  This paper concerns the self-assembly of scaled-up versions of arbitrary
finite shapes. We work in the multiple temperature model that was introduced by
Aggarwal, Cheng, Goldwasser, Kao, and Schweller (Complexities for Generalized
Models of Self-Assembly, SODA 2004). The multiple temperature model is a
natural generalization of Winfree's abstract tile assembly model, where the
temperature of a tile system is allowed to be shifted up and down as
self-assembly proceeds. We first exhibit two constant-size tile sets in which
scaled-up versions of arbitrary shapes self-assemble. Our first tile set has
the property that each scaled shape self-assembles via an asymptotically
"Kolmogorov-optimum" temperature sequence but the scaling factor grows with the
size of the shape being assembled. In contrast, our second tile set assembles
each scaled shape via a temperature sequence whose length is proportional to
the number of points in the shape but the scaling factor is a constant
independent of the shape being assembled. We then show that there is no
constant-size tile set that can uniquely assemble an arbitrary (non-scaled,
connected) shape in the multiple temperature model, i.e., the scaling is
necessary for self-assembly. This answers an open question of Kao and Schweller
(Reducing Tile Complexity for Self-Assembly Through Temperature Programming,
SODA 2006), who asked whether such a tile set existed.
