"During the harvest month, Tamil women in South India draw designs in front of the thresholds of their houses. In order to prepare their drawings, they set out a rectangular reference frame of equidistant points. Then curves are drawn in such a way that they surround the dots without touching them. The (culturally) ideal design is composed of a single closed line."

P.P.J.Gerdes: On ethnomathematical research and symmetry,
Symmetry: Culture and Science 1, 2 (1990), 154-170.

"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.

P.P.J.Gerdes: On ethnomathematical research and symmetry,
Symmetry: Culture and Science 1, 2 (1990), 154-170.

"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..."

G.Bain: Celtic Art - the Methods of Construction, Dower, New York, 1973.


              Mirror curves are present in ethnical art, as Tamil threshold designs or Tchokwe sand drawings. Historically, they are to be found in the art of most peoples surrounding the Mediterranean, the Black and Caspian Seas, in the art of Egyptians, Greeks, Romans, Byzantines, Moors, Persians, Turks, Arabs, Syrians, Hebrews and African tribes. Highlights are Celtic interlacing knotworks, Islamic layered patterns and Moorish floor and wall decorations. In this paper mirror curves are considered from the point of view of geometry, tiling theory, graph theory and knot theory. After the enumeration of mirror curves in a rectangular square grid, and a discussion of mirror curves in polyominoes and uniform tessellations, the construction of mirror curves is generalized to any surface.  

Introduction  and preliminaries

Let  a connected edge-to-edge tiling of some part of a plane by polygons be given. Connecting the midpoints of adjacent edges we obtain a 4-regular graph: the graph where in every vertex they are four edges, called steps. 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 we have three possibilities to continue our path: to choose left, middle, right edge. If the middle edge is chosen such vertex will be called a crossing. By introducing in every crossing the relation over-under, every mirror curve can be converted into a knotwork design.

The term “mirror curve” could be simply justified if we take a rectangular square grid RG[a,b] with the sides a and b, where that sides are mirrors, and the additional internal two-sided mirrors are placed between the square cells, coinciding with an edge, or perpendicular to it in its midpoint. In this case, a ray of light, emitted from one edge-midpoint, making with that edge a 45° angle, after the series of reflections will close a component. Beginning from different edge-midpoints, till exhausting the complete step graph, 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 polyamond, polyomino or polyhexe, respectively.

After the historical remarks about Tamil and Tchokwe mirror curves and knotwork designs created by Leonardo and Dürer, the rules for the construction of one-component mirror curves are given and mirror curves are generalized to any surface. Those algorithm rules are used for the combinatorial enumeration of mirror curves obtained from RG[a,b] with a minimal number of internal mirrors. After that are considered black-white designs derived from mirror curves (so called Lunda designs), the use of mirror curves for a polyomino shape 0-1 notation, and Lunda polyominoes; then, the mirror curves are considered from the point of view of knot theory.