In this thesis we study self-assembly systems, in particular, we focus on the tile complexity of shapes. Using the standard tile assembly model proposed by Rothemund and Winfree, we give two lower bounds on the tile complexity of arbitrary shapes in terms of the radius and diameter of the shape. Applying our results to a square yields a partial answer to a problem of Rothemund and Winfree. We also introduce a new model of self assembly—the step assembly model—which can significantly reduce the number of tile types needed to assemble a given shape. For this model, we give an upper bound on the tile complexity of arbitrary shapes and exhibit a family of shapes with constant tile complexity. Furthermore, we relate the tile complexity of a shape in the step assembly model to the well known node search number of its underlying spanning tree.
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