  We investigate the power of the Wang tile self-assembly model at temperature
1, a threshold value that permits attachment between any two tiles that share
even a single bond. When restricted to deterministic assembly in the plane, no
temperature 1 assembly system has been shown to build a shape with a tile
complexity smaller than the diameter of the shape. In contrast, we show that
temperature 1 self-assembly in 3 dimensions, even when growth is restricted to
at most 1 step into the third dimension, is capable of simulating a large class
of temperature 2 systems, in turn permitting the simulation of arbitrary Turing
machines and the assembly of $n\times n$ squares in near optimal $O(\log n)$
tile complexity. Further, we consider temperature 1 probabilistic assembly in
2D, and show that with a logarithmic scale up of tile complexity and shape
scale, the same general class of temperature $\tau=2$ systems can be simulated
with high probability, yielding Turing machine simulation and $O(\log^2 n)$
assembly of $n\times n$ squares with high probability. Our results show a sharp
contrast in achievable tile complexity at temperature 1 if either growth into
the third dimension or a small probability of error are permitted. Motivated by
applications in nanotechnology and molecular computing, and the plausibility of
implementing 3 dimensional self-assembly systems, our techniques may provide
the needed power of temperature 2 systems, while at the same time avoiding the
experimental challenges faced by those systems.
