What is a quasicrystal?

This post was originally a talk given in Pizza Seminar on October 15, 2014. Most of the mathematical content is from Wikipedia and this paper by Niizeki.

Abstract

The crystallographic restriction theorem says a finite-order automorphism of \(\mathbb{Z}^3\) (in chemistry language, a symmetry of a crystal) has order 1, 2, 3, 4, or 6. But \(\mathbb{Z}^3\) is small potatoes; so let’s go to \(\mathbb{Z}^n\), do some Galois theory in cyclotomic fields, and get us some icosahedral symmetry! (Icosahedrite is a real thing, say serious scientists.) Pretty pictures will be provided, including demonstrations of self-similarity. Self-similarity is always cool, right?

Contents

1. Abstract
2. Contents
3. The crystallographic restriction theorem
4. X-ray crystallography: peeking inside real crystals
   1. Setup
   2. A troublesome photo
   3. Demo: Making waves
5. Slicing higher-dimensional lattices
   1. The problem: locating atoms
   2. The explicit recipe
   3. Demo: Atoms in 2-D quasicrystals
6. Self-similarity
   1. A number-theoretic model of quasicrystals
   2. A mathematical description of zooming
7. 3-D quasicrystals
   1. The n-dimensional viewpoint
   2. The search for 3-D quasicrystals
8. Footnotes

The crystallographic restriction theorem

A crystal is a regular grid (like \(\mathbb{Z}^3\)) of atoms. There is a remarkable theorem about what symmetries a crystal can admit:

Theorem (crystallographic restriction theorem)

If \(A \in \operatorname{Aut} \mathbb{Z}^3\) has finite order, then the order is 1, 2, 3, 4, or 6.

Proof

Properties of \(A\) imply properties of its eigenvalues:

\(A\)	eigenvalues
finite order	on the unit circle \(S^1\)
real	in complex conjugate pairs (so one is \(\pm 1\))
integer	their sum, \(\operatorname{tr} A\), is an integer

So if \(e^{i\theta}\) is an eigenvalue of \(A\), then \[ \operatorname{tr} A = e^{i\theta} + e^{-i\theta} \pm 1 = 2 \cos \theta \pm 1. \] Therefore \(\cos \theta \in \frac{1}{2} \mathbb{Z}\). The eigenvalues \(e^{i\theta}\) satisfying this are of order 1, 2, 3, 4, and 6.

X-ray crystallography: peeking inside real crystals

Setup

Here’s how scientists figure out the internal structure of a crystal. To start, shine an x-ray on it and record the scattered radiation on film. For our purposes, an x-ray is a function \(f\) satisfying \[ f(x+?) = e^{i k \cdot x} f(?) \] for some vector \(k\) that depends on the ray direction and wavelength. This lets us calculate the factors by which \(f\) changes from source \(x_1\) to atom \(x\) and from atom to film \(x_2\).

The total change in \(f\) is by a factor of \( e^{i[(k_1 - k_2) \cdot x - k_1 \cdot x_1 + k_2 \cdot x_2]} . \) Integrating this over all atom positions \(x\) in the crystal yields the total radiation bounced in the direction of \(k_2\). So…

Upshot

The photo records (part of) the Fourier transform of the distribution of atoms.¹

A troublesome photo

Since rotational symmetries of a lattice are inherited by its Fourier transform, photos like the one below would seem to violate the crystallographic restriction theorem.

Demo: Making waves

To get a picture of what kind of crystal we’re looking at, let’s approximate the inverse transform by adding waves at \(n\) regularly spaced angles. Try it for 5, 8, 12, and 7.

Demo
Sums of waves

You can animate these too; and Jason Davies has a WebGL adaptation of the animation.

Slicing higher-dimensional lattices

The problem: locating atoms

The last demo showed you the graph of \(f(x) = \displaystyle\sum_{i=1}^n \cos (k_i \cdot x) \) in \(\mathbb{R}^2\) for evenly spaced \(k_i\) on the unit circle. This is all very nice and smooth, but we want to actually place atoms—discrete points—in a grid. Let’s put them in the hotspots—i.e. where \(f\) hits some threshold.

To turn this into a discrete process, use the following trick: realize the waves \(\cos (k_i \cdot x)\) as 2-D slices of independent waves in \(\mathbb{R}^n\)—then hotspots are where the slice passes near lattice points.

The explicit recipe

Let \(f\) be the restriction of the function \[ \mathbb{R}^n \ni x \mapsto \sum_{i=1}^n \cos (e_i \cdot x) \] to a 2-plane on which the basis vectors \(e_i\) project to \(k_i\). To choose the 2-plane, let \(A \in \operatorname{GL}_n \mathbb{Z}\) act by cycling the basis vectors \(e_i\). Diagonalizing over \(\mathbb{C}\) gets you the matrix \[ \begin{pmatrix} \zeta & & & & \\ & \zeta^{-1} & & & \\ & & \zeta^2 & & \\ & & & \zeta^{-2} & \\ & & & & \ddots \end{pmatrix} \] where \(\zeta = e^{2 \pi i / n}\). Over \(\mathbb{R}\), the eigenvalues pair into 2×2 blocks, corresponding to 2-planes on which \(A\) acts as rotation by a multiple of \(\frac{2\pi}{n}\). Pick any one of them.

Upshot

• The 2-plane is a 2-dimensional subrepresentation \(V\) of \(\langle A \rangle \curvearrowright \mathbb{R}^n\).
• Atoms (hotspots) are points of \(V\) that lie kinda near points of \(\mathbb{Z}^n\).

Demo: Atoms in 2-D quasicrystals

With this in hand, a computer can be convinced² to find the atoms for you, drawing larger dots for better hotspots. Try it for several values of \(n\); I suggest 5, 7, 8, 6, 2, 12, and 11.

Demo
2-D quasicrystals

Remark

An alternative method is to slice fundamental domains (\(n\)-cubes) with the 2-plane, which yields pictures like the Penrose tiling below. Jason Davies has an applet you can play with, based on an older one by Greg Egan.

Self-similarity

At this point, we could go home—we have a description of quasicrystals from which we can produce both interactive computer renderings and theorems (like quasicrystals not actually having translational symmetry, due to \(V\) containing no integer points besides 0).

Of course, there is more—the “quasi”-ness of quasicrystals allows a sort of self-similarity that honest crystals can’t have. To see it in action, go back to that atoms demo and try zooming in or out for a while.

To explain this self-similarity—the apparent indistinguishability of different zoom levels—we will need a bit of number theory.

A number-theoretic model of quasicrystals

Theorem (crystallographic restriction in arbitrary dimensions)

\(\operatorname{GL}_m \mathbb{Z} \) contains a matrix \(A\) which has \(\zeta_n = e^{2\pi i/n}\) as an eigenvalue (so it rotates some 2-plane by \(\frac{2 \pi}{n}\)) if and only if \(m \geq \varphi(n)\), where \(\varphi\) is Euler’s totient function.

Proof

The Galois conjugates of \(\zeta_n\) must also be eigenvalues of \(A\).

• Since \(\zeta_n^n = 1\), they’re \(n\)th roots of 1.
• Since \(\zeta_n\) generates all \(n\)th roots of 1, they’re primitive—i.e. \(\zeta_n^k\) where \(k\) and \(n\) are coprime.

The number of such \(k\) is \(\varphi(n)\). To prove that \(\varphi(n)\) works, observe that \[ \mathbb{Z}[\zeta_n] \hookrightarrow \mathbb{Q}(\zeta_n) \cong_{\mathbb{Q}\text{-vect}} \mathbb{Q}^{\varphi(n)} \subset \mathbb{R}^{\varphi(n)} \] as a discrete subgroup, since the field norms of the leftmost terms are in \(\mathbb{Z}\).

Remark

If \(p\) and \(q\) are large enough primes, you can produce an integer matrix \(A\) of order \(pq\) and size \(\varphi(p) + \varphi(q) < \varphi(pq)\), but it won’t rotate any 2-plane by \(\frac{2\pi}{pq}\).

A mathematical description of zooming

Here’s how we implement³ zooming: scale the 2-plane \(V\) by some factor and the unseen \(m-2\) dimensions by some other factor to preserve \(m\)-dimensional volume. That way the total visual density of dots remains about the same. Here’s how we interpret zooming and self-similarity:

• Zooming acts by scalars on \(V\) and the other irreducible sub-representations of \(\langle A \rangle \curvearrowright \mathbb{R}^{\varphi(n)}\), so zooming is a 1-parameter subgroup \(\gamma\) in \(\operatorname{SAut}_A \mathbb{R}^{\varphi(n)} \) (volume preserving automorphisms).
• Visually, transforming by anything in \(\operatorname{Aut} \mathbb{Z}[\zeta_n]\) isn’t detectable.
• Self-similarity means that the image of \(\gamma\) in \(\operatorname{SAut}_A \mathbb{R}^{\varphi(n)} / \operatorname{Aut} \mathbb{Z}[\zeta_n]\) accumulates on itself (or is periodic).

The existence of self-similarity rests on one key fact from number theory.

Theorem (Dirichlet’s Unit Theorem)

Suppose a number field has \(r\) embeddings in \(\mathbb{R}\) and \(c\) embeddings in \(\mathbb{C}\). Then its group of units has rank \[ r + \frac{c}{2} - 1 . \]

Corollary

\(\operatorname{SAut}_A \mathbb{R}^{\varphi(n)} / \operatorname{Aut} \mathbb{Z}[\zeta_n]\) is compact.

Proof

Since automorphisms act by scalars on all irreducible subrepresentations,

\[\begin{align*} \operatorname{Aut}_A \mathbb{R}^{\varphi(n)} &\cong (\mathbb{C}^\times)^{\frac{\varphi(n)}{2}} \cong (\mathbb{R} \times S^1)^{\frac{\varphi(n)}{2}} \\ \operatorname{SAut}_A \mathbb{R}^{\varphi(n)} &\cong \mathbb{R}^{\frac{\varphi(n)}{2} - 1} \times (S^1)^{\frac{\varphi(n)}{2}} . \end{align*}\]

Since \(\mathbb{Z}[\zeta_n] = \mathcal{O}_{\mathbb{Q}(\zeta_n)}\), Dirichlet’s Unit Theorem implies that \(\operatorname{Aut} \mathbb{Z}[\zeta_n]\) contains a copy of \(\mathbb{Z}^{\frac{\varphi(n)}{2} - 1}\). Then \(\operatorname{Aut} \mathbb{Z}[\zeta_n]\), being discrete (it’s conjugate to a subgroup of \(\operatorname{SL}(\varphi(n), \mathbb{Z})\)), is cocompact.

Self-similarity now follows, having been reduced to the statement that a 1-parameter subgroup of a torus is recurrent.

Remark

The image of \(\gamma\) is periodic when (and only when) \(\varphi(n) = 2\)—so when \(n\) is 5, 8, 10, or 12. This makes the self-similarity exact—i.e. different zoom levels can look identical (not just suspiciously similar)—and allows one to build games like Richard Schwartz’s Lucy and Lily based on facts like \[ (2 + \sqrt{5})(2 - \sqrt{5}) = -1. \]

Remark

Michael Rule’s collection of quasicrystals graphics includes an alternative description of zooming that sidesteps the number theory, instead using Shepard Tones to produce self-similarity.

3-D quasicrystals

The n-dimensional viewpoint

A quasicrystal with irreducible symmetry \(G\) is an irreducible-over-\(\mathbb{R}\) subrepresentation \(V\) of an irreducible-over-\(\mathbb{Z}\) representation \(G \hookrightarrow \operatorname{SL}(m,\mathbb{Z})\). The atoms are points of \(\mathbb{Z}^m\) near \(V\), and the Galois conjugates of \(V\) (the other subrepresentations over \(\mathbb{R}\)) allow the possibility for self-similarity.

The search for 3-D quasicrystals

To look for 3-dimensional quasicrystals, one first finds a list of the seven irreducible finite subgroups of \(\operatorname{SL}(3,\mathbb{R})\). Most of these preserve an ordinary 3-D lattice—i.e. the representation can be defined over \(\mathbb{Z}\). The two exceptions are the two icosahedral groups—the alternating group \(A_5\) and the product \(A_5 \times \{\pm 1\}\)—so up to finite index subgroups, there’s only one irreducible 3-D quasicrystal, which you can explore in the following demo.

Demo
3-D quasicrystals

The standard representation of \(A_5\) has one Galois conjugate, which can be found in a list of representations of \(A_5\). The two representations can be described explicitly as follows: in the standard representation, \(A_5\) acts by rotation on the icosahedron with vertices \(\pm(\sqrt{5}, 0)\) and \(\pm(1, 2\zeta_5^k)\) in \(\mathbb{R} \times \mathbb{C}\). To obtain the other representation, merely conjugate the action on the set of vertices by the permutation induced by \(\zeta_5 \mapsto \zeta_5^2\).

Oh, and—yes, they’re real.

Footnotes