Triangular Self-Assembly

TL;DR

This paper explores the computational capabilities of triangular tile self-assembly systems, demonstrating their Turing universality and comparing their expressive power to square tile systems, with specific results on shape assembly complexity.

Contribution

It establishes that triangular tile systems are Turing universal and compares their power to square systems, providing counterexamples and complexity bounds.

Findings

Triangular tile systems are Turing universal.

They are not directly comparable to square systems in shape production.

Assembling a triangle of size O(N^2) requires O(log N / log log N) tiles.

Abstract

We discuss the self-assembly system of triangular tiles instead of square tiles, in particular right triangular tiles and equilateral triangular tiles. We show that the triangular tile assembly system, either deterministic or non-deterministic, has the same power to the square tile assembly system in computation, which is Turing universal. By providing counter-examples, we show that the triangular tile assembly system and the square tile assembly system are not comparable in general. More precisely, there exists square tile assembly system S such that no triangular tile assembly system is a division of S and produces the same shape; there exists triangular tile assembly system T such that no square tile assembly system produces the same compatible shape with border glues. We also discuss the assembly of triangles by triangular tiles and obtain results similar to the assembly of squares, that is to assemble a triangular of size O(N^2), the minimal number of tiles required is in O(log N/log log N).