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

Atanu Manna, Suvra Kanti Chakraborty

TL;DR

This work addresses extending Hridaya Kolam designs to non-coprime dot-arm configurations by introducing a modular-arithmetic framework. It defines residue-class sequences $a_k^{(r)} \equiv (kn+r) \bmod m$ across $d=\gcd(m,n)$ residues, producing $d$ disjoint loops whose structure depends on the coprimality of $m/d$ and $n$. A clear completeness criterion is established: all $m\times n$ dot-arm positions are visited exactly once if $\gcd\left(\frac{m}{d},n\right)=1$; otherwise the pattern partially covers the space, yielding richer but non-exhaustive motifs. The paper provides an algorithm and illustrative examples, enabling systematic construction of non-coprime Hridaya Kolams with potential architectural and decorative applications.

Abstract

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.

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

TL;DR

This work addresses extending Hridaya Kolam designs to non-coprime dot-arm configurations by introducing a modular-arithmetic framework. It defines residue-class sequences across residues, producing disjoint loops whose structure depends on the coprimality of and . A clear completeness criterion is established: all dot-arm positions are visited exactly once if ; otherwise the pattern partially covers the space, yielding richer but non-exhaustive motifs. The paper provides an algorithm and illustrative examples, enabling systematic construction of non-coprime Hridaya Kolams with potential architectural and decorative applications.

Abstract

This paper extends Hridaya Kolam patterns to cases where the number of dots () and arms () are not coprime, i.e., . 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 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.
Paper Structure (10 sections, 2 theorems, 18 equations, 17 figures, 1 algorithm)

This paper contains 10 sections, 2 theorems, 18 equations, 17 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Sequences Generated by Modular Arithmetic
  3. Defining generating sequences
  4. Multi-Loop Kolam Structures via Modular Sequences
  5. Algorithm to Generate Modular Sequences and Draw Multiple Loops
  6. Generated sequences for various values of $m$ and $n$
  7. Multiple loop visualization
  8. Exhaustive Coverage of Dot-Arm Structures
  9. Partial (Non-Exhaustive) Coverage of Dot–Arm Structures
  10. Conclusion

Key Result

Proposition 1

Let $m,n$ be positive integers and let $d=\gcd(m,n)>1$. For each residue $r\in\{0,1,\dots,d-1\}$ define the sequence where residues modulo $m$ are represented by $\{0,1,\dots,m-1\}$. Then the $m$ integers $0,1,\dots,m-1$ are divided into exactly $d$ disjoint repeating sequences of this form. Hence, the number of disjoint loops produced by the construction equals $d=\gcd(m,n)$.

Figures (17)

  • Figure : Figure: $m=4, n=6$
  • Figure : Figure: $m=6, n=4$
  • Figure :
  • Figure :
  • Figure :
  • ...and 12 more figures

Theorems & Definitions (9)

  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Example 3