Angola-Zambia Sand Geometry-gctid29269

[Kala Kambon]

  Their symmetry is distinguished by Paulus   Gerdes, discussing so-called mirror curves resulting from knot   and link sand drawings from Lunda region (Eastern Angola and   Northwestern Zambia). If we have a some polyomino in a regular   plane tiling, with the set of (two-sided) mirrors incident to   the edges or perpendicular in their midpoints, the ray of light   starting from such midpoint, after series of reflections will   return to it, forming a closed path: a mirror curve. If the polyomino   is completely covered by a singular curve, it always represents   a knot projection; otherwise, if it is exhausted by several components,   it is a link projection. The drawing will be symmetrical or asymmetrical,   depending on the placement of internal mirrors, so the symmetry   is not a necessary property of mirror-curves. Anyway, they possess   the other remarkable property: the modularity. The possibilities   for the modular design of such structures are unlimited. The   variety could be obtained by using topological variations of   the prototiles, but also if we use different basic polyominoes,   resulting from Archimedean (uniform) plane tilings.
   Analyzing mirror-curves, Gerdes discovered   Lunda designs: if the successive smaller squares through which   the curve passes are colored in a color-alternating manner (black-white),   we obtain the black-and-white mosaic. Such designs possess the   local equilibrium property: each edge-midpoint is equally surrounded   by black and white small squares. Certainly, from the local equilibrium   results the global equilibrium in every row and in every column.   Every square Lunda design is a modular black-and-white design,   formed by only three kinds of prototiles (two kinds of internal   and one kind of border prototiles).
 

Example of a Tchokwe sand drawing

Similar ancient Egyptian pattern

        One section of Zaslavsky (1973a, 105-109)   was dedicated to networks, based on Torday's information (1925)   on the Bushongo (actual Zaire) and Bastin's study (1961) of decorative   art of the Tchokwe (Angola) (For educational use of Zaslavsky's   analysis of the Bushongo networks, see NCTM (1984) and Whitcombe   & Donaldson (1988)). She had not had access to the ethnographical   information on such networks published by Baumann (1935, 222-223),   Hamelberger (1952) and Dos Santos (1961). Since the publication   of "Kmt Counts" large ethnographical collections   of networks have become available: Pearson (1977: 'sandgraphs'   observed in the 1920s in the Kwandu-Kuvanga and Muxiku provinces   of Angola); Fontinha (1983: 'sona' or 'sanddrawings' collected   principally among the Tchokwe of northeastern Angola during the   1940s and 1950s); and Kubik (1986, 1987a, 1987b, 1988: networks   observed among the (Va)luchazi in northwestern Zambia during   the 1970s). In order to facilitate the memorisation of their   standardised 'sona', the drawing experts used the following mnemonic   device. After cleaning and smoothing the ground, they first set   out with their fingertips an orthogonal net of equidistant points.   Now one or more lines are drawn that 'embrace' the points of   the reference frame. By applying their method the drawing experts   reduce the memorisation of a whole drawing to that of mostly   two numbers (the dimensions of the reference frame) and a geometric   algorithm (the rule of how to draw the embracing line(s)). Most   drawings belong to a long tradition (cf. Redinha, 1948). They   refer to proverbs, fables, games, riddles, animals, etc. and   play an important role in the transmission of knowledge and wisdom   from one generation to the next. In Kubik's view the 'sona' "transmit   empirical mathematical knowledge" (1987a, 450). The geometry   of the 'sona' is a "non-euclidean geometry": "The   forefathers of the Eastern Angolan peoples discovered higher   mathematics and a non-Euclidian geometry on an empirical basis   applying their insights to the invention of these unique configurations"   (Kubik, 1987b, 108). He calls attention to the symmetry of many   'sona', the implicit rules for construction and rules for anchoring   figures of the same type. The ethnographical publications of   collections of 'sona' drew the attention of mathematicians. Ascher   and Gerdes conducted research on the 'sona', independently one   of another. Ascher's study (1988, 1991) deals with geometrical   and topological aspects of 'sona', in particular, with symmetries,   extension, enlargement through repetition, and isomorphy. Gerdes   (1989a, 120-189) analyses symmetry and monolinearity (i.e. a   whole figure is made up of only one line) as cultural values,   classes of 'sona' and corresponding geometrical algorithms for   their construction, systematic construction of monolinear groundpatterns,   chain and elimination rules for the construction of monolinear   'sona'. It is suggested that the 'drawing experts' who invented   these rules probably knew why they are valid, i.e. they could   prove in one or another way the truth of the theorems that these   rules express. He advances also with the reconstruction of lost   symmetries and monolinearities by means of an analysis of possible   drawing errors in reported 'sona' (for an introductory summary   of his research findings, see: Gerdes, 1990d, 1991d and 1991e).   Inspired by his historical research findings, Gerdes experimented   with possibilities to use the 'sona' in mathematics education,   in order to value and revive a rich scientific tradition that   had been vanishing (Cf. Gerdes, 1988a, b; 1989a, b, c; 1990a;   1991a and g; cf. Ratteray, 1991). He also initiated a mathematical   exploration of the properties of some (extended) classes of 'sona'   (see Gerdes, 1989a, 288-297). In a similar way, Kubik's research   stimulated a mathematical investigation by Jaritz on a particular   class of 'sona' (1983).
   For more see:
   Art   and symmetries
   Uncovering 'hidden' mathematical ideas: geometrical   form
   Networks,   graphs or 'sanddrawings'
   
 


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