<![CDATA[Ethnomathematics research has largely examined static artifacts such as buildings, ornaments, and textiles, while patterned human movement has received little attention. This study analyzed the spatial-geometric and kinematic structure of tawaf and saʿi, the two movement-based pillars of umrah, using the Extended Theory of Mathematical Connections (ETMC) within an emic–etic–dialogical (ethnomodelling) framework. Official manasik guidebooks, scholarly texts, and publicly available visual records were analyzed through document and visual content analysis in three phases: emic description of the prescribed rules, etic geometric and kinematic modelling, and dialogical coding of the connections between them. Tawaf was modelled as a positively oriented closed path with winding number seven around a non-circular convex obstacle formed by the Kaʿbah and Ḥijr Ismāʿīl; by Steiner’s formula, the length of each circuit grows linearly with distance from the Kaʿbah, L(r) = PK + 2πr, regardless of the building’s exact shape. Saʿi was modelled as a reversible open path in which the ritual rule of ending at Marwah follows from the parity of seven one-way passages. The shared emic counting unit, shawṭ, denotes a full period in tawaf but a half period in saʿi. Raml and harwalah add piecewise-uniform motion that supports graphs of motion, distance–displacement reasoning, and an optimization model. All three ETMC categories and all seven intra-mathematical connection types were identified. The study extends ethnomathematics to collective ritual movement, offers a replicable procedure for grounding models in emic rules, and proposes a learning trajectory for geometry education in Islamic schools.]]>
Copyrights © 2026
