TITLE: What one really can do with fractals built from L-systems?
QUESTION [8 upvotes]: For any L-system one can naturally associate a fractal. Why these fractals are (mathematically) useful apart that they are a source of nice pictures?

REPLY [5 votes]: The Wikipedia page for L-systems links to the Thue-Morse sequence for which there are a large number of applications.  In particular it can be interpreted as the sequence of distances between atoms in a 1D quasicrystal.  Hence I suppose you might think about the work on substitution tilings and higher dimensional quasicrystals as an application of similar ideas.
Thinking even more broadly, the recent work by Borrelli, Jabrane, Lazarus and Thibert on implementing convex integration for a $C^1$ embedding of a flat torus in $R^3$ uses explicitly an L-system-like rewriting process to add corrugations.