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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?

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  • $\begingroup$ $\mathbb Z \cup r\mathbb Z$ for $r\in \mathbb R\setminus\mathbb Q$ (or, as a density, $\rho(x) = \sum_n \delta(x-n)+ \sum_n \delta(x-rn)$) is quasiperiodic but not self-similar. $\endgroup$ Sep 7, 2018 at 16:31
  • $\begingroup$ sciencedirect.com/science/article/abs/pii/S0022024804004014 I'm not sure if this is useful to you or not. Perhaps your Fibonacci example is one of non-linear self similarity $\endgroup$
    – Kai
    Apr 22, 2023 at 19:02

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