Scaled pier fractals do not strictly self-assemble

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.