标题： 将Hridaya Kolam扩展到多环：非互质点-臂结构的研究 显示英文标题

标题： Extending Hridaya Kolam to Multiple Loops: A Study of Non-Coprime Dot--Arm Structures

摘要： 本文将Hridaya Kolam图案扩展到点数($m$)和臂数($n$)不互质的情况，即$\gcd(m, n) \ne 1$。这些配置会产生多个不相交的闭合环路。我们提出了一种基于模运算的算法，以系统地生成此类图案，并提供了各种非互质$(m, n)$对的示例图案，以展示所产生的多环结构。这些多环Kolam设计可以激发平面图、立面和装饰元素中的建筑图案和装饰图案。 显示英文摘要

摘要： This paper extends Hridaya Kolam patterns to cases where the number of dots ($m$) and arms ($n$) are not coprime, i.e., $\gcd(m, n) \ne 1$. Such configurations give rise to multiple disjoint closed loops. We propose a modular-arithmetic-based algorithm to systematically generate such patterns, and illustrative patterns for various non-coprime $(m, n)$ pairs are provided to demonstrate the resulting multi-loop structures. These multi-loop Kolam designs can inspire architectural motifs and ornamental patterns in floor plans, facades, and decorative elements.