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.
- Skeleton of the kolam with eight arms of length five each,
- The kolam is drawn following the tracing sequence <1, 3,5,2,4 > .
- 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,...,an} 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 xk = (ai, j) is the kth point then the next point to be joined is obtained as
xk+1 = f(xk ) = f(ai, j) = (al, 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..... xmn 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 xr+1 = x1 in the sequence x1 x2 x3 ...xr xr+1. xmn. Also x2r+1 = xr+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 xr+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).
The Dream-Catcher
The spiral growth stylization is in line with the growth of lotus petals in nature and since Indian yogic thought considers the yogic hridayakosha to be a lotus kept inverted in the middle of the chest, hrdayakamalam can indeed be considered as symbolizing this kosha, in which param brahma is believed to reside.
South Indian women are also known to represent the lotus in simpler non spiraling, but circularly symmetrical kolams, not unlike spiderwebs. In this connection, I thought I would give an extract from the Wikipedia account of the Dream-Catcher', an American Indian charm, which is believed to save children from bad dreams and allows them to sleep in peace.
In Ojibwa (Chippewa) culture, a dream catcher (Ojibwe asabikeshiinh, the inanimate form of the word for "spider"[1][2] or bawaajige nagwaagan meaning "dream snare"[2]) is a handmade object based on a willow hoop, on which is woven a loose net or web. The dreamcatcher is then decorated with personal and sacred items such as feathers and beads.
Origin and legends
While dream catchers originated in the Ojibwa Nation, during the Pan-Indian Movement of the 1960s and 1970s they were adopted by Native Americans of a number of different Nations. They came to be seen by some as a symbol of unity among the various Indian Nations, and as a general symbol of identification with Native American or First Nations cultures. However, some Native Americans have come to see them as "tacky" and over-commercialized due to their acceptance in popular culture.[3]
Traditionally, the Ojibwa construct dream catchers by tying sinew strands in a web around a small round or tear-shaped frame of willow (in a way roughly similar to their method for making snowshoe webbing). The resulting "dream-catcher", hung above the bed, is then used as a charm to protect sleeping children from nightmares. Dreamcatchers made of willow and sinew are not meant to last forever but instead are intended to dry out and collapse over time as the child enters the age of adulthood.
The Ojibwa believe that a dreamcatcher changes a person's dreams. According to Terri J. Andrews in the article "Legend of the Dream Catcher," about the Ojibwa nation in the magazine World & I, Nov. 1998 page 204, "Only good dreams would be allowed to filter through . . . Bad dreams would stay in the net, disappearing with the light of day."
It's recommended to hang the dream catcher above someone sleeping to guard against bad dreams. Good dreams pass through and slide down the feathers to the sleeper.
Another legend "Good dreams pass through the center hole to the sleeping person. The bad dreams are trapped in the web, where they perish in the light of dawn."
Popularization
In the course of becoming popular outside of the Ojibwa Nation, and then outside of the pan-Indian communities, "dreamcatchers" are now made, exhibited, and sold by some New age groups and individuals. According to Philip Jenkins, this is considered by most traditional Native peoples and their supporters to be an undesirable form of cultural appropriation.[4]
The official portrait of Ralph Klein, former Premier of the Canadian province of Alberta and whose wife Colleen Klein is Métis, incorporates a dreamcatcher.[5]
References
- Freelang Ojibwe Dictionary
- Prindle, Tara. "NativeTech: Dream Catchers". Retrieved on September 23, 2007.
- Native American Dreamcatchers
- Jenkins, Philip (September 2004). Dream Catchers: How Mainstream America Discovered Native Spirituality. New York: Oxford University Press. ISBN 0195161157.
- "Ralph Klein breaks tradition in legislature portrait". Canadian Broadcasting Corporation (2007-08-31).
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written by T.Robinson, 2010-08-29 10:33:44
Recently efforts have been made at Madras Christian College (MCC) on presenting these intricate, geometrical kolam patterns to students with vision impairment. Tactile (raised line) kolam sheets using microcapsule papers provided the access to the visually impaired students to appreciate the ethno-mathematical aspects of this wonderful art form. - T.Robinson (MCC).
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written by narensomu, 2009-08-06 03:34:52
With th images in place, it is possible to appreciate the similarities between the Indian and " Indian" patterns.
Yes, tamilnation.org should be checked out. These articles sound interesting.
Regards
ns
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written by P. Desikan, 2009-08-05 21:45:55
But how are you to access the journal Forma?
By Ctrl+ click at appropriate places in the references, not as given in my previous comment, but as part of a lovely less mathematical article on Kolams, for which the link is
http://www.tamilnation.org/culture/kolam.htm
Am I forgiven?
Regards. Partha
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written by P. Desikan, 2009-08-05 21:26:29
If Dr Siromoney's little thesis on hridayakamalam touched your mathematical funny nerves and they ask for more, here are a few references to mathematical abstractions on other kolams and kolams in general. Have fun.
• Solving Infinite Kolam in Knot Theory
Y. Ishimoto
Forma, Vol. 22 (No. 1), pp. 15-30, 2007
[Abstract] | [Full text] (PDF 848 KB) | [Attractive image Fig.3, Fig.4]
• Fundamental Study on Design System of Kolam Pattern
K. Yanagisawa and S. Nagata
Forma, Vol. 22 (No. 1), pp. 31-46, 2007
[Abstract] | [Full text] (PDF 568 KB) | [Attractive image Fig.11, Fig.13, Fig.20]
• P Systems for Array Generation and Application to Kolam Patterns
K. G. Subramanian, R. Saravanan and T. Robinson
Forma, Vol. 22 (No. 1), pp. 47-54, 2007
[Abstract] | [Full text] (PDF 96 KB) | [Attractive image Fig.4, Fig.5, 6]
• Extended Pasting Scheme for Kolam Pattern Generation
T. Robinson
Forma, Vol. 22 (No. 1), pp. 55-64, 2007
[Abstract] | [Full text] (PDF 796 KB) | [Attractive image Fig.6, Fig.7
Regards. Partha
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written by P. Desikan, 2008-12-28 07:26:08
Sreeparna or Dwai will soon be able to restore the images in the article and I would like you to look at the first figure.
It has three parts a, b and c.
You must be familiar with the symmetrical Thamaraikolams which you can draw with eight arms as given in the part a of the figure,
You can make eight petals by joining each pair of adjacent tips. You can also connect alternating tips when you will get eight outer petals and eight inner petals. The sixteenpetals thus form the shodasadala kamalam which is an uncomplicated version of the hridayakamalam, depicting our hridaya kosam. But our ancients knew that kamalams grew their petals in a continuous spiral, whether one considers the elementary 16 petal version or the 1008 petal one similar to the the sahasraara kosha. You will notice that part c of the figure is obtained by spiralling from inside (though spirally drawn from outside conventionally) and in effect gives two sets of eight petals attached to an 8 star in the middle. Thus the 8 arm 5 pulli lotus is depiction of the heart lotus.
My reason for including the dream catchers is because the spider web is symmetrical thaamarai-like in design and the american Indians intuitively related to the pacific potential of this design, though the spider may be using it for its own ends.
Thank you for your interest.
Regards. Partha
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written by narensomu, 2008-12-28 01:41:55
Thankyou for the article.Hridhayakamalam is a popular kolam and unlike lesser kolams the dots arent placed in a square or rectangular pattern.
This was my Mother's favorite kolam ,[ but one I dont make very often, should take it up]
Regarding the n number of ways to do a kolam on a set number of dots,could it that the ancients knew there were different ways ,[even without using a computer] but decided to use this particular pattern for some reasons?
The information about the dream catchers is very interesting. We can understand the reactions of the people to the commercialisation of what they hold sacred...
Regards
ns
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Please convey my congratulations to the group at MCC who undertook this wonderful exercise of teaching kolam design appreciation to the visually impaired.
It is such a thoughtful thing to do.
Warm regards. Partha.