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Abstract
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Introduction
Methodology
Results and Discussion
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Kolam Simulation using Angles at Lattice Points

Profile image of Shailaja D SharmaShailaja D Sharma

2023, arXiv (Cornell University)

Abstract

Kolam is a ritual art form practised by people in South India and consists of rule-bound geometric patterns of dots and lines. Single loop Kolams are mathematical closed loop patterns drawn over a grid of dots and conforming to certain heuristics. In this work, we propose a novel encoding scheme where we map the angular movements of Kolam at lattice points into sequences containing 4 distinct symbols. This is then used to simulate single loop Kolam procedure via turtle moves in accordance with the desired angular direction at specific points. We thus obtain sequential codes for Kolams, unique up to cyclic permutations. We specify the requirements for the algorithm and indicate the general methodology. We demonstrate a sample of Kolams using our algorithm with a software implementation in Python.

Figures (38)
Figure 1: Deciding where to go while drawing a Kolam - Part 1.
Figure 1: Deciding where to go while drawing a Kolam - Part 1.
Figure 2: Deciding where to go while drawing Kolam - Part 2.
Figure 2: Deciding where to go while drawing Kolam - Part 2.
Figure 5: Lattice structures imposed on 1 — 3 — 1 and 3 x 3 dot patterns. Figure 4: Cardinal directions of a Kolam dot and intercardinal directions of a lattice point.
Figure 5: Lattice structures imposed on 1 — 3 — 1 and 3 x 3 dot patterns. Figure 4: Cardinal directions of a Kolam dot and intercardinal directions of a lattice point.
Figure 3: A square unit cell imposed on a Kolam dot.
Figure 3: A square unit cell imposed on a Kolam dot.
Table 1: Coding lattice angles
Table 1: Coding lattice angles
Figure 7: Direction of a point with respect to its predecessor point. Figure 6: Direction of point B from point A.
Figure 7: Direction of a point with respect to its predecessor point. Figure 6: Direction of point B from point A.
Figure 8: Convention for sequence start-point of square and rhombic Kolams.
Figure 8: Convention for sequence start-point of square and rhombic Kolams.
‘igure 9: A 1 — 3 — 1 Kolam with angle changes while drawing.
‘igure 9: A 1 — 3 — 1 Kolam with angle changes while drawing.
Figure 10: Angle changes in Fig. 9 replaced with their corresponding symbols.
Figure 10: Angle changes in Fig. 9 replaced with their corresponding symbols.
Figure 11: Shapes involved in single loop Kolams.
Figure 11: Shapes involved in single loop Kolams.
Figure 12: “0 changes in symbol” shapes. Figure 13: “1 change in symbol” shapes in vertical direction.
Figure 12: “0 changes in symbol” shapes. Figure 13: “1 change in symbol” shapes in vertical direction.
conditions, the loop will be terminated after iterating through these z’s. The Kolam drawing will be finished. For shape-III and shape-IV, even if there are no z’s, no problem will arise. For example, if the sequence ends with ‘dcab’, the shape-IV (dcab) will be drawn and the loop will be terminated without any errors. The Kolam drawing will be finished. But if z’s are not added, it raises an error or selects a wrong shape when the final shape is either shape-I or shape-II. For example, if the sequence ends with ‘aab’, there is no change from a—a and this is a shape-I (a) pattern but there is a change from a—+b. It is the first condition with no shape specified in section 6. Now ‘a’ becomes the current element. There is a change from a—+b and it needs to take second step to determine the shape. As there is no element after ‘b’, it will raise an error. To avoid these errors and discrepancies and to encode all Kolams in a standard way, we add z’s in the end of a Kolam sequence. The loop will always be terminated after iterating through these z’s.
conditions, the loop will be terminated after iterating through these z’s. The Kolam drawing will be finished. For shape-III and shape-IV, even if there are no z’s, no problem will arise. For example, if the sequence ends with ‘dcab’, the shape-IV (dcab) will be drawn and the loop will be terminated without any errors. The Kolam drawing will be finished. But if z’s are not added, it raises an error or selects a wrong shape when the final shape is either shape-I or shape-II. For example, if the sequence ends with ‘aab’, there is no change from a—a and this is a shape-I (a) pattern but there is a change from a—+b. It is the first condition with no shape specified in section 6. Now ‘a’ becomes the current element. There is a change from a—+b and it needs to take second step to determine the shape. As there is no element after ‘b’, it will raise an error. To avoid these errors and discrepancies and to encode all Kolams in a standard way, we add z’s in the end of a Kolam sequence. The loop will always be terminated after iterating through these z’s.
Figure 14: “1 change in symbol” shapes in horizontal direction.
Figure 14: “1 change in symbol” shapes in horizontal direction.
Figure 15: “2 changes in symbol” shapes in clockwise direction.
Figure 15: “2 changes in symbol” shapes in clockwise direction.
Figure 16: “2 changes in symbol” shapes in anti-clockwise direction.
Figure 16: “2 changes in symbol” shapes in anti-clockwise direction.
Figure 17: “3 changes in symbol” shapes in anti-clockwise direction.
Figure 17: “3 changes in symbol” shapes in anti-clockwise direction.
Figure 18: “3 changes in symbol” shapes in clockwise direction.
Figure 18: “3 changes in symbol” shapes in clockwise direction.
Figure 19: Iterating through the sequence: an example. 6 Kolam Simulation Algorithm
Figure 19: Iterating through the sequence: an example. 6 Kolam Simulation Algorithm
We can also use the sequence in a cyclic iteration without z’s and adding ‘b’ in the third position from the last of the sequence. In this case it will draw the Kolam infinite times without stopping. The output of this Kolam is given in Figure 29.
We can also use the sequence in a cyclic iteration without z’s and adding ‘b’ in the third position from the last of the sequence. In this case it will draw the Kolam infinite times without stopping. The output of this Kolam is given in Figure 29.
Table 2: Navagraha Kolams and corresponding digital sequences
Table 2: Navagraha Kolams and corresponding digital sequences
Table 3: Rhombic Kolams and corresponding digital sequences
Table 3: Rhombic Kolams and corresponding digital sequences
Figure 21: Navagraha (3 x 3 square) Kolam 2.
Figure 21: Navagraha (3 x 3 square) Kolam 2.
Figure 22: Navagraha (3 x 3 square) Kolam 3.
Figure 22: Navagraha (3 x 3 square) Kolam 3.
Figure 26: Navagraha (3 x 3 square) Kolam 7. Figure 25: Navagraha (3 x 3 square) Kolam 6.
Figure 26: Navagraha (3 x 3 square) Kolam 7. Figure 25: Navagraha (3 x 3 square) Kolam 6.
Figure 24: Navagraha (3 x 3 square) Kolam 5. Figure 23: Navagraha (3 x 3 square) Kolam 4.
Figure 24: Navagraha (3 x 3 square) Kolam 5. Figure 23: Navagraha (3 x 3 square) Kolam 4.
Figure 30: A 1 — 5 — 1 rhombic Kolam. Figure 29: A 1 — 3 — 1 rhombic Kolam.
Figure 30: A 1 — 5 — 1 rhombic Kolam. Figure 29: A 1 — 3 — 1 rhombic Kolam.
Figure 28: Navagraha (3 x 3 square) Kolam 9.
Figure 28: Navagraha (3 x 3 square) Kolam 9.
Figure 27: Navagraha (3 x 3 square) Kolam 8.
Figure 27: Navagraha (3 x 3 square) Kolam 8.
Figure 31: A 1 — 7 — 1 rhombic Kolam.
Figure 31: A 1 — 7 — 1 rhombic Kolam.

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