Triangular Tile Self-assembly Systems

We discuss theoretical aspects of the self-assembly of triangular tiles; in particular, right triangular tiles and equilateral triangular tiles. Contrary to intuition, we show that triangular tile assembly systems and square tile assembly systems are not comparable in general. More precisely, there exists a square tile assembly system S such that no triangular tile assembly system that is a division of S produces the same final supertile. There also exists a deterministic triangular tile assembly system T such that no square tile assembly system produces the same final supertiles while preserving border glues. We discuss the assembly of triangles by triangular tiles and show triangular systems with Θ(logN/loglogN) tiles that can self-assemble into a triangular supertile of size Θ(N 2). Lastly, we show that triangular tile assembly systems, either right-triangular or equilateral, are Turing universal.