A Topological Approach to Creating Any Pulli Kolam, an Artform from South India

Pulli kolam is a ubiquitous art form drawn afresh every morning on the threshold of most homes in South India. It involves drawing a line looped around each dot of a collection of dots (pullis) placed on a plane in accordance with three mandatory rules, namely, all dots should be circumscribed, all line orbits should be closed, and two line segments cannot 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 method by which one can systematically draw all possible kolams for any number of dots N arranged in any spatial configuration on a surface. For a given N, there exist a set of parent kolams from which all other possible kolams can be derived. All parent kolams arising from different spatial arrangements of N dots can be classified into parent kolam types; within each type, all parents are topologically equivalent, or homotopic. The number of kolams for agiven N is shown to be infinite if only the three mandatory rules stated above are followed; it becomes finite as more optional rules and restrictions are imposed. This intuitive method can be mastered by anyone to create countless kolams with no prior knowledge or the need for a detailed mathematical understanding. It is also amenable to developing apps and educational games that introduce the concepts of symmetry and topology.