  In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an
accretion-based model which only allows for a single tile to attach to a
growing assembly at each step, there are no tile assembly systems capable of
self-assembling the discrete self-similar fractals known as the "H" and "U"
fractals. We then show that in a related model which allows for hierarchical
self-assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile
assembly systems which self-assembles the "U" fractal and conjecture that the
same holds for the "H" fractal. This is the first example of discrete self
similar fractals which self-assemble in the 2HAM but not in the aTAM, providing
a direct comparison of the models and greater understanding of the power of
hierarchical assembly.
