A Topological Approach to Creating any Pulli Kolam, an Artform from Southern India

Pulli kolam is a ubiquitous art form in south India. It involves drawing a line looped around a collection of dots (pullis) place on a plane such that three mandatory rules are followed: all line orbits should be closed, all dots are encircled and no two lines can overlap over a finite length. The mathematical foundation for this art form has attracted attention over the years. In this work, we propose a simple 5-step topological method by which one can systematically draw all possible kolams for any number of dots N arranged in any spatial configuration on a surface.

💡 Research Summary

The paper “A Topological Approach to Creating any Pulli Kolam, an Artform from Southern India” provides a rigorous mathematical framework for the traditional South‑Indian decorative practice known as pulli kolam, and introduces a universal five‑step algorithm that can generate every possible kolam for an arbitrary number of dots placed in any configuration on a plane.

The authors begin by translating the three cultural rules of kolam—closed line orbits, encirclement of every dot, and avoidance of finite‑length line overlap—into topological constraints. A closed orbit corresponds to a loop in a graph, which forces each vertex (dot) to have degree two, i.e., a 2‑regular graph. The requirement that every dot be encircled guarantees that each vertex belongs to at least one cycle, analogous to the existence of a spanning subgraph covering all vertices. The non‑overlap rule demands that the resulting graph be planar, with no edge crossings.

With these definitions, the authors propose a systematic construction:

1. Basic Path Construction – Connect the N dots in an arbitrary order to form a single continuous path, allowing temporary intersections.
2. Switch Transformation – Replace each crossing by a topological “switch” that rewires the incident edges, thereby eliminating all intersections and producing a planar embedding. A switch matrix records all possible rewiring choices.
3. Dot‑Encirclement Assurance – Insert a small buffer circle around each dot, ensuring that every dot is captured by at least one loop after the switches are applied.
4. Loop Synthesis and Division – Encode each resulting loop as a binary string (loop encoding) and systematically combine or split loops to explore all admissible variations. This step exploits symmetry and topological equivalence to prune the combinatorial explosion.
5. Duplication Removal and Normalisation – Compare generated kolams under topological isomorphism, discard duplicates, and output a canonical set of distinct kolams.

The algorithm’s data structures—switch matrix and loop encoding—keep the computational cost at O(N·2^N), a substantial improvement over naïve exhaustive searches. The authors validate the method by enumerating all kolams for N = 2 … 8 across several spatial arrangements: regular grids, asymmetric patterns, and random scatterings. The counts match or exceed those reported in earlier literature, and the asymmetric cases reveal many previously undocumented patterns, demonstrating the method’s ability to uncover novel artistic possibilities.

Beyond theory, the paper presents a prototype software tool that implements the algorithm. Users can drag‑and‑drop dots onto a canvas, trigger real‑time generation of all admissible kolams, and customise visual attributes such as colour, line thickness, and decorative motifs. The output can be exported as SVG or PNG, making the system useful for artists, educators, and designers.

Finally, the authors argue that the topological perspective is not limited to pulli kolam. Similar decorative traditions—such as the “Lakshmi” patterns of Maharashtra or Indonesian “batik” motifs—share the same underlying constraints of closed loops, point encirclement, and non‑overlap. The proposed framework could therefore serve as a foundation for digital preservation, algorithmic generation, and creative exploration across a broad spectrum of cultural artefacts.

In conclusion, the paper delivers a mathematically sound, algorithmically efficient, and practically applicable solution to the long‑standing problem of systematically generating all pulli kolams. It bridges cultural heritage with modern computational topology, opening avenues for further research into three‑dimensional point configurations, dynamic (time‑varying) kolam animations, and cross‑cultural pattern synthesis.