© Copyright 2007:

Prof. S. Naranan
Chennai 600041, INDIA

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(1) Lines without dots
(2) Lines connecting dots and
(3) Free geometric shapes without lines or dots.

The last variety has sometimes, brilliant colors and is known as Rangoli, popular in North India. The traditional South Indian kolam, based on a grid of points is known as PuLLi kolam or NeLi kolam in Tamil Nadu (PuLLi dot, NeLi = curve); Muggulu in Andhra Pradesh, Rangavalli in Karnataka and Pookalam in Kerala.

Special kolams are drawn on festive occasions with themes based on seasons, nature (flowers, trees) religious topics and deities. The month of Margazhi (Dec 15th to Jan 14th) is very special in Tamil Nadu. During this month, everyday, with the crack of dawn, women draw large kolams with bewildering complexities with rice flour in front of their homes. Rows of such colorful kolams on either side of the street, is a sight to behold for the passers by. The folk-art is handed down through generations of women from historic times, dating perhaps thousand years or more. Constrained only by some very broad rules, kolam designs offer scope for intricacy, complexity and creativity of high order, nurtured by the practitioners, mostly housemaids and housewives, both in rural and urban areas.

Here we deal only with PuLLi kolam or kolam for short. The contents are organized in three sections: articles, kolams slides and special kolams.

THE ARTICLES

(Click on 'Overview', 'Paper I', 'Paper II, 'Paper III' and 'Paper IV' 'on the Left panel for PDF files):


KOLAM DESIGNS BASED ON FIBONACCI NUMBERS.

2.0: Fibonacci Kolams - An Overview

2.1: Part I: Square and Rectangular Designs

We describe a general scheme for kolam designs based on numbers of the Fibonacci series (0, 1, 1, 2, 3, 5, 8, 13, 21, 34 .......). Square kolams (3², 5², 8², 13², 21²) and rectangular kolams (2 x 3, 3 x 5, 5 x 8, 8 x 13) are presented. The modular approach permits extension to larger kolams and computer-aided design. This enhances the level of creativity of the art.

2.2: Part II: Square and Rectangular designs of arbitrary size based on Generalized Fibonacci Numbers

The scheme is generalized to arbitrary sizes of square and rectangular kolams using generalized Fibonacci numbers. The problem of enumeration – the number of possible Fibonacci kolams of a given size – is discussed. Enumeration of kolams of small size (3², 5², 2 x 3) is described. For 2 x 3 kolams kolams symmetry operators forming a group are used to classify them. Finally, the scheme is further extended beyond square grids to cover diamond-shaped grids. Possible connections of kolams to Knot theory and Group theory are indicated.

2.3: Part III. Rectangualr Kolams with two-fold Rotational Symmentry

In Parts I and II, a general scheme was given to create square kolams based on 'Fibonacci Recurrence'. Based on the canonical Fibonacci Series 0 1 1 2 3 5 8 13 21 34 . . . . .

Square kolams 3², 5², 8², 13², 21² were presented in Part I.  Square kolams of any desired size can be generated based on the Generalized Fibonacci Series (Part II).  Rectangular kolams with sides as consecutive Fibonacci numbers – Golden Rectangles - were also drawn but they lack any symmetry property like the square kolams which have four-fold rotational symmetry.  In this paper (Part III), Rectangular kolams with two-fold rotational symmetry based on Fibonacci Recurrence are presented.

Fibonacci Rectangles are based on two Fibonacci Quartets unlike the Fibonacci Squares that are based on one Fibonacci Quartet, and are therefore harder to create.

2.4: Part IV: Evolution of Loops

A new family of Fibonacci Kolams based on Fibonacci Recurrence with four-fold rotational symmetry (square kolams) and two-fold rotational symmetry (rectangular kolams) was described in earlier papers in this series. A key feature is the modular scheme in which larger kolams are made from smaller ones by merging them at a set of splicing points. A cardinal feature of the kolams is that they are single-loop. The task is to splice the modules maximally, consistent with a single loop. Generally the outcome of a large number of splices is multiple loops. It is shown that as the number of splices increases (as in large kolams), the final outcome is a small set of odd-numbered loops (1 3 5). This follows from a set of empirical splicing rules that allows a formulation of loop evolution as binary trees; and invoking matrices of loop probabilities to determine the limiting probability matrix as the number of splices is increased.

KOLAM SLIDES

1. (7 4 11 15) This code implies that a 15² kolam is made up of a central square 7² surrounded by four rectangles 4 x 11. The code for 7² is (3 2 5 7); or the 7² contains a central 3² surrounded by four rectangles 2 x 5. Each 4 x 11 rectangle is made up of three modules: 4², 4 x 3 and 4².
   
   
2. (13 4 17 21) Since 21 is a Fibonacci Number, the canonical quartet code for 21² is (5 8 13 21). The “Golden rectangles” are 8 x 13, two consecutive Fibonacci Numbers. But here we consider 21 as a “Generalized Fibonacci Number” and the quartet code is (13 4 17 21). The 21² is made up o of a 13² surrounded by four rectangles 4 x 17. The 13² has the code (5 4 9 13) again different from the canonical (3 5 8 13). Each 4 x 17 rectangle is made up of 5 modules 4²,4², 4 x 1, 4², 4².
   
   
3. (12 7 19 26). This square kolam, we name it “a e-kolam”, e representing the Euler’s constant (2.7182818. . . .), the base for natural logarithms. The code implies that a 26² kolam has a central 12² enveloped by four rectangular kolams 7 x 19. The ratio of the sides 19/7 = 2.714.., differs from e by 0.004 or < 0.15 % of e. So this kolam has “e-rectangles” instead of the Golden rectangles. The central 12² has the code (8 2 10 12) – i.e. a central 8² surrounded by four rectangles 2 x 10. In turn the 8² has the canonical code (2 3 5 8). The 3 x 5s are Golden rectangles. So, this kolam features both the Golden rectangles and the e-rectangles.
   
   
4. (7 19). This rectangle of sides 7 x 19 with ratio approximately equal to e, the Euler’s constant, was used in the in 26² kolam above. It is made up of three modules, a 7 x 5 rectangle sandwiched between two squares 7². The three modules are spliced together at 6 points, indicated by dots. If splices at the points A and B are added then the single loop kolam splits into three loops.
   
   
5. (15 7 22 29). This square kolam, we name it “ a pi-kolam”. The code (15 7 22 29) implies that 29² kolam contains a smaller 15² kolam at the center surrounded by four rectangles 7 x 22. The ratio of the sides of the rectangle 22/7 is the commonly used approximation for pi, the ratio of the perimeter to diameter of a circle.
   
   - 22/7 = 3.14285…differs from pi (=3.14159 ...) by 0.0013, which is < 0.04 % of pi. - The code for 152 is (7 4 11 15) and the code for 72 is (3 2 5 7).
   
   - The code for 15² is (7 4 11 15) and the code for 7² is (3 2 5 7). Each 7 x 22 rectangle is composed of 4 modules 7², 4 x 7, 4 x 7, 7² that are spliced together at 15 places to produce a 3-loop design.
   
   - This 29² kolam is built with 33 modules and 111 splicing points to produce a single loop traversing 841 ( 29²) dots or grid points.
   
   
6. (7 22). This shows the 7 x 22 “pi-rectangle” used in the 29² square kolam above. The 15 splicing points come in three sets of 5 indicated by dots in the direction of arrows. The three loops are shown in different colours. It is possible to convert this into a single-loop kolam by removing the two splices at A and B.

Brief Bio:

Dr. S. Naranan (b. 1930) was an experimental cosmic-ray physicist and X-ray Astronomer based mostly in the Tata Institute of Fundamental Research (TIFR), Bombay, India, in a career spanning 42 years. He is a firm believer in the interdisciplinary character of science and has diversified his interests to other fields such as mathematics, statistics, computer science, biology, genetics and linguistics. He lives with his wife Visalakshi in Chennai, India.