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

Abstract

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 kolam sf or 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 kolam sf or ag ivenN 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. Ke yw ords: Kolam, Art, South India, Topology, Homotopy 1. What is a Kolam? Figure 1 depicts an example of a kolam ,a nancient and still popular South Indian art form. This particular type of kolam is called the pulli kolam in Tamil, which consists of a series of dots (called pullis) placed on a surface, each of which is then circumscribed by lines that form closed orbits. It is a very common sight on the threshold of homes in the five southern states with a combined current population of ∼252 million. They are called by varied names in the respective regional languages of these states: kolam in Tamil spoken in Tamil Nadu, golam in Malayalam spoken in Kerela, rangole in Kannada spoken in Karnataka, and muggulu in Telugu spoken in Andhra Pradesh and Telangana. With every sunrise, women wash the floor in front of the houses, and using rice flour, place the dots and draw a kolam largely from memory. Learning how to draw kolams from an early age is an important aspect of growing up in southern India, especially for girls. As they continue to learn from other women in their family, the kolams become increasingly complex, with a larger number of dots and more intricate line orbits. Remembering the dot configurations and line orbits is a daily exercise in geometric thinking. The process is immensely pleasurable, especially when a kolam is successfully completed with no loose ends. While the conventional kolam si mpose several rules, here we begin with three simple rules in order to give ourselves greater room for discovery and creativity. Given an arbi