Kolams and Sona

Contents of this web page

What are Kolams, what are Sona?
Kolams
Sona
Some Mathematics

Miscellaneous
References
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.

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
Background
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. These kolams often have the name rangoli.

Simple Example

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

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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

The structure of a kolam is understood by looking for closed lines. They become visible by a yellow coloration.
- 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.

Drawing Exercises
Just like mandalas of the Far Eastern cultures, the line kolams lend themselves to your own activities.
As a suggestion, I show some pictures drawn with MSPaint. The programme is available to anyone who uses Windows.

The following pictures were taken from five cells like nearly all the pictures on this page.

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2 ...

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Of course you can do it without a computer, all you need is a pencil with an eraser or a biro and paper.
- Draw a square frame as above and design a pattern inside.
- Draw a simple dot pattern and try to find as many figures as possible for it.
- Watches a video of simple kolam, stop it, draw the half-finished figure and then finish it without the video.
- Give a simple kolam and change it.
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Sona top
Background
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.

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As the Sona complement stories, they often depict animals.
Antelope

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The compact lusona becomes an antelope when corners, head with horn, tail and legs are add.

Do you also feel the irony in this picture?

Leopard

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.

(1)

Three Birds

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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 (!).

(2)

A Fable

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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.

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"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."
(3)

Drawing Exercises
Drawing sona is difficult because of the closed line.
This will be discussed in the next chapter.

Some Mathematicstop
Mirror Model

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

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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.

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

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

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

What remains is a grid of squares standing on top.

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

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It is easy to recognise the 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.

In Search of Sona
The question arises which rectangular figures m*n is a lusona.
Here are four examples:

So not all rectangular figures are a lusona.

These four examples illustrate the following rules.
- 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.

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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.

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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.

With the insertion of a mirror, a figure can no longer created by a closed line.

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You need two closed lines to capture the figure.

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

The question is, where mirrors must be placed so that the figure remains a lusona.
For this purpose, all possible positions of up to five mirrors are played through.

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Result: Among the 42 figures examined, eight are a lusona.

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By the way, there are more sona if you allow two-sided mirrors even at the edge.

In the case of one interior mirror, there is a simple rule.
For this, a pattern of dark and light fields must be introduced for the 4x3 lusona.

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You put the grid on the 4x3 lusona.................................................................................

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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.

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

These are the three mirrors drawn above and, for reasons of symmetry, four more are added.

This is confirmed by the examination of 36 figures above.

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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.

Apparently, the rule does not apply to compact figures that cannot be drawn from a closed line. This is proven by the following example.
The coloring is also no longer clear.
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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.

Miscellaneous top
Celtic knots
The 3x3-Kolam and the 4x3-Lusona become Celtic knots, if the lines are chosen thicker and along a line alternate underpasses and bridges.

Indian Sand Painting

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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 :-)

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Knight's Tour

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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.