Symmetry groups and kinetic potential of Tash Khauli Palace patterns

This study explores the profound relationship between Islamic geometric patterns, symmetry groups, and kinetic systems, using patterns from Tash Khauli Palace in Uzbekistan, situated in the UNESCO-designated Itchan Kala. The investigation employs a parametric model with Rhinoceros® and Grasshopper software to analyze the kinetic potential within the symmetry group structures of these patterns. The research involves decomposing geometric shapes, identifying symmetry groups, and scrutinizing parameterized kinetic movements. Seventeen two-dimensional repetitive patterns—classified under what are commonly referred to as "wallpaper groups" in mathematical symmetry theory—are categorized, revealing a notable association between the P6M symmetry group —a classification featuring six-fold rotational and mirror symmetry—and heightened kinetic potential. This leads to the research question: 'Which symmetry groups possess the capacity for transformation into kinetic motion?' The findings emphasize the specific relevance of these patterns for design applications and contribute insights into the historical and architectural significance of Tash Khauli Palace. The study also demonstrates how kinetic exploration can enrich design possibilities in geometric patterns.