© Copyright 2007:
Prof. S. Naranan
This website demonstrates with theory and examples the connection between the beauty of Indian kolams with the elegance of the well known Fibonacci Numbers.
Prof S. Naranan, a Physicist, in course of drawing these Kolams as part of his many hobbies, discovered the elegant connection and has come up with some fascinating designs. The articles and actual drawings of the Kolams can be found by clicking the links on the left margin.
Kolams are decorative geometrical patterns that adorn the entrances of households
and places of worship especially in South India. Kolam is a line drawing of curves
and loops around a regular grid of points. Usually kolams have some symmetry (e.g.
four-fold rotational symmetry). There are variants:
(1) Lines without
dots THE ARTICLES
(Click on 'Overview', 'Paper I', 'Paper II, 'Paper III', 'Paper IV' and 'Paper V'on the Left panel for PDF files):
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.
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
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.
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.
A Fibonacci Kolam is made up of unit square tiles on a grid of unit square
cells. The tiles come only in 8 different shapes (A B C D E F G H). This feature renders possible design of a Boardgame for kolams, either as a solitaire or as a two-person game. A board of size 14 x 14 and 300 tiles have been produced. The game can be adapted for play on the computer.
KOLAM SLIDES SPECIAL KOLAMS
KOLAM SLIDESThis Power Point file has 49 slides. The first 16 slides contain interesting facts about Fibonacci Numbers. The remaining deal with the Kolam designs based on Fibonacci Numbers. The kolams are duplicated from the two articles mentioned above. Explanations for the kolam slides can be found in the articles.
SPECIAL KOLAMSThe Generalized Fibonacci Numbers permit a wide choice for the rectangles that go into the square designs. With the standard “Fibonacci Numbers” the rectangles are “Golden rectangles” with sides equal to consecutive Fibonacci numbers and their ratio approaching the Golden Ratio ? (=1.61803...). The explanations for the 6 slides are given below; All these kolams have a single loop except the last one.
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.
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