3.2  Mirror curves

Any connected edge-to-edge tiling of some part of a plane by polygons can be used as the basis for the following construction. Connect the midpoints of adjacent edges to obtain a 4-regular graph: a graph where there are four edges, called steps, at every vertex. A path in that graph is a connected series of steps, where each step appears only once. Every closed path in that graph is called a component. The set of all components of such graph is called a mirror curve. In every vertex there are three possibilities to continue the path: to choose the left, middle, or right edge. If the middle edge is chosen that vertex will be called a crossing. By introducing the relation "over-under" in every crossing, every mirror curve can be converted into a knotwork design. The justification for the term "mirror curve" is apparent if we visualize a square grid RG[a,b] whose sides a and b are mirrors, and additional internal two-sided mirrors are placed between the square cells, coinciding with an edge, or perpendicular to it at its midpoint. In this grid, a ray of light, emitted from one edge-midpoint, at an angle of 45°, will close a component after a series of reflections. Beginning from a different edge-midpoint, and continuing until the complete step graph is exhausted, we obtain a mirror curve. It is easy to conclude that the preceding description could be extended to any connected part of a regular triangular, square or hexagonal tessellation, this means to any polyiamond, polyomino or polyhexe, respectively. 

3.2.1  Tamil treshold designs

"During the harvest month of Margali (mid-December to mid-January), the Tamil women in South India used to draw designs in front of the thresholds of their houses every morning. Margali is the month in which all kinds of epidemics were supposed to occur. Their designs serve the purpose of appeasing the god Siva who presides over Margali. In order to prepare their drawings, the women sweep a small patch of about a yard square and sprinkle it with water or smear it with cow-dung. On the clean, damp surface they set out a rectangular reference frame of equidistant dots. Then the curve(s) forming the design is (are) made by holding rice-flour between the fingers and, by a slight movement of them, letting it fall out in a closed, smooth line, as the hand is moved in the desired directions. The curves are drawn in such a way that they surround the dots without touching them." (Gerdes, 1989). 

The (culturally) ideal design is composed of a single continuous line. Names given to designs formed of a single "never-ending" line are normally pavitram, meaning "ring" and Brahma-mudi or "Brahma's knot". The purpose of the pavitram is to scare away giants, evil spirits, or devils. 

Is it not strange that the design composed of two or several superimposed closed paths, are nevertheless called pavitram? Maybe the designs formed of a few never-ending lines are just degraded versions of originally single closed path figure? Is it possible to construct a design rather similar to them, but made out of only one line? Slightly changing them, we may transform some imperfect, multi-linear designs into the ideal ones. 

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