In keeping with other developments in the natural sciences, more and more researchers use mathematics to analyze the geometric and constructive principles hidden in works of art and then apply these principles to generate new patterns. This thesis is in line with trend. It develops tools for the analysis of the geometric patterns hidden in Bakuba textiles and Zillij mosaics. As such, it contributes to the interdisciplinary Generative Design project, which explores the mathematical structure of artifacts using groups, shape grammars, and other techniques. The thesis consists of six parts: (1) A brief introduction to background of the thesis. After introducing the project and geometric patterns in Bakuba textiles and Zillij mosaics, we discuss the structure of the Generative Design systems, discuss results the project team has obtained, and explore the possibility of applying groups to implement our goals. (2) An introduction to wallpaper groups, related theorems and results. After introducing the Crystallographic Restriction theorem, we provide a proof that establishes that there are precisely seventeen groups for creating wallpaper patterns from given geometric patterns. To understand these seventeen groups and to explain the relationship between groups and Bakuba textiles and Zillij mosaics, we illustrate these groups using motifs extracted from Bakuba and Zillij. (3) An illustration of the mathematical reconstruction of Bakuba and Zillij motifs using groups. After analyzing patterns in Bakuba Textiles and Zillij Mosaic, we show that we can use certain groups to reconstruct patterns with different motifs involving various parameters. This brings us closer to our goal. The groups considered at this stage are not necessarily wallpaper groups. (4) An illustration of the mathematical reconstruction of Bakuba and Zillij motifs using grammars. In this part, we use shape grammars defined by Stiny to reconstruct patterns in Bakuba and Zillij by defining suitable grammars and applying appropriate sequences of shape grammar rules. (5) A new method of creating new pattern. In this section, we compare and discuss the methods of group theory and shape grammars to extract motifs from classical artifacts and to generate new patterns. Then we develop a new method that enhances the two classical methods by generating new patterns from Bakuba and Zillij motifs using wallpaper groups. (6) The thesis ends with appendices in which technical definitions and basic proofs are given to make the thesis self-contained. In particular, the proof that there are precisely seventeen wallpaper groups is presented in detail for completeness and accessibility.
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