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ABSTRACT
Kolam is a form of traditional Indian folk art that is widely used in Southern part of India as threshold decoration in front of dwellings. Kolam Practitioners, mostly women, memorize the complicated kolam designs using some syntactic rules. There are different types of kolam patterns in which dots or pullis and lines or curves are used. In this paper, we examine a kolam pattern called Hridaya Kamalam in which five pullis are marked on eight converging arms in radial form and they are joined by lines using certain rules.
Hridaya Kamalam kolam is generalized to contain m arms and n pullis in each arm. The number of unending lines (kambis) needed to complete the design is also obtained. For a design with m arms and n pullis, the number of kambis required to complete the pattern is given by the HCF of (m,n). When m and n are prime to each other, the pattern contains only one unending line.
A class of Hridaya Kamalam kolam is generated by choosing different values for the number of arms m and the number of pullis n. An algorithm for generating these designs is implemented on a Genie I Computer. The pullis can be joined by straight lines, circular arcs or any other form of curves. Curves that will be more pleasing to the eyes can be generated for getting attractive designs.
I. INTRODUCTION
Kolam or rangoli is a form of traditional Indian folk art used widely in Tamil Nadu , Karnataka and Andhra Pradesh as threshold decorations in front of dwellings. There are different types of kolam patterns in which dots or pullis, and lines or curves are used. The pullis are marked on the floor first and then using certain rules these pullis are joined either by straight lines, or smooth curves [1]. Kolam Practitioners ( KPs ) , mostly women , memorize different kolam patterns and draw them in their dwellings. Narasimhan [2] has drawn the attention of computer scientists to study how the KPs memorize complicated kolam patterns and examine whether they make use of any syntactic rules that underlie kolam designs. Formal language theory has been successfully applied and the properties of certain types of kolam designs have been studied extensively by Sirornoney , Siromoney and Kritihivasan [3,4,5]. In this paper we examine a particular kolam pattern called Hridaya Kamalam which is a stylized form of lotus flower, and study the variations of this design often completed by a single unending line (kambi)
II. Hridaya Kamalam Kolam
Hridaya Kamalam kolam in its most common form has eight converging arms or axes and each arm is of
'length' five units. KPs
memorize this design by marking the five
pullis on the eight converging arms in radial form. In practice, the directions of the arms are memorized and only
pullis are marked along the directions. Then they memorize a sequence of numbers which they apply repeatedly to join the
pullis. This sequence of numbers is the rule that is used to form the petals of the
Hridaya Kamalam kolam (Figure 1).
Let the pullis be marked as 1,2,3,4 and 5 on each of the arms from the center 0. The sequence of pullis to be joined is given by <1,3,5,2,4>. This sequence of pullis are joined from one arm to the next, starting from any one of the arms arbitrarily. The same sequence is repeated until the design is completed, that is, no pullis left out in any arm. This pattern requires only one kambi . The points can be joined either in the clock-wise or counter clock-wise direction. The shape of the kolam drawn in the clock-wise direction will be the mirror image of the kolam drawn in the counter clock-wise direction.
III. GENERALIZATION
The common Hridaya Kamalam kolam is generalized to have m arms and n pullis in each arm. We examine the general rules that will produce designs resembling the Hridaya Kamalam kolam with varying number of arms and pullis (arms are of constant length in each design, but varying between different designs). We also find the number of kambis that are required to complete a generalized design.
Let m arms of certain length 'n' units emit from a point 0 (center) with an angle between any two consecutive arms. The arms are numbered as l,2,3,..,,m in the clock-wise direction. Each arm is divided into n equal parts and they are marked as l,2,3,,..,n from the center
0. Let P denote the permutation
group of the set N = {1,2,3,...,n} and let A = {a_{1},a_{2},...,a_{n}}
We start with the initial point x_{1} = (a_{1},1), where the first element in the ordered pair represents the pulli and the second represents the arm. Successive points to be joined are determined using the following transformation.
If
x_{k} = (a_{i,} j) is the kth point then the next point to be joined is obtained as
x_{k}_{+1} = f(x_{k} ) = f(a_{i,}
j) = (a_{l}, J)
where I = i (mod n) + 1 and J = j (mod m) +1.
In figure 1, the Hridaya Kamalam kolam with eight arms and five pullis that is, m = 8 and n = 5, is shown. The sequence of pullis used for tracing the kolam is A = <a_{1}, a_{2}, a_{3}, a_{4}, a_{5} > = < 1, 3, 5, 2, 4>.
Thus a Hridaya Kamalam kolam is characterized in terms of the number of arms, the number of pullis and the tracing sequence, that is, (m,n,A).
It is also possible to obtain a closed loop or kambi before completing the
kolam. This situation arises when the starting point is reached before all the
pullis are traced in the pattern. In such a situation, we start again with an arbitrary starting point in the next arm and continue to trace the
kolam. This process is continued until no
pulli is left out in any arm. This process leads to the following interesting question. "For a given
(m,n,A) what is the number of kambis required to complete the
Therefore, mn/r = c
=> mn/c = r = um = vn
=> mn= (uc)m = (vc)n.
(a). (m,n,A) = (6,2,(1,2)) and
(b). (m,n,A) = (9,3,(1,3,2)) .
1. Archana and Gita Narayanan, The Language of Symbols, Crafts Council of India, Madras (1985).
2. R. Narasimhan, The oral literacy in the Indian context (personal
communication).
3. G. Siromoney, R. Siromoney and K. Krithivasan, Abstract families of matrices and picture
languages, Computer Graphics and Image Processing, 1:284-307
(1972).
4. G. Siromoney, R. Siromoney and K. Krithivasan, Picture languages with array
rewriting rules, Information and Control, 22:447-470 (1973).
5. G. Siromoney, R. Siromoney and K. Krithivasan, Array grammars and
kolam, Computer Graphics and Image Processing, 3:63-82 (1974).
6. P.K. Ghosh and S.P. Mudur, Parametric curves for graphic design
systems, The Computer Journal, 26:312-319 (1985).