Application of combination concept and two-dimensional crystallographic groups in the generation of woven motifs

Abstract

Woven has various geometric motifs. Geometric motifs with square and rectangular elements are often found in the basic pattern of woven motifs. Mathematically, the repetition of patterns in woven is a transformation consisting of shifting, rotation, and reflection. Repeated and symmetrical patterns in the plane, which are formed from the transformation, belong to the two-dimensional crystallographic group. This study aims to generate woven motifs using the basic pattern of binary images. This basic pattern is generated using the concept of binary numbers which are then combined to form a number of binary images of size k x k. All of these basic patterns are then transformed into various motifs using the concept of two-dimensional crystallographic groups. The number of archetypes produced for k=2 is 6 basic patterns and for k=3 as many as 56 basic patterns. The two-dimensional crystallographic group used is a group that uses 11 basic square patterns, namely p1, p2, pm, cm, cmm, pg, pgg, pmg, pmm, p4, and p4g. The number of motifs for k=2 is 66 and for k=3 as many as 616. The woven motifs produced have many similarities with each other so that selection is necessary. The final selection results for k=2 obtained 32 unique motifs and for k=3 obtained 408 unique motifs.