Strictly Self-Assembling Discrete Self-Similar Fractals Using Quines

The abstract Tile-Assembly Model (aTAM) was initially introduced as a simple model for DNA-based self-assembly, where synthetic strands of DNA are used not as an information storage medium, but rather a material for nano-scale construction. Since then, it has been shown that the aTAM, and variant models thereof, exhibit rich computational dynamics, Turing completeness, and intrinsic universality, a geometric notion of simulation wherein one aTAM system is able to simulate every other aTAM system not just symbolically, but also geometrically. An intrinsically universal system is able to simulate all other systems within some class so that $m\times m$ blocks of tiles behave in all ways like individual tiles in the system to be simulated. In this paper, we explore the notion of a quine in the aTAM with respect to intrinsic universality. Typically a quine refers to a program which does nothing but print its own description with respect to a Turing universal machine which may interpret that description. In this context, we replace the notion of machine with that of an aTAM system and the notion of Turing universality with that of intrinsic universality. Curiously, we find that doing so results in a counterexample to a long-standing conjecture in the theory of tile-assembly, namely that discrete self-similar fractals (DSSFs), fractal shapes generated via substitution tiling, cannot be strictly self-assembled. We find that by growing an aTAM quine, a tile system which intrinsically simulates itself, DSSF structure is naturally exhibited. This paper describes the construction of such a quine and even shows that essentially any desired fractal dimension between 1 and 2 may be achieved.