  This paper gives a (polynomial time) algorithm to decide whether a given
Discrete Self-Similar Fractal Shape can be assembled in the aTAM model.In the
positive case, the construction relies on a Self-Assembling System in the aTAM
which strictly assembles a particular self-similar fractal shape, namely a
variant $K^\infty$ of the Sierpinski Carpet. We prove that the aTAM we propose
is correct through a novel device, \emph{self-describing circuits} which are
generally useful for rigorous yet readable proofs of the behaviour of aTAMs.We
then discuss which self-similar fractals can or cannot be strictly
self-assembled in the aTAM. It turns out that the ability of iterates of the
generator to pass information is crucial: either this \emph{bandwidth} is
eventually sufficient in both cardinal directions and $K^\infty$ appears within
the fractal pattern after some finite number of iterations, or that bandwidth
remains ever insufficient in one direction and any aTAM trying to self-assemble
the shape will end up either bounded with an ultimately periodic pattern
covering arbitrarily large squares. This is established thanks to a new
characterization of the productions of systems whose productions have a
uniformly bounded treewidth.
