Chronological Advancement of Symmetry Groups in Seljuk Geometric Patterns: A Computational Analysis

Zeynab Nazer, Péter Rabb

Abstract


This study investigates the evolution of symmetry groups across three distinct phases of Seljuk architecture revealing a compelling transition of geometric patterns from structural pragmatism to cosmological allegory. Integrating computational symmetry analysis of pattern symmetries with cultural historiography, and applying four novel, quantitatively validated metrics. We decode Seljuk architects’ synthesis of mathematical rigor, material innovation, and spiritual symbolism. These novel metrics: Tiling Efficiency (TE), Ornament Density (OD), Symmetry Complexity Index (SCI), and Chronological Advancement Score (CAS), quantify this evolutionary progression, providing a replicable framework for mapping geometric evolution in Islamic architecture. The Early phase (428–455 AH / 1037–1063 CE), exemplified by Gunbad-i Qabus Monument, prioritized p4 tetragonal symmetry with austere brickwork reflecting Turkic asceticism. The Middle phase (455–493 AH / 1063–1100 CE) marked a pivotal turning point, evident in the Kharāqan Twin Towers’ dynamic p4g glide reflections and hybrid stucco-brick motifs encoding Sufi allegories. By the Late phase (493–590 AH / 1100–1194 CE), structures like the Mausoleum of Mu’mina Khatun epitomized this geometric zenith. Its intricate geometric patterns executed in turquoise-glazed tiles, representing celestial sophistication through p10 decagonal symmetry and turquoise-glazed tessellations (TE = 92%, OD = 280–320 joints/m², CAS = 2750), embodying Neoplatonic unity. A key contribution to demonstrating this evolutionary progression is the reclassification of the NNW tympanum panel (Kharāqan West Gonbad) to p4g symmetry, resolving ambiguities through detailed symmetry analysis with computational tools and visual verification. Material analysis reveals the brick’s structural resilience compared with stucco’s vulnerability, with degradation reducing TE by 10–20% in Middle-phase monuments. Expert validation of metrics confirms their relevance to architectural and art historical interpretation. Bridging computational rigor and art-historical scholarship, this work repositions Seljuk geometry as a dynamic mediator of science, spirituality, and cultural exchange, offering a validated model for analyzing architectural evolution globally.

Keywords


Symmetry Groups; Seljuk Architecture; Computational Metrics; Geometric Evolution; Cosmological Symbolism; Cultural Historiography

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DOI: https://doi.org/10.18860/jia.v9i1.32568

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