Celebration of Scholarship and Creative Activity 2021
In this project, we study the problem of determining the minimum number of unique tile types required for the self-assembly of thin rectangles in the abstract Tile Assembly Model (aTAM). The aTAM is the simplest, yet most popular, discrete mathematical model of nano-scale DNA tile self-assembly. In this field, we try to minimize the number of unique tile types because it is expensive, time-consuming, and difficult to experimentally build many different types of molecules (tile types), but it is trivial to experimentally copy an existing tile type. Our objective to give an improved upper bound on the tile complexity of “just-barely” 3D thin rectangles at temperature-1, where tiles are allowed to be placed at most one step into the third dimension. Our self-assembling computer algorithm, which produces a unique terminal assembly, implements a just-barely 3D, zig-zag counter, whose base depends on the dimensions of the target rectangle, and whose digits are encoded geometrically, vertically oriented and in binary.