Mathematics of kolam: folkloric graph theory
We are happy to present you our with with Shanthi Chandrasekar about folkloric graph theory of kolams. This collaboration started during SciArt mixer and is in process since then.
Section 1: Language of Kolam
Kolam is a language, graphically encoded and spiritually rich in traditions [1,2]. Kolam drawings are drawn by women in the South Indian state of Tamil Nadu, on the floor at the entrances to their houses everyday over centuries using rice flour, chalk, chalk powder and many other materials. Through kolam drawings one could convey complicated messages and emotions, even though the original message and language of kolam drawings may have been forgotten. Kolam drawings are steeped in philosophy and life lessons. Though these deep messages are inherent in the drawings and the process, they might not be obvious to the kolam artist or the viewer.
Kolam drawings have deep connections with other folkloric types of drawings from other places around the globe: sona drawings, rangoli (in Northern India), celtic drawings, and malekula [1,3]. Interestingly, kolam drawings are quite different by the set of rules from other similar drawings. For instance, from sona or malekula drawings: in pulli kolam (comes from the word “pulli”, a dot) the drawing has a continuously closed curve that partitions the planar space into as many bounded regions as there are dots, so that each bounded region contains exactly one dot (as in the drawing below).
One of the traditions connected to kolams is that this art form from the south of India involves geometric patterns made using dots and lines without any tools, but drawn freehand. Originally the material used for this was rice flour called “kolam”, which gives a name to the drawing itself. Another more general name of kolam in the literature is just “sand drawings”.
What is especially interesting for us from a mathematical point of view is that the geometric patterns that constitute the kolam and manner of constructing them vary from one geographical region to another [3]: “a kolam could be made up of a single, unsegmented, closed thread of line or it could be made up of superimposition of two or more closed threads of lines, each constituting one component of the global kolam pattern.”
These limitations of kolam drawings also provide interesting questions of their mathematical properties.
While various graph theoretical properties of kolam and sona drawing were studied [1,2], such as the number of possible combinations of different sona drawings and other combinatorial properties of such drawing, many other properties of kolam remain uncovered. To add to this, there are numerous physical and psychological properties of kolam which have not been studied or observed.
In our study we will particularly talk about:
- encoding kolams using mathematical objects of graph theory;
- fractal properties of kolam drawings;
- list some links between kolams, tilings and knot theory;
Section 2: Encoding kolam with graphs
Kolam’s complexity can reach really high scales, therefore one can either come to some sort of classification scheme or to learn how to encode kolams into some simpler objects.
Therefore we focus on two main objects here: kolam graphs and kolam maps. We define kolam maps as follows.
Def. Kolam map is a mathematical object of a graph, denoted here by letters Gm. Every kolam map Gm corresponds to a kolam, which we denote here by a letter G. Kolam map Gm consists of nodes and edges. Nodes of kolam map Gm are dots, around which the initial kolam G is drawn. Edges of kolam map Gm are defined between each two nodes of kolam map if two closed kolam cells are neighboring cells of kolam, see Figure below. Therefore instead of a complex object of kolam we get its simpler representation as a kolam map graph: G → Gm.
We note here that the same procedure of mapping complex drawings into simpler forms of mathematical graphs can be defined for sona drawings [1].
In our study we will talk about geometrical properties of yet another object from graph theory [4], such as dual graphs, which turns out to have beautiful connections with kolam drawings. Dual graphs are mathematical objects, studied in the mathematical disciplines of graph theory and topology [2]. Below we describe a non-trivial bridge between kolam drawings and dual graphs. Dual graphs often allow us to look into properties of a graph from another point of view, hence kolams, additionally to educational components [3], shed light on the mathematical properties of graphs themselves.
Kolams can be drawn given a set of nodes. Depending on the locations and symmetry properties of the dots set we get into different properties of kolam as well [6]. For example, given N nodes, where N dots form a set of square points we can generally get kolam of different nature and of different form.
How many kolams are there?
We can map each kolam to kolam map, which is a graph, then by counting number of various graphs — kolam maps, we can get the number of different kolams. Using simple combinatorial techniques one can simply get enumeration of all graphs on N points, as it was done in Harary [4]. Such classification of kolam map graphs therefore also allows to classify kolam drawings. Below we show one of the possible classification of all tree graphs on N nodes, where N=[1,8], demonstrated by Harary.
In kolam drawings one can also use adjacent kolam properties for kolam classifications, such as a. which type of symmetry does kolam drawing obeys: mirror, rotational or translational symmetries or a combination of them, b.what is the way of connecting the dots of kolam.
The main idea of kolam maps is taken from the book of Harary on graph theory, where he introduced the concept of intersection graphs. The notion of graph of intersections relies on a simple idea of representation of any intersections of sets as a graph. There is a theorem, which states the following.
Theorem: every graph is an intersection graph.
Proof. For each point v of G, let St be the union of points {v} with the set of lines incident with r. Then it is immediate that G is isomorphic with Q(F) where F = (SJ).
This also proves, that any kolam can be represented as an intersection graph and therefore there can be kolam map Gm found (which is an intersection graph of kolam).
Section 3: Other connections
One may say about kolam drawings (as also about sand drawings in general) that they are related to the origins when mathematical studies and art were having much stronger interrelations and when one could not distinguish between traditions of making art and making mathematical calculations. Remember examples when Greek philosophers were making geometrical drawings and finding some geometrical proportions, which were related to some fundamental constants such as value of pi. Also the value of pi was calculated by various other early civilizations including India in order to understand circles.
Connection between Kolams and Harary graphs
In the mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges (wikipedia source). It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph. For kolam maps there is an important connection with graphs, which are planar.
References
[1] Eric Demaine article https://erikdemaine.org/papers/Sona_BRIDGES2006/paper.pdf
[2] On Kolam and K12 education https://condor.depaul.edu/ppereira/courses/geo/pagany.pdf
[3] Kolam drawings https://www.cmi.ac.in/gift/Kolam.htm
[4] F. Harary, “Graph theory”
[5] Kolam website https://www.shanthic.com/
