Contents of this web page
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
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
A kolam is a South Indian floor
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
These kolams are more precisely
called stroke kolams on the English Wikipedia page, neli kolam, kambi kolam
or sikku kolam in Tamil.
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
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.
||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.
of a Kolam
The structure of a kolam
is understood by looking for closed lines. They become visible by a yellow
- 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.
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.
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
- Give a simple kolam and
Sona are traditional drawings of
some Bantu people like the Chokwe in an area of Angola and Zambia in southern
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.
The figure to a 4x3 grid is a lusona. It therefore consists
of a closed line without beginning and end.
As the Sona complement stories,
they often depict animals.
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?
||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.
||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
||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."
Drawing sona is difficult because
of the closed line.
This will be discussed in the next
||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
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 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.
Search of Sona
The question arises which rectangular figures m*n is
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
- 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
||The rectangles overlap in such a way that together they
are further created by a closed line.
||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.
With the insertion of a mirror,
a figure can no longer created by a closed line.
||You need two closed lines to capture the figure.
||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.
Result: Among the 42 figures examined, eight are a lusona.
||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.
||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 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.
||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
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.
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.
The 3x3-Kolam and the 4x3-Lusona
become Celtic knots, if the lines are chosen thicker and along a line alternate
underpasses and bridges.
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.
Beach Painting :-)
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
(1) South African History Online (www.sahistory.org.za)
(2) https://www.zukunft-irular.de/neue-seite/ (Sona-Geometrie:
und Sona on the Internet top
Die flüchtige Kunst der Inderinnen
Henning Krause (Spektrum)
Dr. Henning Krause (Spektrum)
- dargestellt am Beispiel der Sona Geometrie
M. Weber, A. Mischau (Mathematisches
und (bzw. in) andere(n) Kulturtechniken (.pdf-Datei)
Experiments with African Sona Designs
Symmetric Chokwe Sand Drawings
Marcia Ascher (spektrum.de)
Die Kolam-Figuren Südindiens
Mattia De’ Michieli Vitturi
drawings, mirror curves and pattern designs
- Sand Drawings from Africa
Sona de Angola Matemática duma Tradição Africana
Portuguese, 191 pages, lots of illustrations
Slavik Jablan, Ljiljana Radovi,
Radmila Sazdanovi Ana Zekovi
South African History Online
Patterns - Revisiting the Contributions of the People in Sub-Saharan Africa
to Modern Mathematics
theme for Chennai's Kolam and Rangoli competition
A meeting of women artists
Some drawings and stories
KOLAM WITH 10-2 DOTS | HOW TO DRAW A SIKKU KOLAM | NELI KOLAM
Example of an Indian floor painting
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
page is also available in German.
3/2023 Jürgen Köller