Pictures worth a thousand tiles, a geometrical programming language for self-assembly

We present a novel way to design self-assembling systems using a notion of signal (or ray) akin to what is used in analyzing the behaviour of cellular automata. This allows purely geometrical constructions, with a smaller specification and easier analysis. We show how to design a system of signals for a given set of shapes, and how to transform these signals into a set of tiles which self-assemble into the desired shapes.We show how to use this technique on two examples: squares (with optimal assembly time and a small number of tiles) and general polygons with arbitrarily good resolution.