The Tchokwe people of south-central Africa have a tradition of storytelling using geometric drawings in the sand such as the one on the right, called Sona, as memory aids. Typically these drawings would be traced in the sand winding around a rectangular grid of dots using a finger, while telling the tale. Ideally the entire figure would be traced without having to remove the finger from the sand so that the entire diagram is constructed with a single curve that does not retrace itself. Often the figure could be drawn by following a simple geometric algorithm on a grid with particular dimensions. Below is an example of one of the simplest types of Sona – the plated mat design. To draw the figure one imagines that the edge of the rectangular grid is like a mirror with lines hitting the mirror and rebounding at 45^o .

Try tracing each design, making sure you don't change direction at any intersection. The figure on the left can be traced by a single continuous curve but the one on the right requires two curves. Is it possible to determine whether a given diagram can be traced with only one loop? If more than one loop is required how many loops are needed? You will address both these questions in this activity.

Gather enough stones so that you can make a rectangular grid of stones on the sand up to dimension 7x7. Fill in the following table by completing sand drawings for all possible rectangular grids with c columns and r rows and recording in the table the number of loops that were need to complete the diagram.

By observing the patterns in your table describe how you could determine the number of loops required to trace a rxc rectangular grid for any given number of rows r and columns c. Without drawing determine how many loops are required to trace a (a) 13 by 26 rectangle; (b) a 100 by; and (c) a 105 rectangle, a n by 3n rectangle.

More complicated drawings can be made by choosing different arrangement of stones to trace as shown in the diagrams below.

Investigate whether the number of loops required to trace a diagram changes when you add square arrays of stones to your rectangular grid. Write down your observations.

Some Sona designs correspond to light-ray patterns created with the addition of horizontal mirrors between the dots. The result is a pattern known as the lion's stomach. The example shown on the right can be drawn with a single loop.

Trace the diagram on the right. Next try using sticks in the sand to represent mirrors and try to trace diagrams using different sized grids with sticks arrange in an analogous pattern. Does the number of loops required to trace the grid change? Can you always trace the diagram? Write down your observations?

Diagrams are often modified somewhat in order to tell a story. As a final piece of work create your own Sona drawing to tell a simple story. An example is given below

The Hunter and the Dog