Measuring Symbolic Sophistication in Geometric Ornament: The Symmetry Complexity Index (SCI) and its Art Historical Validation in Islamic Geometric Patterns

This study introduces the Symmetry Complexity Index (SCI), a computational metric to quantify the symbolic sophistication of Islamic geometric patterns (IGPs). Grounded in group theory, the SCI assesses symmetry complexity through weighted operations, hypothesizing that higher scores reflect greater symbolic depth. We analyzed 41 IGPs from key Islamic dynasties (Umayyad to Mughal, 660–1737 CE) and validated the SCI through expert consultation (n = 5, r = 0.87, p < 0.001). The results revealed a clear evolutionary trend, with SCI scores rising from basic 6-point patterns (SCI = 40–50) in the Abbasid era to complex 16-point patterns (SCI = 70–85) during the Mamluk period, alongside notable regional variations, such as Mughal simplicity and Mamluk intricacy. The findings of this study highlight the SCI’s effectiveness in capturing historical developments in symbolic sophistication. This study recommends the broader application of the SCI for analyzing geometric ornamentation and exploring the relationship between decorative complexity and urban identity in Islamic architecture.