3.2.3  Mirror curves

"The imitation of the three-dimensional arts of plaiting, weaving and basketry was the origin of interlaced and knotwork interlaced designs. There are few races that have not used it as a decoration of stone, wood and metal. Interlacing rosettes, friezes and ornaments are to be found in the art of most people surrounding the Mediterranean, the Black and Caspian Seas, Egyptians, Greeks, Romans, Byzantines, Moors, Persians, Turks, Arabs, Syrians, Hebrews and African tribes. Their highlights are Celtic interlacing knotworks, Islamic layered patterns and Moorish floor and wall decorations." (Bain, 1973

The common geometrical construction principle of these designs, discovered by P. Gerdes, is the use of (two-sided) mirrors incident to the edges of a square, triangular or hexagonal regular plane tiling, or perpendicular to its edges in their midpoints (Gerdes, 1990, 1996, 1997, 2000). In the ideal case, after the series of consecutive reflections, the ray of light reaches its beginning point, defining a single closed curve. In other cases, the result consists of several curves of that type. For example, to the Celtic designs from G. Bain's book Celtic Art (1973), correspond the following mirror-schemes.

Trying to discover their common mathematical background, two questions appear: how to construct such a perfect curve- a single line placed uniformly in a regular tiling, that is, how to arrange the set of mirrors generating it, and how to classify the curves obtained. In principle, any polyomino (polyiamond or polyhexe) (Golomb, 1994) with mirrors on its border, and two-sided mirrors between cells or perpendicular to the internal cell-edges in their midpoints, could be used for the creation of the corresponding curves. 

For their construction in some polyomino (polyiamond or polyhexe), we propose the following method. First, construct all the different curves in it containing lines that connect different cell-edge midpoints till the polyomino is exhausted, i.e., uniformly covered by k curves. After that, in order to obtain a single curve we introduce internal mirrors and "curve surgery", according to the following rules

  1. any mirror introduced in a crossing point of two distinct curves connects them into one curve; 

  2. depending on the position of a mirror, a mirror introduced into a self-crossing point of an (oriented) curve either does not change the number of curves, or breaks the curve into two closed curves.

In every polyomino we may introduce k-1, k, k+1,..., 2A-P/2 internal two-sided mirrors, where A is the area and P is the perimeter of the polyomino. Introducing the minimal number of mirrors k-1, we first obtain a single curve, and we try to preserve that when we add new mirror.