Self-Assembly of Squares and Scaled Shapes

The abstract Tile Assembly Model (aTAM) [3] is a mathematical model of self-assembly in which system components are four-sided Wang tiles with glue types assigned to each tile edge. Any pair of glue types are assigned some nonnegative interaction strength denoting how strongly the pair of glues bind. An aTAM system is an ordered triplet .T; ; / consisting of a set of tiles T , a positive integer threshold parameter called the system’s temperature, and a special tile 2 T denoted as the seed tile. Assembly proceeds by attaching copies of tiles from T to a growing seed assembly whenever the placement of a tile on the 2D grid achieves a total strength of attachment from abutting edges, determined by the sum of pairwise glue interactions, that meets or exceeds the temperature parameter . The pairwise strength assignment between glues on tile edges is often restricted to be “linear” in that identical glue pairs may be assigned arbitrary positive values, while non-equal pairs are required to have interaction strengths of 0. We denote this restricted version of the model as the standard aTAM. When this restriction is not applied, i.e., any pair of glues may be assigned any positive integer strength, we call the model the flexible glue aTAM. Given the aTAM’s model of growth, we may consider the problem of designing an aTAM system which is guaranteed to grow into a target shape S , given by a set of 2D integer coordinates, and stop growing. Such systems are guaranteed to exist for any finite shape S , but solutions will typically vary in the number of tiles jT j used. For a given shape S , an interesting problem is to design a system that assembles S while using the fewest, or close to the fewest, number of tiles jT j possible. This fewest possible number of tiles required for the assembly of a given shape S is termed the program-size complexity of S .