Scaled tree fractals do not strictly self-assemble

TL;DR

This paper proves that scaled versions of discrete self-similar tree fractals cannot strictly self-assemble in Winfree's abstract Tile Assembly Model, regardless of temperature, highlighting limitations in the assembly process.

Contribution

It establishes a fundamental limitation on the self-assembly of scaled tree fractals within the tile assembly framework.

Findings

Scaled tree fractals do not strictly self-assemble.

The result holds at any temperature in the model.

Limits the applicability of self-assembly for certain fractals.

Abstract

In this paper, we show that any scaled-up version of any discrete self-similar {\it tree} fractal does not strictly self-assemble, at any temperature, in Winfree's abstract Tile Assembly Model.