Scaled pier fractals do not strictly self-assemble
Abstract
A pier fractal is a discrete self-similar fractal whose generator contains at least one 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 scaled pier fractals, where a scaled fractal is the shape obtained by replacing each point in the original fractal by a c × c block of points, for some c ∈ Z+. We prove that no scaled discrete self-similar pier fractal strictly self-assembles, at any temperature, in Winfree's abstract Tile Assembly Model.
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