Centralised connectivity-preserving transformations for programmable matter

We study a model of programmable matter systems consisting of n devices lying on a 2-dimensional square grid which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B . We investigate the class of shapes which can be constructed in such a scenario under the additional constraint of maintaining global connectivity at all times. We focus on the scenario of transforming nice shapes , a class of shapes consisting of a central line L where for all nodes u in S either u ∈ L or u is connected to L by a line of nodes perpendicular to L . We prove that by introducing a minimal 3-node seed it is possible for the canonical shape of a line of n nodes to be transformed into a nice shape of n − 1 nodes. We use this to show that a 4-node seed enables the transformation of nice shapes of size n into any other nice shape of size n in O ( n 2 ) time. We leave as an open problem the expansion of the class of shapes which can be constructed using such a seed to include those derived from nice shapes. • We study theoretical models of programmable matter systems. • One minimal mechanical operation is allowed: rotation. • Focus on transformability questions: can a given shape be transformed into another? • With the help of three additional nodes can convert a line of nodes into any nice shape with constant waste. • With the help of four additional nodes can convert a nice shape into any other nice shape via the line without waste.