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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).

Figure 1.
1. Skeleton of the kolam with eight arms of length five each,
2. The kolam is drawn following the tracing sequence <1,3,5,2,4 > .
3. Acompleted Hridaya Kamalam design.

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 = {a₁,a₂,...,a_n} be a member of P. In fact, A represents the sequence of pullis to be joined from one arm to the next. We call A the tracing sequence. The Hridaya Kamalam kolam is now traced as follows.

We start with the initial point x₁ = (a₁,1), 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 = <a₁, a₂, a₃, a₄, a₅ > = < 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 kolam ?" . To answer this question we give the following proposition.

PROPOSITION : 1

For a Hridaya Kamalam kolam (m,n,A), the number of kambis or unending lines required to complete the kolam is given by the highest common factor of (m,n).

Proof : Let x₁ x₂ x₃….. x_mn be the totality of points in the design. Let us assume that we come to the starting point after tracing r points. If r is equal to mn then the pattern contains only one unending line. However, when r is less than mn then we have x_r+1 = x₁ in the sequence x₁ x₂ x₃ …x_r x_r+1. x_mn. Also x_2r+1 = x_r+1 = x₁ and so on. Hence the remaining points will be traced again on the same closed loop. However, since there are mn points in the sequence, the remaining (mn-r) points can be traced by starting at the point x_r+1 = (a₁,l), where I is the arm in which the pullis is not yet traced, and using the transformation of that gives successive points. After tracing r points, it will come into a closed loop again by symmetry. Hence proceeding in the same manner we get the number of closed loops required as mn/r = c where r is the minimum number of points required to obtain one closed loop. We now show that r divides both m and n . Since we come to the same point on an arm after tracing r points, r must be a multiple of number of arms, that is, r = un. Also since we pass through the sequence of all pullis in the tracing sequence A, and come to the same pullis after tracing r points, r must be a multiple of n, that is, r = vn, where u and v are positive integers.
Therefore, mn/r = c
=> mn/c = r = um = vn
=> mn= (uc)m = (vc)n.

This implies that both m and n are multiples of c and hence c is a factor of m as well as n . Since we have chosen r as the minimum number of points required to complete one closed loop, the c obtained must be the largest of the possible common factors of m and n . This completes the proof.

Thus we establish that when m and n are prime to each other, the pattern contains only one unending line.

Since the tracing sequence A is taken as an arbitrary member of P, the above result holds for any member of P.

This also implies that the kolam patterns obtained for the members of P are isomorphic to one another as each pair of members of P have one-to-one correspondence.

Figure 2 illustrates the Hridaya Kamalam kolam patterns for the following specifications,
(a). (m,n,A) = (6,2,(1,2)) and
(b). (m,n,A) = (9,3,(1,3,2)) .

The number of unending lines required for the first specification is 2 and for the second specification it is 3.

Figure 2. a) A kolam pattern with six arms of length two each, with the tracing sequence <1,2> . 

b) A kolam pattern with nine arms of length three each, with the tracing sequence <1,3,2>.

A computer program is written for generating a class of Hridaya Kamalam kolam for any given specification (m,n,A). Straight lines and circular arcs are used for joining the points. However, it is also possible to have the points joined by curves that will be pleasing to the eyes [6].

IV. CONCLUSION

A threshold design called Hridaya Kamalam kolam is generalized to contain m arms and n pullis in each arm. A class of Hridaya Kamalam kolam is generated by choosing different values for m and n, and using a tracing sequence A that specifies the order of points to be joined. Hence these types of kolam are characterized in terms of m,n and A. It is also shown that the number of kambis or unending lines required to complete the kolam with the specification (m,n,A) is the highest common factor of (m,n).

A computer program has been written to simulate the drawings of the Hridaya Kamalam kolam for a given specification (m,n,A). In the first version the points are joined by straight lines. Improved versions include joining points using circular arc segments and the future version will have smooth curves that will be more pleasing to the eyes .

It is now possible to generate a variety of new designs of the Hridaya Kamalam kolam type which can be used as threshold designs . The program for generating the Hridaya kolam has been implemented on a Genie I computer.

 REFERENCES

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).