|
CURVES by
"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,
"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,
"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.
Abstract
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.
|