Danzer's 7-fold
A substitution tiling with three triangles as prototiles, based on 7-fold symmetry. The four different edge lengths occurring are $\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$, $\sin(\frac{3\pi}{7})$, $\sin(\frac{2\pi}{7}) + \sin(\frac{3\pi}{7})$, The inflation factor is $1+{\sin(\frac{2\pi}{7})}/{\sin(\frac{\pi}{7})}$ , which is not a PV number. There are simple matching rules for the tiling. In fact, the list of all vertex stars occurring in the substitution tiling serves as one. This is stated in [ND96], but never really published, up to my knowledge.

Without Decoration
Finite Rotations
Polytopal Tiles
Self Similar Substitution
Matching Rules