Scaled pier fractals do not strictly self-assemble

TL;DR

This paper proves that scaled pier fractals, a class of self-similar fractals with specific generator properties, cannot strictly self-assemble in Winfree's abstract Tile Assembly Model at any temperature.

Contribution

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

Findings

Scaled pier fractals do not strictly self-assemble in the model.

The result holds at any temperature setting.

This imposes constraints on the design of self-assembling fractal structures.

Abstract

A \emph{pier fractal} is a discrete self-similar fractal whose generator contains at least one \emph{pier}, that is, a member of the generator with exactly one adjacent point. Tree fractals and pinch-point fractals are special cases of pier fractals. In this paper, we study \emph{scaled pier fractals}, where a \emph{scaled fractal} is the shape obtained by replacing each point in the original fractal by a $c \times c$ block of points, for some $c \in \mathbb{Z}^+$. We prove that no scaled discrete self-similar pier fractal strictly self-assembles, at any temperature, in Winfree's abstract Tile Assembly Model.