## Mathematics of quasicrystals

I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.

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Have a look at the papers of my colleague, Robert Moody. He was formerly an algebraist (Kac-Moody), but has spent the last 10-15 years developing tools to study quasi-crystals. – Anthony Quas Jun 18 at 17:04
See mathoverflow.net/questions/98171/… – Spice the Bird Jun 18 at 17:15
Chapter 17 of Automatic Sequences: Theory, Applications, Generalizations by Allouche and Shallit (Cambridge University Press, 2003) looks at quasicrystals from the viewpoint of automatic and morphic sequences. – Joel Reyes Noche Jun 19 at 13:38
A reference-request tag is needed. – Mahmud Jun 20 at 1:10
Done {}{}{}{}{} – Yemon Choi Jun 20 at 1:38

The trouble with the quasicrystals is that the literature in this area is dominated by non-mathematical or pseudo-mathematical papers and books. In particular, just extracting a mathematical definition of a quasicrystal from this literature is not so easy. This situation is well-illustrated by the wikipedia article on quasicrystals and the MO discussion of this topic at http://mathoverflow.net/questions/98171/what-is-the-relation-between-quasicrystals-riemann-hypothesis-and-pv-numbers

The two papers that I found most enlightening and mathematical, address this problem head-on (more on this below):

[1]. A. Hof, "On diffraction by aperiodic structures", Commun. Math. Phys., 169 (1995), p. 25-43.

[2]. J-B. Gouere, "Quasicrystals and almost periodicity", Commun. Math. Phys., 255 (2005), p. 655-681.

Both papers prove some nontrivial mathematical theorems, on the basis of these theorems one can then form two, somewhat different, mathematical definitions of a (quasi)crystal. Senchal's 2-page long survey paper (see Mahmud's answer) or, better, her book (see Joseph's answer) is a good introduction, the trouble is that she does not prove anything in the book and that she could be sloppy with her definitions, for instance, she conflates functions, measures and distributions, which are needed for defining crystals.

If you look in Senchal's book, you first get the following physical definitions of crystals: "A crystal is any solid with essentially discrete diffraction diagram." (This includes both traditional crystals and quasicrystals.) The word "essentially" will be the difference between different mathematical definitions, which one derives from [1] and [2].

A (mathematical) quasicrystal is a tiling $T$ of ${\mathbb R}^n$ by convex polytopes satisfying certain properties:

1. Since general tilings are hard to work with, pretty much everybody assumes that $T$ is a Voronoi tiling of ${\mathbb R}^n$ based on a certain discrete subset $N\subset {\mathbb R}^n$, which a geometer would call a separated net.

2. Unfortunately, just having a separated net is not enough in order to deal with the "diffraction" issue. There is a disagreement which tilings are allowed as crystals and which are not. There is a general agreement that at least all periodic tilings (the ones where $N$ is a finite union of orbits of a discrete group of translations) and certain tilings constructed by Penrose and others via projection method, should be counted as crystals and quasicrystals respectively. I will call these two classes of tilings as "standard."

a. Here is Hof's definition of a crystal (Senchal's definition is taken from Hof's paper). Hof in [1] takes the auto-correlation function (actually, a distribution) $\gamma$ of $N$ and computes the (appropriately defined) Fourier transform $\hat\gamma$ of $\gamma$ (this is a mathematical interpretation of the "diffraction diagram"). Then $\hat\gamma$ is a measure $\mu$ which, in general, splits as a sum of two measures $\mu_d+\mu_c$: Discrete part $\mu_d$, which is supported on a certain countable subset of ${\mathbb R}^n$ and continuous part $\mu_c$. He proposes that "essentially discrete diffraction" means that $\mu_d$ is nonzero. Hof then proves that "standard" tilings indeed have nontrivial $\mu_d$ (According to [2], Hof even proves that $\mu_c=0$ in this case, but I did not check this). The trouble with this definition is that, as far as I can tell, there is no known purely geometric interpretation of the condition $\mu_d\ne 0$ in terms of the next $N$ itself (at least, none existed 6 years ago).

b. Gouere [2] (his work is an extension of Hof's approach and of a work by Lagarias) works with a slightly different definition, i.e., that $\mu_c=0$ (such sets $N$ are called Patterson sets). His main result is a purely geometric interpretation (actually, several interpretations) of this condition, see Theorem 1.1 in [1]: Patterson sets are the sets which are almost periodic with respect to Besikovitch's metric.

Remark 1. As far as I can tell from reading Freeman Dyson's paper here, definition of a quasicrystal that Dyson proposes is the one with $\mu_c=0$.

Remark 2. Gouere does not propose that a Patterson set is the right definition of a crystal, this is just my take on his paper. Condition $\mu_c=0$ is more limited, but, in view of [2], is geometric and also covers "standard" examples, while the condition $\mu_d\ne 0$ is more general, but is nongeometric.

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 Just few comments: 1) $\mu_d \neq 0$ is not a good condition and I don't think Hof used it. The problem with it is that $\mu_d(\{ 0 \}) = (\dens (N) )^2$ so, unless N is trivial, $\mu_d \neq 0$. The usual two conditions we use are (i) $\mu_c =0$ or the weaker (ii) $\supp(\mu_d)$ is relatively dense. Also, in that paper Hof showed that thermal motion always induces some nontrivial $\mu_c$, thus $\mu_c=0$ is not really the right definition unless you work at 0K. – Nick S May 7 at 18:59 2) While characterizing all point sets with these properties is hard, "geometric" conditions which imply each of them are known. Regular model sets are obtained from higher dimensional lattices and they always have $\mu_c=0$... Their subsets, Meyer sets can be characterized by one of the following two equivalent definitions (there are actually more, but I left the rest out): (i) $\Lambda$ is relatively dense and $\Lambda-\Lambda:=\{ x-y| x,y \in \Lambda\}$ is uniformly discrete. – Nick S May 7 at 19:04 (ii) $\Lambda$ is relatively dense, uniformly discrete and $\Lambda-\Lambda \subset \Lambda+F$ for some finite set $F$..... It is somehow surprising, but one (hence both) of these conditions imply that $\supp(\mu_d)$ is relatively dense. In general Meyer sets have also non-trivial $\mu_c$. – Nick S May 7 at 19:06 Dea Nick: Maybe you should expand your comments into an answer (and correct TeX in the process). For part 1: This is what I got from reading Hof's paper, but I could have missed something, of course. – Misha May 10 at 13:00

If you are interested in the non-commutative geometry side of things, there is an overview article, The Noncommutative Geometry of Aperiodic Solids (pdf link) by Jean Bellissard. He writes the paper building up from the most basic possible physical concepts and makes the use of noncommutative geometry to study quasicrystals seem quite natural, and it is done in a mathematically rigorous manner. Edit: I should emphasize that this paper is about the physical side of quasicrystals, written from a mathematical perspective. I wasn't sure if that was what you wanted, as opposed to just the mathematical study of quasicrystals without regard for any related physics.

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You might want to consider reading the following excellent introduction to the subject:
Charles Radin, "Miles of tiles", AMS, 1999.

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Quasicrystals and Geometry by Marjorie Senechal. A bit dated (1996) but still a place to start. (Amazon link).

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Here is a hard-to-find but worthy book from the point of view of statistical mechanics:

• Jacek Miękisz, Quasicrystals - Microscopic models of nonperiodic structure, Louven University Press, 1993.

There has been some progress since the writing of the book, but the main question (the construction of a lattice-gas model with translation-invariant finite range interactions admitting a quasi-crystalline phase) remains open.

If you don't like statistical mechanics, there is enormous literature on aperiodic tilings. The book

has two chapters on aperiodic tilings. For a more up-to-date account, I would recommend the lecture notes of Jarkko Kari.

If you tell us more specifically, which aspect of it you would like to study, maybe we could help better.

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My favorite paper on this stuff is Applications of Group Cohomology to the Classification of Quasicrystal Symmetries (by Fisher and Rabson)
http://iopscience.iop.org/0305-4470/36/40/005/pdf/0305-4470_36_40_005.pdf
They take the work of David Mermin and recast it in terms of cohomology, lattices, and fourier space. So they start from scratch and define what a quasicrystal is. At the end they give some examples where electron-degeneracy and diffraction patterns arise as certain homology classes.

If you want to study symmetries, then there is Howard Hiller's note Crystallography and Cohomology of Groups
which handles exactly that. Associated to this is the notion of a crystallographic group $\mathbb{Z}^n\rtimes\mathbb{Z}_p$.