Self-similar substitution in the SEH_00 tiling

Determine whether the SEH_00 hexagonal quasiperiodic tiling admits a self-similar substitution system that recursively generates the connected curves formed by rhombus tiles, reproducing successive generations of the associated fractal skeleton.

Background

In the paper, the authors construct fractal “skeletons” via a metallic-mean L-system and decorate them with rhombus tiles to obtain tilings. For j = 3, they observe that the connected curves of rhombus tiles within the SEH_00 tiling correspond to early generations of the fractal skeleton, suggesting a possible inflation/deflation structure.

They explicitly state that whether a self-similar substitution system exists for SEH_00 remains open, motivating a precise investigation into substitution rules that could reproduce these connected rhombus curves across scales.