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

For more see:

Art and symmetries

Uncovering 'hidden' mathematical ideas: geometrical form

Networks, graphs or 'sanddrawings'

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