A new model of geometric chirality for two-dimensional continuous media and planar meta-materials

Abstract

We have, for the first time, identified ten tenets of two-dimensional (2D) chirality that define and encapsulate the symmetry and scaling behaviour of planar objects and have used them to develop three new measures of geometric 2D chirality. All three models are based on the principle of overlap integrals and can be expressed as simple analytical functions of the two-dimensional surface density, rho(r). In this paper we will compare the predicted behaviour of these models and show that two of them are fully integrable and scalable and can therefore be applied to both discrete and continuous 2D systems of any finite size, or any degree of complexity. The only significant difference in these two models appears in their behaviour at infinite length scales. Such differences could, however, have profound implications for the analysis of chirality in new generations of planar meta-materials, such as chiral arrays, fractals, quasi-periodic 2D crystals and Penrose tiled structures.