The FASS-curve of the pentagon bases on an aperiodic substitution tiling with four substitution rules and appropriate decorations.
The substitution tiling was derived from the Robinson Triangle Tiling.
Its inflation factor is the golden mean $\frac{\sqrt{5}}{2} + \frac{1}{2} = 1.618033988\ldots$
.
[Pau2021]
Pautze, S
Space-Filling, Self-Similar Curves of Regular Pentagons, Heptagons and Other n-Gons
Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture
2021,
38,
157-164,
bridges2021-157.pdf