Kolams and Sona
Contents of this web page
What are Kolams, what are Sona?
Some Mathematics
In Search of Sona
Drawing Exercises
Kolams und Sona on the Internet
Final Remark.

What are Kolams, what are Sona?
Kolams are Indian floor paintings; sona (singular lusona) are African sand paintings.
The only thing they have in common is that dot patterns are given as a mnemonic and then lines are drawn around the dots so that each dot is circled. The result is a great variety of attractive symmetrical figures. 

You can get an impression of kolams and sona by watching the first two Youtube videos in the link list below.

Kolams and sona are the subject of ethnomathematics. There are links at the end of this web page.

Example of a kolam, example of a lusona

Kolams    top
A kolam is a South Indian floor painting. 
In certain regions in South India, women decorate the floors in front of houses with elaborate, rotationally symmetrical figures. They draw dots 
with rice flour as a guide and then draw lines around the dots. To do this, they skillfully let powder trickle between their index finger 
and middle finger and portion it out with their thumb. 
The often complicated patterns are passed on from generation to generation in a family. - They have a religious background. 
These kolams are more precisely called stroke kolams on the English Wikipedia page, neli kolam, kambi kolam or sikku kolam in Tamil.

In other regions, point grids and lines are dispensed with. Instead, coloured areas or carpets of flowers are laid out. Rice flour is replaced by stone powder or chalk powder, often together with natural or synthetic colour powders. 
The pictures, which always have rotational symmetry, are presented together with the stroke kolams at celebrations, at festivals, in competitions and on the internet. 

Simple Example
...... If you give nine points in square form, a kolam could look like this. 

...... Kolams are created by closed, overlapping lines. 

In this case, the image is created from the three edge curves of the yellow pieces.

Example of a Kolam

This kolam is created by seven closed curves, marked by the lines in red (4x), green, blue and violet. 

The structure of the kolam is better recognized by yellow colorings.
- There are four outer figures and a central circle.
- In the center is a rounded cross.
- A square with tines surrounds the figure. 
The kolam is created by drawing the closed lines. 
Several closed lines, this is typical for kolams, although they sometimes get by with one closed line.

Sona     top
Sona are traditional drawings of some Bantu people like the Chokwe in an area of Angola and Zambia in southern Africa. 
Storytellers draw sona with their finger in the smooth sand while telling a story. Without putting the finger down, a closed line is drawn. The line leads around points of a predefined pattern, crosses again and again and finally returns to the starting point. 
The drawings illustrate the stories.

Simple Example 
The figure to a 4x3 grid is a lusona. It therefore consists of a closed line without beginning and end. The figure is monolineal.

As the sona complement stories, they often depict animals. 
The compact lusona becomes an antelope...................................

Do you feel the irony in this picture? I do

The picture shows a leopard with extended paws, on the left the little head, on the right the little tail. 

More precisely: Two cubs are also visible in the drawing. 
They are lying next to each other and opposite each other.


Three Birds
...... There are also sona in which images are repeated. They are connected in such a way that a 
closed line is maintained. 

Here there are three birds, in my reference there are ten (!).


A Fable
...... This  lusona is also drawn with a closed line. 
Four almost identical figures are linked together. 
The abstract figure comes alive when you know the fable that is told while drawing. 

"The following sand drawing illustrates a fable: Sambálu, the rabbit (positioned a point B), discovers a salt mine (point A). Immediately, the lion (point C), the jaguar (point D), and the hyena (point E) demand possession, asserting the rights of the strong. The rabbit, affirming the inviolable rights of the weak, then quickly makes a fence to isolate the mine from all usurpers.
Note that only from B can one go to point A, without going beyond the line that represents the fence."

Exceptionally, when the story requires it, sona consists of several closed lines.
But on this page, for simplicity, a figure is called a lusona when they arise from a closed line.

Some Mathematics top
Mirror Model
... This lusona 4x3 is used to describe how the figure is "mathematised" on the basis of a model conception. 

Place a rectangle around the 4x3 points and imagine that the rectangle is mirrored on the inside. If a ray of light is sent into the rectangle from the upper left, it is reflected many times and describes a path as started on the left. 

The ray drawn in is not allowed. The path is controlled by grids. 

A grid is placed in the rectangle............................................................................................

...... A ray of light is sent into the rectangle in such a way that it makes its way exactly between the predefined points.

...... Finally, it returns to the starting point; it crosses all the squares on diagonals. 

What remains is a grid of squares standing on top.

...... When you delete the black grid, then the red grid is more clearly visible. .................................................

...... It is easy to recognise the 4*3-lusona above in the grid of light rays. 

The curves can be explained by the fact that the walls of the rectangle act as mirrors. 

First Rule
The question arises which rectangular figures m*n is a lusona.
Here are four examples:

These four examples illustrate the following first rule.
 - A closed line only results if m and n have no common divisor.
 - If several closed lines are necessary to create the figure, the number of curves is equal to the greatest common divisor. 

The rectangles to the leopard family have the data 10*3 and 2x3, their dimension numbers have no common divisor. 
... ...... The rectangles overlap in such a way that together they are further created by a closed line.

Inner two-sided mirrors
... The compact lusona becomes more interesting when you create patterns inside.

...... In the simplest case, you dissolve a crossed line and replace the crossing with roundings.

This can be explained in the mirror model by placing a two-sided mirror on a crossing.

...... You can also put several mirrors inside. 
                                                                                          (The figure on the right is a lusona.)

With the insertion of a mirror, the following figure can no longer created by a closed line. 
...... You need two closed lines to capture the figure. ..................................,,.,......................

...... From this figure, by the way, you can create a lusona again by resolving a crossing in which the two different lines meet.

Second rule
There is a simple rule for m*n-sona with one interior mirror. 
For this, a pattern of dark and light fields must be introduced. 
...... You put the grid on the 4x3 lusona.................................................................................

...... You follow a line and colour every second square grey. 

For instance you start with a grey square at the upper left corner. Then you go two squares to the right and one square down. This square becomes grey again and so on.

The second rule says that there can always be a mirror where 2x2 squares meet. 

These are the three mirrors and, for reasons of symmetry, four more can be added.

This is confirmed by the study of the 42 figures in the next chapter by the first three sona. 

Another example, the 5*4-Lusona

From a Lusona to a Matrix
...... ...... If you write number 1 in place of the dark fields and a zero in place of the light fields, you get a matrix of zeros and ones. 

It can be assigned to a 4x3 figure and uniquely identifies it.

Conversely, you can design new patterns from 0 and 1 and create at a new lusona. 

In Search of Sona    top
Question:  Where must be placed mirrors inside the 4x3-lusona, so that the figure remains a lusona?
For this purpose, all possible positions of up to five mirrors are played through..
Result: Among the 42 figures examined, eight are a lusona.

Question: In the 3x3 kolam, where must the double mirrors (blue) be inside and on the edge for the figure to become a lusona?
Result: There are 36 sonas, nine of which are symmetrical. 

Question: How do you get to the following kolam of five closed curves?
... You draw a point-symmetrical figure with ballpoint pen on checked paper in the hope that from it an 
 handsome kolam will be created. Draw it - as described in the next chapter - with MS Paint or the online program. 

... You also find sonas...................................................................................................................................

Drawing Exercises      top
Five basic figures
Above it says that kolams are created by drawing lines around points. If you have a field of 2x2 points, there are many ways to circle dots. 
Kolams as a subject of mathematics are composed of the following basic figures.
Thus the first four figures are omitted. Only the next three figures are called kolams. 
Strictly speaking, the fifth figure must have a dot in the center. 

Just like the mandalas of Far Eastern cultures, the line kolams lend themselves to their own activities. 
Drawing with pen and paper
For example, if you have a 3x2 dot pattern and want to draw Kolams or even Sona, this is the best way to go.
 - Draw a 3x2 rectangle and double mirrors at suitable places.
 - Then you sketch the kolams. It is difficult to do this, because you have to have the five basic figures in your head. 
 - Then the sketches are transferred into a final drawing. Use MS-Paint or the online program described below. 

Drawing with MS-Paint
The drawings on this website were created with the drawing program MS-Paint. It is available to anyone who uses Windows on the Internet. 
...... The figures are, as already indicated above, composed of five building blocks together. A circle is added.

I show two sequences of pictures as a suggestion for your own activities. 

You can color the kolams.

Drawing with a program
There is a program available on the Internet that makes it easy to draw kolams. 
You can get to the online program by calling it in the link list below with the name Forkphorus. 
The user interface is clear. On the right is the tool, on the left is the drawing area for a maximum of 5x5 Kolams. 
At the top the six basic figures are available. You can bring a figure to the drawing area by tapping on it and then on the target square. There are fields for turning and deleting the figure. The field at the bottom right deletes the drawing. The field at the bottom left has no effect on me. 

The program helped to find the sona among the 3x3 kolams in the previous chapter. If you specify the compact 3x3 figure, you can "create" kolams by turning basic figures. You can recognizes the position of the double mirrors as gaps and can make out sona, even if the shapes are not correct. 

Perhaps an open question is how many kolams and then sona there are to the compact, relatively simple 3x2 and 4x2 rectangles.
So for this, you have to look for the possible positions of the double mirrors with the help of notes on squared paper systematically. 
In a second step, you can recognize the sona with the help of the online program. 

Kolam Game
If you glue the basic figures on squares of cardboard, you get a game to lay. 

Miscellaneous top
Paulus Gerdes
When researching, it becomes clear that the lusona research was founded and advanced by the scientist Paul Gerdes. He wrote numerous papers on the subject and gathered many students around him who studied lusona. He was Dutch, a professor of mathematics in Mozambique from 1976 and also took up citizenship there. 

Celtic knots
The 3x3-Kolam and the 4x3-Lusona become Celtic knots, if you make the lines thicker and place along a line alternating underpasses and bridges.

Indian Sand Painting
Sand image of the North American natives, 

fixed with hair setting lotion, 

outside the circle four-sided symmetry,

made for tourists,

we bought somewhere near the Grand Canyon, USA.

German Beach Painting :-)

Knight's Tour
The knight's tour is about the knight moving on the chessboard in such a way that it (also) describes a closed line. 
On the left is a solution for the smaller 6*5 square.

More on my page House of Santa Claus

References   top
(1) South African History Online (www.sahistory.org.za)
(2) https://www.zukunft-irular.de/neue-seite/ (Sona-Geometrie: Zehn Vögel)
(3) http://www.math.buffalo.edu/mad/special/african-geometry.html
(4) https://www.researchgate.net/publication/256932750_Sona_drawings_mirror_curves_and_pattern_designs

Kolams und Sona on the Internet       top


Alban Da Silva (Spektrum)
Mathematik einer Südseetradition (Sandzeichnungen des südpazifischen Archipels)

Kolam: Die flüchtige Kunst der Inderinnen

Henning Krause  (Spektrum)
Dr. Henning Krause  (Spektrum)
Ethnomathematik - dargestellt am Beispiel der Sona Geometrie

pulli-kolams, Was sind Sona?

M. Weber, A. Mischau (Mathematisches Panorama)
Mathematik und (bzw. in) andere(n) Kulturtechniken   (.pdf-Datei)

Kolam, Sona (Mathematik), Paulus Gerdes, Ethnomathematik ,    Sandbild, Sandanimation


Darrah Chavey
Mathematical Experiments with African Sona Designs

Darrah Chavey
Constructing Symmetric Chokwe Sand Drawings

Drawing program

Ljiljana Radovic

Marcia Ascher (spektrum.de)
Ethnomathematik: Die Kolam-Figuren Südindiens

Mattia De’ Michieli Vitturi
Sona drawings, mirror curves and pattern designs

Sona - Sand Drawings from Africa

Paulus Gerdes 
m-Canonic mirror curves

Paulus Gerdes
Geometria Sona de Angola  Matemática duma Tradição Africana
Portuguese, 191 pages, lots of illustrations

Slavik Jablan, Ljiljana Radovi, Radmila Sazdanovi Ana Zekovi
Knots in Art

South African History Online
Sona Patterns - Revisiting the Contributions of the People in Sub-Saharan Africa to Modern Mathematics

Kolam, Rangoli, Lusona, Sandpainting, Sand animation, Sand mandala, Ethnomathematics, Zenigata Sunae

Angolan Sand Drawings
Some drawings and stories

Example of an Indian floor painting

Pongal' theme for Chennai's Kolam and Rangoli competition
A meeting of women artists

Kolam designs
Collection of videos

Final Remark  top
A visitor to my pages, Volker Sayn, pointed out the kolams and sona to me, which to my own astonishment I had never heard of before. 
I have used his extensive documents for this website and thank him. 

Feedback: Email address on my main page

This page is also available in German.

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© 3/2023 Jürgen Köller