# Proposing a 2D quasicrystal; what are the necessary and sufficient conditions? (If it looks like a duck and quacks like a duck, or...?)

Quasicrystals show repeating motifs or patterns but are not periodic. And yet their Fourier transforms invariably demonstrate a pattern of sharp peaks revealing that there is still an underlying... something.

Generally if one works hard one can show that a quasicrystal proper of $$n$$ dimensions is actually a manifestation of periodic lattice of at least $$n+1$$ dimensions. For example, some 3D quasicrystals with 10 and 12 fold symmetry have been shown to come from 5 dimensional lattices sliced at "funny" (irrational) angles.

The actual atomic locations of the quasicrystal in our world will often slightly offset themselves to minimize energy (cf. phason) and so back-solving for the higher dimensional mathematical lattice from a given real-world quasicrystal atomic positions is a fun mathematical challenge.

Question: If one wants to propose that an observed structure1 could be a 2D quasicrystal these days, what would be considered as the necessary and sufficient conditions? Is "if it looks like a duck and quacks like a duck" sufficient (i.e. repeating motifs or patterns that are not periodic + strong, sharp peaks in a Fourier transform) or must one actually solve for the higher dimensional lattice to which it is associated first?

1In order to make this harder but more useful to me, let us stipulate that the Fourier transform has sixfold symmetry and not ten- or twelve-fold. Thus we can't use the shortcut of "there aren't any 2D Bravais lattices with this rotational symmetry, therefore it's a quasicrystal Q.E.D."

An example of a 2D quasicrystal with 6-fold symmetry would be twisted bilayer graphene at say several degrees but not 30° where it becomes 12-fold. (For more on the twelvefold 30° case Dodecagonal bilayer graphene quasicrystal and its approximants).

note: There's a potentially quite helpful discussion in Twisted Graphene Bilayers and Quasicrystals: A Cut and Projection Approach.

A simple but convenient illustration of how a 1D quasicrystal originates from a 2D regular lattice, from Hyperuniformity of quasicrystals

Projection of lattice points to create the 1D point set of interest. The red dots lie in the physical space X.

Independent of facing a 3D sample, or one of 2D/2.5D, IUCr's definition applies to both of them when stating

«There are two definitions of the notion of quasicrystal.

1. A quasicrystal is an aperiodic crystal that is not an incommensurate modulated structure, nor an aperiodic composite crystal. Often, quasicrystals have crystallographically 'forbidden' symmetries. These are rotations of order different from 1, 2, 3, 4 and 6. In three dimensions a lattice periodic structure may only have rotation symmetry of an order equal to one of these numbers. However, presence of such a forbidden symmetry is not required for a quasicrystal. A system with crystallographically allowed rotation symmetry that is locally similar to one with forbidden symmetries is also a quasicrystal.

2. The term quasicrystal stems from the property of quasiperiodicity observed for the first alloys found with forbidden symmetries. Therefore, the alternative definition is: a quasicrystal is an aperiodic crystal with diffraction peaks that may be indexed by n integral indices, where n is a finite number, larger than the dimension of the space (in general). This definition is similar to that of aperiodic crystal.»

IUCr's Online Dictionary of Crystallography, entry edited by 17 November 2017.

Perhaps the review by Steurer about the current state of affairs is helpful for your work, too (Fig. 6 bears similarity to the figure in your question):

Steurer, W. Quasicrystals: What do we know? What do we want to know? What can we know? in Acta Cryst. 2018, A74, 1-11; doi 10.1107/S2053273317016540 (open access).

• Thanks! I think when a referee questions if a paper proposing that a 2D quasicrystal has been produced gets pushback from referees, citing a few sentences from an online dictionary of crystallography may not be very helpful in a rebuttal. But it's certainly a start. :-)
– uhoh
Nov 1, 2021 at 12:25
• Note, I added a fairly recent review article (open access) which in turn may offer «material» to argument in favour (or not) if the sample in question is described well enough as quasicrystal or not. Equally possible, to enter +quasicystal +2D in the search box of «search IUCr Journals» on journals.iucr.org to constrain the search (then yielding 10 instead of 312 publications with quasicrystal alone) to draw analogies with other work in the field. Re dictionary: It is a joint effort by IUCr (not just anonymous Wikipedia contributors). Nov 1, 2021 at 12:50
• The Steure topical review is going to be an extremely helpful read (as are most of those in the search results). Thank you for the update!
– uhoh
Nov 1, 2021 at 12:59
• By very superficial screening, the Feuerbacher paper «Moiré, Euler and self-similarity – the lattice parameters of twisted hexagonal crystals» (Acta Cryst. A, open access) has similarities to the Wikimedia commons' deposit figure you link to, too. Nov 1, 2021 at 12:59
• Another HNQ for uhoh. Nov 1, 2021 at 15:51