This paper presents an ethnomathematical analysis of the pattern generation system in Kanjeevaram silk Sarees, a major cultural artifact of Tamil Nadu. We propose that intricate, highly structured buttas (motifs) and pallu (end-piece) designs are not merely artistic expressions but the output of a sophisticated, rule-based geometric system. This system can be formally modeled as a one-dimensional cellular automaton (CA) for border patterns and a two-dimensional CA for the pallu. Through computational implementation and comparative analysis with traditional sarees, we demonstrate that characteristic patterns emerge from simple local rules governing thread placement. By analyzing historical design ledgers and collaborating with master weavers in Kanchipuram, we have codified these rules into a computational model that achieves 89% pattern accuracy compared to traditional specimens. This work provides a formal framework for preserving the design logic of Kanjeevaram Sarees, demonstrates the inherent computational thinking in this traditional craft, and offers a novel tool for both cultural heritage preservation and contemporary design innovation.