Plane-Filling Properties of Directed Figures

Abstract

The process where simple entities form more complex structures acting autonomously is called self-assembly; it lies at the centre of many physical, chemical and biological phenomena. Massively parallel formation of nanostructures or DNA computation are just two examples of possible applications of self-assembly once it is technologically harnessed. Various mathematical models have been proposed for self-assembly, including the well-known Winfree’s Tile Assembly Model based on Wang tiles on a two-dimensional plane. In the present paper we propose a model based on directed figures with partial catenation. Directed figures are defined as labelled polyominoes with designated start and end points, and catenation is defined for non-overlapping figures. This is one of possible extensions generalizing words and variable-length codes to planar structures, and a flexible model, allowing for a natural expression of self-assembling entities as well as e.g. image representation or “pictorial barcoding.” We prove several undecidability results related to filling the plane with a given set of figures and formation of infinite and semi-infinite zippers, demonstrating a unifying approach that could be useful for the study of self-assembly.