Fractal nature of growth forms

Ascertain whether any tiling growth form can be a fractal set; if not, demonstrate that growth forms must be boundaries of star bodies.

Background

Growth forms observed in periodic and regular grid settings are polytopal or smooth, but the broader class of tilings could, in principle, exhibit fractal boundaries. The authors explicitly state uncertainty and propose a negative conjecture, suggesting growth forms are star-body boundaries.

Resolving this would clarify fundamental geometric limitations of growth forms across tiling classes.