  We contribute results for a set of fundamental problems in the context of
programmable matter by presenting algorithmic methods for evaluating and
manipulating a collective of particles by a finite automaton that can neither
store significant amounts of data, nor perform complex computations, and is
limited to a handful of possible physical operations. We provide a toolbox for
carrying out fundamental tasks on a given arrangement of tiles, using the
arrangement itself as a storage device, similar to a higher-dimensional Turing
machine with geometric properties. Specific results include time- and
space-efficient procedures for bounding, counting, copying, reflecting,
rotating or scaling a complex given shape.
