Chiral and achiral square-cell configurations; the degree of chirality

Abstract

The shape features, in particular, the chirality properties of the patterns of molecules adsorbed on various surfaces, can be modeled by square cell configurations in the plane. Square-cell configurations can be represented by planar graphs. Various families of planar graphs are chiral in two dimensions. A drawing of a tree in the plane, for example, the letter F, is called a plane tree. Chiral plane trees were counted by Harary and Robinson; this result was extended to the enumeration of chiral alkanes in 3-space by Robinson, Harary and Balaban. The shape of a Jordan curve in the plane can be modeled by square cell configurations, which we call animals. Many of the animals show two-dimensional chirality. This suggests the dichotomy of chiral and achiral animals for the shape characterization of Jordan curves, such as molecular curves, and cross-sections of molecular contour surfaces. Chirality of animals can be analysed in terms of the animal codes introduced earlier, leading to the concept of the degree of chirality of Jordan curves and to the conjecture that almost all animals are chiral.