About Algorithms, Braids, and Kolam Figures
Children love to make friendship bracelets. In order to make a patterned friendship bracelet, it is necessary to follow a set of steps that is repeated over and over again. Many children also enjoy drawing figures with symmetry. The focus of Algorithms, Braids, and Kolam Figures module is using algorithms to make patterns.The two "patterns from algorithms" activities are braiding string and drawing "kolam figures". The inspiration for drawing kolams comes from a fascinating article by Dr. Marcia Ascher, "The Kolam Tradition", in the January 2002 issue of American Scientist. Women of Tamil Nadu in southern India decorate the thresholds of their homes daily with these figures, following algorithms which are passed down from generation to generation. Computer scientists Gift and Rani Siromoney of Madras Christian College took an interest in kolam figures and applied the rules for constructing them to writing algorithms for computer picture languages. The children who do this activity can experience algorithms producing predictable drawings and patterns.
A secondary theme of this module is the mathematics of braids. Some very basic mathematical ideas can be
illustrated using braids instead of numbers and an operation called "glue"--joining two braids together. The basic
ideas are closure, the associative law, identity, and inverses. You are familiar with these properties
from operations with integers:
1. Closure: if you add two integers, you always get another integer. Instead of adding integers, we're gluing braids. The idea is the same, though.
2. Associativity: (2 + 4) + 5 is the same as 2 + (4 + 5)
3. Identity: if you add zero to a number you get the same number back again.
4. Inverse: If you add a number to its negative, you get zero. An inverse is something which undoes what you did--the unraveller. When you combine something with its inverse, you get the identity. 5 + (-5) = 0
Most elementary children won't have had experience working with negative numbers. They may be able to appreciate the unraveller, though. There's a demonstration of a braid and its inverse on the table. You may have to set it up again after a child experiments with it. The second half needs to be an EXACT mirror image of the first (same number of crossings) in order for this to work.
The Activities
Materials Needed
An extra copy of the four pages of braid instructions which are on the display trifold.
An extra copy of the instructions for the Anklets of Krishna which is on the display trifold.
A copy of the instructions for the other kolam figure.
A copy of square dot paper for every child who wants to draw a kolam figure.
Colored pencils, markers, or crayons (we're using colored pencils so that erasing is easier).
Scissors for cutting strings.
String to braid with--we're using colored craft cord, about 1 or 2 mm in diameter, in various colors.
An assembled unravelling braid demonstration--see photo.
Making braids
One or two copies of each of the braid instructions should be spread out around the table. You should supply about 9" of each color of string the child chooses. (The child could choose to use two strands of the same color). You should control the distribution of string (and keep hold of the scissors). Each braid needs to start with a knot--assist child in knotting all of the strands together in an overhand knot if necessary.
Drawing kolam figures
There is a supply of square grid paper to draw the figures on, and a collection of colored pencils to draw with. Kids can follow the algorithms and draw the figures. "Go straight" means continue in the direction your pencil was headed when it completed the loop or half loop. The pencil lines go around the dots rather than dot-to-dot.
Experimenting with the inverse
The unravelling braid demonstration needs to be put together before the Math Night. Tie or glue together the ends of three thick cords. Find something to put pegs in--a styrofoam block and paper clips would work, or a wood block with holes and short dowels to put in the holes would work. The pegs are there just to keep the first half of the braid from unravelling while you braid the mirror image.
To set up the demonstration, braid the three strings partway, slip the middle strand in between two pegs, and continue braiding, but braid the mirror image of the part before the pegs. The children at the display can hold the ends of the braid, pull the pegs, and shake the braid a little. If the inverse braid is done correctly it should shake apart so the three strands are untangled.
Sources
The mathematics of braids: "Group Theory and Braids" in New Mathematical Diversions in Scientific American by Martin Gardner"The theory of braids" by Emil Artin in The Mathematics Teacher, May 1959 p. 328
Kolam figures: "The Kolam Tradition" by Marcia Ascher in American Scientist Jan-Feb 2002 p. 56
Web Resources
Braid mathematics:A website by Djun M. Kim at UBC including a connection between braids and dances. It is fairly technical.
Braids, Curve Diagrams, and Orderings
Kolam Figures:
Webpages of Tamil Electronic Library (containing some simple, beautiful figures which would be very suitable for
these activities):
Written Mar 21, 2002 by Janet A. Dixon Revised Jan 10, 2006