  In the abstract Tile Assembly Model, self-assembling systems consisting of
tiles of different colors can form structures on which colored patterns are
``painted.'' We explore the complexity, in terms of the numbers of unique tile
types required, of assembling various patterns. We first demonstrate how to
efficiently self-assemble a set of simple patterns, then show tight bounds on
the tile type complexity of self-assembling 2-colored patterns on the surfaces
of square assemblies. Finally, we demonstrate an exponential gap in tile type
complexity of self-assembling an infinite series of patterns between systems
restricted to one plane versus those allowed two planes.
