  We prove that if a subset X of the integer Cartesian plane weakly
self-assembles at temperature 1 in a deterministic (Winfree) tile assembly
system satisfying a natural condition known as *pumpability*, then X is a
finite union of doubly periodic sets. This shows that only the most simple of
infinite shapes and patterns can be constructed using pumpable temperature 1
tile assembly systems, and gives strong evidence for the thesis that
temperature 2 or higher is required to carry out general-purpose computation in
a tile assembly system. Finally, we show that general-purpose computation is
possible at temperature 1 if negative glue strengths are allowed in the tile
assembly model.
