  The well-studied Two-Handed Tile Assembly Model (2HAM) is a model of tile
assembly in which pairs of large assemblies can bind, or self-assemble,
together. In order to bind, two assemblies must have matching glues that can
simultaneously touch each other, and stick together with strength that is at
least the temperature $\tau$, where $\tau$ is some fixed positive integer. We
ask whether the 2HAM is intrinsically universal, in other words we ask: is
there a single universal 2HAM tile set $U$ which can be used to simulate any
instance of the model? Our main result is a negative answer to this question.
We show that for all $\tau' < \tau$, each temperature-$\tau'$ 2HAM tile system
does not simulate at least one temperature-$\tau$ 2HAM tile system. This
impossibility result proves that the 2HAM is not intrinsically universal, in
stark contrast to the simpler (single-tile addition only) abstract Tile
Assembly Model which is intrinsically universal ("The tile assembly model is
intrinsically universal", FOCS 2012). However, on the positive side, we prove
that, for every fixed temperature $\tau \geq 2$, temperature-$\tau$ 2HAM tile
systems are indeed intrinsically universal: in other words, for each $\tau$
there is a single universal 2HAM tile set $U$ that, when appropriately
initialized, is capable of simulating the behavior of any temperature-$\tau$
2HAM tile system. As a corollary of these results we find an infinite set of
infinite hierarchies of 2HAM systems with strictly increasing simulation power
within each hierarchy. Finally, we show that for each $\tau$, there is a
temperature-$\tau$ 2HAM system that simultaneously simulates all
temperature-$\tau$ 2HAM systems.
