3.2.2  Tchokwe sand drawings

"The Tchokwe people of northeast Angola are well known for their beautiful decorative art. When they meet, they illustrate their conversations by drawings on the ground. Most of these drawings belong to a long tradition. They refer to proverbs, fables, games, riddles, etc. and play an important role in the transmission of knowledge from one generation to the other." (Gerdes, 1990

"...Just like the Tamils of South India, the Tchokwe people invented a similar mnemonic device to facilitate the memorization of their standardized drawings. After cleaning and smoothing the ground, they first set out with their fingertips an orthogonal net of equidistant points. The number of rows and columns depends on the motif to be represented. By applying their method, the Tchokwe drawing experts reduce the memorization of a whole design to that of mostly two numbers and a geometric algorithm. Most of they drawings display bilateral and/or rotational (90°or 180° ) symmetries. The symmetry of their pictograms facilitates the execution of a drawing. This is important, as the drawings have to be executed smoothly and continuously. Any hesitation or stopping on the part of the drawer is interpreted by the audience as an imperfection and lack of knowledge, and assented with an ironic smile." (Gerdes, 1990

Tchokwe sand drawings called sona (singular: lusona) played an important role in transmitting knowledge and wisdom from one generation to the next. Young boys enjoyed making sand drawings with their fingers, followed by stories about them. They learned a meaning and the execution of easier drawings during their period of intensive schooling, the mukanda initiation rites. The more difficult sona were only known by those story tellers, who were real akwa kuta sona (those who know how to draw), highly estimated and forming a part of an elite in Tchokwe society (Gerdes, 1993). 

"Leonardo spent much time in making a regular design of a series of knots so that the cord may be traced from one end to the other, the whole filling a round space..." (Bain, 1973). 

The construction of knot designs, closely connected with mirror curves, occupied the attention of two of the greatest painters-mathematicians: Leonardo and Dürer (Bain, 1973). In some of their constructions, they very efficiently used the following geometrical property: for a rectangular square grid RG[a,b] with the sides a and b, if a, b are relatively prime the result is always a single closed curve covering uniformly the square tiling of a rectangle. 

Let us notice one more beautiful geometrical property: mirror curves can be obtained by using only a few different prototiles. For the construction of all the curves, with internal mirrors incident to the cell-edges, three prototiles are sufficient in the case of a regular triangular tiling, five in the case of square, and 11 in the case of hexagonal regular tiling (Jablan, 1995). 

Using the combinations occurring in the 11 uniform Archimedean tilings (Grünbaum and Shephard, 1986), or prototiles producing the impression of space structures and colored prototiles, we may obtain very artistic interlacing patterns which are examples of modular design: the use of a few initial elements ( modules - prototiles) for creating an infinite collection of designs. The mirror curves obtained from Archimedean tilings remind us of the optical phenomenon: change of the direction of a light ray transferring from one physical environment to the other. 

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