## Sona Geometry

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### What are Sona?

The Chokwe people of northeastern Angola, and adjacent countries, are well known for their beautiful decorative art, ranging from the ornamentation of plaited mats and baskets, iron work, ceramics, engraved calabash fruits and tatooings, to paintings on house walls and sand drawings. Their drawings in the sand are called **sona** (sing. **lusona**). These drawings, examples of which are shown on the right, are used both in their decorative artwork and in their story-telling tradition.

To draw a lusona, the Chokwe first clean and smooth the ground or an area of sand, and then set out with their fingertips a net of points, which are then used to help in drawing the figure. The distances between two horizontal or vertical neighboring points should all be the same. The number of these points depends on the motif to be represented. When telling their stories through these pictures, the drawing process often parallels the development of the story, and these points may represent characters in the stories. The pictures on the right show a traditional lusona called "Lioness with Two Cubs" as it would appear drawn in the sand, followed by a geometric interpretation of the drawing. The mathematics of this particular lusona will be discussed later.

In general, a lusona is drawn by imagining a "bouncing ball" traveling through the net of points, and "bouncing" off an invisible wall whenever it leaves the area covered by those points. For more interesting patterns, we can place "walls" inside the layout of points to cause the bouncing ball to bounce off these internal obstacles. (The exact location of these "walls" is shown here, but not in an actual sand drawing -- the artist needs to remember where they are, or remember something equivalent to that.) For most sona, the goal is that the entire drawing should be able to be drawn using a single continuous line, without picking the finger up out of the sand, and without re-drawing any portion of the line. While this description can lead to a surprisingly large number of fascinating patterns, it still gives only an introduction to the variety of these patterns as drawn by the Chokwe.

### What were Sona used for?

The Tchokwe people are decorators and used these types of patterns as ornamentation. Specific lusona refer to proverbs, fables, games, riddles, animals, etc. and used to play an important role in the transmission of knowledge from one generation to the next. Every boy learned the meaning and execution of the easier sona during the intensive schooling phase of the circumcision and initiation rites. The meaning and execution of the more difficult sona was only known by specialists, the so called **akwa kuta sona** (those who know how to draw), who transmitted their knowledge to their sons.

With the colonial penetration and occupation the lusona tradition began to disappear. When researchers began collecting these in the 1940's and 50's, only a few old men knew the "secrets" of the more complicated drawings, and it was not easy to convince them to execute those sona as they said that formerly they had been more skillful at the art. It is important to emphasize that the designs have to be executed smoothly and continuously, while telling the associated story. Any hesitation or stopping halfway on the part of the drawer is interpreted by the audience as an imperfection and lack of knowledge, and reacted to with an ironic smile.

### The Mathematics of Sona:

A particularly challenging question, which seems to have been investigated extensively, and experimentally, by the Chokwe is "What layouts of dots, or dots and walls, can give rise to one-line drawings, when following the "bouncing ball" rule above?" Mathematicians investigating these drawings have discovered a variety of theorems addressing parts of this question. The two most fundamental theorems discovered seem to have been known, in one form or another, to the Chokwe artists. These theorems are both demonstrated in the "Lioness with Two Cubs" lusona shown above. The theorems are:

**Theorem 1:** A rectangular array of dots (with no internal walls) will give rise to a one-line lusona if and only if the dimensions of that rectangle are relatively prime.

For example, the main shape of the lioness (the mother only) is a 3 X 10 rectangle. Since 3 and 10 have no common factors, a 3 X 10 rectangle will generate a one-line lusona. Our suspicion that this idea, in one form or another, was known to the Chokwe artists gets support from the variety of such relatively prime rectangles that were used as the base shape of one or more sona. In particular, of the 30 smallest relatively prime rectangular shapes, 75% appear in Chokwe sona.

**Theorem 2:** If a square of dots is attached along one side to a one-line lusona, the new arrangement will also generate a one-line sona.

The lion cubs can each be viewed as being created by attaching two different 2 X 2 squares of dots to the main rectangle. At each step, as the artists adds one of these four squares of dots, this theorem tells us that the new shape will remain a one-line lusona. The number of sona that are constructed by this addition of 2 X 2 squares is quite substantial, and hence it seems that the Chokwe artists were aware of this theorem at least for 2 X 2 squares. There are a few examples of the addition of 3 X 3 squares, but not enough other sizes of "added squares" to feel confident that the artists understood this theorem in the generality stated above. However, it does seem clear that they had experimentally discovered this fact for 2 X 2 squares.