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. The mentioned paper focusses on different tilings.

[ND96]

Nischke, K-P and Danzer, L

**A construction of inflation rules based on $n$-fold symmetry**

*Discrete and Computational Geometry*
1996,
15,2,
pp. 221-236,
96j:52035