  In this paper we consider the problem of the strict self-assembly of infinite
fractals within tile self-assembly. In particular, we provide tile assembly
algorithms for the assembly of the discrete Sierpinski triangle and the
discrete Sierpinski carpet within a class of models we term the
\emph{$h$-handed assembly model} ($h$-HAM), which generalizes the 2-HAM to
allow up to $h$ assemblies to combine in a single assembly step. Despite
substantial consideration, no purely growth self-assembly model has yet been
shown to strictly assemble an infinite fractal without significant modification
to the fractal shape. In this paper we not only achieve this, but in the case
of the Sierpinski carpet are able to achieve it within the 2-HAM, one of the
most well studied tile assembly models in the literature. Our specific results
are as follows. We provide a $6$-HAM construction for the Sierpinski triangle
that works at scale factor 1, 30 tile types, and assembles the fractal in a
\emph{near perfect} way in which all intermediate assemblies are finite-sized
iterations of the recursive fractal. We further assemble the Sierpinski
triangle within the $3$-HAM at scale factor 3 and 990 tile types. For the
Sierpinski carpet, we present a $2$-HAM result that works at scale factor 3 and
uses 1,216 tile types. We further include analysis showing that the aTAM is
incapable of strictly assembling the non-tree Sierpinski triangle considered in
this paper, and that multiple hands are needed for the near-perfect assembly of
the Sierpinski triangle and the Sierpinski carpet.
