标题： 扩展Hridaya Kolam到偶数阶点图案及其应用 显示英文标题

标题： Extending Hridaya Kolam to Even-Ordered Dot Patterns and Their Applications

摘要： 这项研究通过将模运算应用于边数为偶数的点排列，且臂数与点数互质，扩展了Hridaya Kolam图案的数学框架。 我们分析了对应的循环序列，这些序列对应于欧拉回路，从而实现了超越经典奇数阶情况的连续单笔Kolam设计。 我们的方法提供了构造这些复杂图案的明确算法，揭示了新的对称性和结构特性。 提升这种传统地面艺术，我们将这些基于数学的图案转化为引人注目的设计，在地毯和纺织行业中的当代Dari艺术中展示了它们的美丽和复杂性。 显示英文摘要

摘要： This study extends the mathematical framework of Hridaya Kolam patterns by applying modular arithmetic to even-ordered dot arrangements with arm counts co-prime to the number of dots. We analyze the resulting cyclic sequences that correspond to Eulerian circuits, enabling continuous single-stroke kolam designs beyond the classical odd-ordered cases. Our method provides explicit algorithms for constructing these intricate patterns, unveiling new symmetries and structural properties. Elevating this traditional floor art, we translate these mathematically grounded motifs into striking designs, showcasing their beauty and complexity in contemporary dari art in the carpet and textile sectors.