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We investigate a fundamental question regarding a benchmark class of shapesin one of the simplest, yet most widely utilized abstract models of algorithmictile self-assembly. Specifically, we study the directed tile complexity of a $k\times N$ thin rectangle in Winfree's abstract Tile Assembly Model, assumingthat cooperative binding cannot be enforced (temperature-1 self-assembly) andthat tiles are allowed to be placed at most one step into the third dimension(just-barely 3D). While the directed tile complexities of a square and ascaled-up version of any algorithmically specified shape at temperature 1 injust-barely 3D are both asymptotically the same as they are (respectively) attemperature 2 in 2D, the bounds on the directed tile complexity of a thinrectangle at temperature 2 in 2D are not known to hold at temperature 1 injust-barely 3D. Motivated by this discrepancy, we establish new lower and upperbounds on the directed tile complexity of a thin rectangle at temperature 1 injust-barely 3D. We develop a new, more powerful type of Window Movie Lemma thatlets us upper bound the number of "sufficiently similar" ways to assign gluesto a set of fixed locations. Consequently, our lower bound,$\Omega\left(N^{\frac{1}{k}}\right)$, is an asymptotic improvement over theprevious best lower bound and is more aesthetically pleasing since iteliminates the $k$ that used to divide $N^{\frac{1}{k}}$. The proof of ourupper bound is based on a just-barely 3D, temperature-1 counter, organizedaccording to "digit regions", which affords it roughly fifty percent moredigits for the same target rectangle compared to the previous best counter.This increase in digit density results in an upper bound of$O\left(N^{\frac{1}{\left\lfloor\frac{k}{2}\right\rfloor}}+\log N\right)$, thatis an asymptotic improvement over the previous best upper bound and roughly thesquare of our lower bound.

Improved lower and upper bounds on the tile complexity of uniquely self-assembling a thin rectangle non-cooperatively in 3D