New bounds on the tile complexity of thin rectangles at temperature-1

TL;DR

This paper establishes new theoretical bounds on the minimum number of tile types needed for self-assembling thin rectangles at temperature-1, introducing novel lower bounds and the first upper bounds for near-3D cases.

Contribution

It provides the first known upper bound for 3D thin rectangles at temperature-1 and improves the understanding of tile complexity limits in 2D.

Findings

Derived a new lower bound on tile complexity for 2D thin rectangles.

Established the first upper bound for 3D thin rectangles at temperature-1.

Introduced a just-barely 3D zig-zag counter construction.

Abstract

In this paper, we study the minimum number of unique tile types required for the self-assembly of thin rectangles in Winfree's abstract Tile Assembly Model (aTAM), restricted to temperature-1. Using Catalan numbers, planar self-assembly and a restricted version of the Window Movie Lemma, we derive a new lower bound on the tile complexity of thin rectangles at temperature-1 in 2D. Then, we give the first known 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 construction, 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.