Self-Assembly of a Statistically Self-Similar Fractal

TL;DR

This paper presents a tile assembly system that constructs a statistically self-similar Sierpinski Triangle, introducing methods to handle infinite coding sequences and a new concept of local determinism in stochastic fractal assembly.

Contribution

It introduces a novel tile assembly mechanism for a random fractal, with techniques for managing infinite sequences and a new form of local determinism in stochastic self-assembly.

Findings

First self-assembly of a random fractal

Mechanism for nested recursion in tile assembly

Definition of almost-everywhere local determinism

Abstract

We demonstrate existence of a tile assembly system that self-assembles the statistically self-similar Sierpinski Triangle in the Winfree-Rothemund Tile Assembly Model. This appears to be the first paper that considers self-assembly of a random fractal, instead of a deterministic fractal or a finite, bounded shape. Our technical contributions include a way to remember, and use, unboundedly-long prefixes of an infinite coding sequence at each stage of fractal construction; a tile assembly mechanism for nested recursion; and a definition of "almost-everywhere local determinism," to describe a tileset whose assembly is locally determined, conditional upon a zeta-dimension zero set of (infinitely many) "input" tiles. This last is similar to the definition of randomized computation for Turing machines, in which an algorithm is deterministic relative to an oracle sequence of coin flips that provides advice but does not itself compute. Keywords: tile self-assembly, statistically self-similar Sierpinski Triangle.