimensional templates, focussing only on the 2 and 3-strand templates. It is found that the symmetric group of two letters, 2 S and the cyclic group of order six, 6 C are isomorphic to the triaxial template of Flock of Pigeons and Sailboats patterns, respectively. These isomorphisms are determined by mapping the elements of the Flock of Pigeons and Sailboats onto the elements of the two groups. Using a software iMac Grapher, several graphs are generated based on the elements of the triaxial template patterns. Next, graph theory is used to analyze the properties of these graphs. The graphs are sorted by the numbers of strands, namely the graphs of block two, graphs of block three up to the graphs of block six. All such graphs are found to feature the characteristics of three types of graphs, namely a complete graph with three vertices, 3 K , a simple graph with six vertices, and an a cyclic graph. Lastly, this research reports on modifications to the template by adding extra colours to the framework strands and the insertion strands. This is done by using new colour ordering in addition to the same colour ordering for the 2-strand template. As a result, a new characteristic on the modified template has been found, namely the existence of different triaxial patterns in one template.