Impossibility and Efficiency Comparisions Between aTAM and 2HAM

Given that the 2HAM can simulate the aTAM (and that the converse is not true), it seems that the 2HAM is more powerful. Thus, it may be somewhat surprising that in it was shown that there is a simple class of shapes (so-called loops) which can be assembled with slightly greater tile type efficiency in the aTAM at temperature 1 than in the 2HAM at temperature 1. (However, this separation disappears at temperature 2.) Nonetheless, in it was also shown that there are shapes called staircases which can self-assemble in the 2HAM using roughly n tile types, while the aTAM requires a number exponential in n (and this can in fact be extended to the busy beaver function, BB(n)). In terms of impossibility, it was shown that there is a class of infinite shapes which self-assembles in the aTAM but not the 2HAM, and also a class of shapes which can self-assemble (in a weaker sense) in the 2HAM but not in the aTAM.