The diffraction patterns of quasicrystals very often display self-similarity ie. similarity under length scaling, thus relating them to fractals.
My question is: Do they always display self-similarity?
Standard literature (e.g. Janot, 'Quasicrystals: A primer') has not been very helpful since it simply states (without explanation) on p. 34
It is worth pointing out that quasiperiodic structures are self-similar, but self-similarity generally does not ensure quasiperiodicity, even though it imposes some sort of long-range order.
On the one hand, I wonder how he knows that quasiperiodic structures are self-similar since I think I have a counterexample. The Fibonacci chain (see my related question) with $S = 1$, $L \neq \textrm{golden mean}$, is quasiperiodic but evidently not self-similar.
On the other hand, all the diffraction images of quasicrystals that I have seen are self-similar.
So, with my counterexample, am I missing the point? Or is this a dimensionality thing, say in 2D and 3D quasiperiodicity implies self-similarity and in 1D not?
