I'm not too familiar with quasi-crystals, but I was recently playing around with a particular discrete function and I got the following neat pattern:


This is just a small segment, but as far as I could tell it does not become periodic. So I am wondering whether this is a quasi-crystal. More exactly:

  1. How would I go about figuring out whether or not this is a quasi-crystal?
  2. Does anyone recognize this pattern?

Knowing of the peculiar diffraction patterns of quasi-crystals, I took the Fourier transform of the above and obtained the following (again just showing a segment):

fourier transformation

(Details: this is in log-scale, with bright colors denoting high values, dark low values)

I find this quite beautiful and hence I'd love to have a better understanding of what's going on. In particular it seems to me that this might be a fractal. Again I am wondering how to formalize this and/or what the best way would be to advance if I want to 'understand' either of these patterns. Basically I'm not sure what to 'look for', which also makes it hard to make my question very concrete.

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    $\begingroup$ it seems your diffraction pattern has four-fold symmetry, which is not what one would expect for a quasicrystal; I also do not see the self-similarity at different scales I would expect for a fractal; you could just have some pattern with short-range order but without any long-range order. $\endgroup$ – Carlo Beenakker Nov 8 '16 at 20:30
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    $\begingroup$ @CarloBeenakker Thank you for your comment! Today I stumbled upon an interesting article about quasicrystals and fourfold symmetry, and I thought you might be interested. Ron Lifshitz ( tau.ac.il/~ronlif/pubs/JAlCom342-186-2002.pdf ) has constructed a 2D Fibonacci tiling which has most properties one would assign to a quasi-crystal (e.g. non-periodic but repetitive, can be built as a cut-and-project from a higher dimensional lattice, Fourier spectrum is a set of Dirac deltas, etc) but it has a fourfold symmetry. In particular he says: [cont] $\endgroup$ – Ruben Verresen Dec 4 '16 at 18:35
  • $\begingroup$ "According to the prevailing (though unofficial) definition, a quasicrystal is required to have some ‘forbidden’ symmetry, incompatible with periodicity—at least one axis of n-fold symmetry with n = 5 or n > 6. I have argued elsewhere that this requirement should officially be removed from the definition of quasicrystals. The existence of the square Fibonacci tiling supports this argument." I would certainly be interested in any thoughts on this. (As for the fractal issue: I agree the lattice doesn't look fractal, but I do think the Fourier spectrum displays a kind of self-similarity.) $\endgroup$ – Ruben Verresen Dec 4 '16 at 18:43
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    $\begingroup$ @CarloBeenakker I think that the chair tiling has 4 fold symmetry, but a dense set of Bragg peaks which is not compatible with periodic crystals. Sphynx tiling is similar in the sense that it is aperiodic but has 6 fold symmetry. This is the reason why mathematicians try to avoid calling things "quasicrystals", we say instead aperiodic order... $\endgroup$ – Nick S Sep 20 '18 at 21:37

You will need to provide some more information, but your pattern has a resemblance with the Penrose tiling, that is indeed a quasi-crystal.

A quasi-crystal is a non-periodic pattern, but any finite pattern is guaranteed to exist in any sufficiently large region. Your picture seem to indicate this. Furthermore, the fact that your pattern is generated by a mathematical function strongly indicate that it is also a fractal, generated by some sort of geometric substitution rule, see e.g., Barnsleys fresh article that was put on arxiv today.


It is not because a quasicrystal has a five-fold axis of rotational symmetry - pentagonal - while yours has four. This is the Penrose tiling and Wang tiling which is the representation of the PHI fractal geometry of atomic arrangements in organic molecules. (https://i.stack.imgur.com/nDmNA.jpg

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    $\begingroup$ The poster has addressed this in a sequence of comments, including this one. $\endgroup$ – LSpice Jan 2 at 0:31

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