Where five-fold becomes legal

The crystallographic restriction forbids five-fold rotation of a lattice, and the prohibition holds in the plane and in space. It does not hold in four dimensions, where a five-fold rotation with whole-number entries exists, has order exactly five, and can be written out in sixteen digits.

Knowing where the boundary is turns the restriction from a prohibition into a measurement of how much room a symmetry needs — and it is the reason a quasicrystal is best described not as a pattern that breaks a rule but as the shadow of a periodic structure in a higher dimension.

The condition, stated exactly #

An $n$-fold rotation of a lattice in $d$ dimensions exists exactly when

$$\varphi(n) \le d$$

where $\varphi$ is Euler’s totient: the count of whole numbers below $n$ sharing no factor with it. That single inequality contains the whole subject.

The reasoning is short. A rotation of order $n$ preserving a lattice is an integer matrix $M$ with $M^n = I$ and no smaller power equal to the identity. Its minimal polynomial divides $x^n - 1$ and must be the $n$-th cyclotomic polynomial $\Phi_n$, the factor of $x^n - 1$ whose roots are the primitive $n$-th roots of unity. The degree of $\Phi_n$ is $\varphi(n)$. A matrix cannot have a minimal polynomial of degree larger than its size, so $M$ must be at least $\varphi(n) \times \varphi(n)$ — and at exactly that size the companion matrix of $\Phi_n$ does the job.

So the question “which rotations fit in $d$ dimensions” is the question “for which $n$ is $\varphi(n) \le d$”, which is arithmetic and finite.

Running it out #

For the first few $n$, $\varphi$ goes $1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4$ — and the pattern is what produces the crystallographic restriction.

In one or two dimensions, $\varphi(n) \le 2$ holds for $n = 1, 2, 3, 4, 6$. Those are the five, and they arrive with no geometry at all.

In three dimensions, $\varphi(n) \le 3$ has the same solutions, because $\varphi(n)$ is even for every $n$ above two, so it is never exactly three. Three dimensions gains nothing, and it gains nothing for a reason about parity.

In four dimensions, $\varphi(n) \le 4$ admits $n = 5, 8, 10, 12$ as well. Five-fold becomes legal, and so do eight-, ten- and twelve-fold, which is exactly the set of symmetries observed in real quasicrystals.

In six dimensions, seven-, nine-, fourteen- and eighteen-fold join them.

The list for four dimensions is the interesting one. Icosahedral quasicrystals have five-fold axes; decagonal ones have ten-fold; octagonal and dodecagonal phases have eight- and twelve-fold. Every quasicrystal symmetry ever reported is on the four-dimensional list, and none is on the two-dimensional one. That is not a coincidence and it is the subject of the cut-and-project construction.

The matrix, written out #

For $n = 5$ the cyclotomic polynomial is $\Phi_5(x) = x^4 + x^3 + x^2 + x + 1$, and its companion matrix is

$$M = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \end{pmatrix}$$

Every entry is a whole number, so $M$ maps the lattice $\mathbb{Z}^4$ onto itself. Raising it to the fifth power gives the identity, and no lower power does — which is checked here by repeated integer matrix multiplication rather than argued, because a companion matrix built with the wrong sign convention has order ten or no finite order at all and looks identical on the page.

What $M$ is, structurally, is multiplication by $\zeta = e^{2\pi i/5}$ in the ring $\mathbb{Z}[\zeta]$, written in the basis $1, \zeta, \zeta^2, \zeta^3$. That is why it is integral: multiplying an algebraic integer by another gives an algebraic integer, and the multiplication is a linear map with integer entries in that basis. The lattice being rotated is the ring of integers of the fifth cyclotomic field, and the “four dimensions” are the four coordinates an element of that ring needs.

What the rotation does to the plane #

A four-dimensional rotation does not have an axis; it has two invariant planes, and it turns each of them by its own angle. For $M$ the two angles are $72°$ and $144°$, and the plane turned by $72°$ is the one that makes the five-fold symmetry visible.

Projecting $\mathbb{Z}^4$ onto that plane sends the $i$-th basis vector to $\zeta^i$ — five directions at $72°$ to one another, four of which are independent. The projected image of the lattice is what the figure at the top of this page draws, and it has two properties worth separating.

It has five-fold symmetry, exactly. Applying $M$ to a lattice point and then projecting gives the same result as projecting and then turning the plane by $72°$, and that identity is asserted here on several lattice points rather than assumed — because a projection chosen slightly wrong still produces a convincing five-pointed star while failing this test.

It is dense. The projection is not a lattice; it fills the plane. That is forced: a discrete five-fold set in the plane would contradict the restriction, so the projection of a five-fold lattice cannot be discrete. What the picture shows is a finite piece of something that has points arbitrarily close together everywhere.

Both facts together are the statement of the problem the next construction solves. Five-fold symmetry is available in the shadow, and the shadow is useless as a pattern until something thins it out.

The second plane, and what it is for #

A four-dimensional rotation turns two planes at once, and the second plane is not a nuisance to be ignored — it is the part of the construction that does the useful work.

The eigenvalues of $M$ are the four primitive fifth roots of unity: $\zeta, \zeta^2, \zeta^3, \zeta^4$. They pair up as conjugates, giving two real planes, one turned by $72°$ and the other by $144°$. Projecting onto the first gives the five-fold picture. Projecting onto the second gives another five-fold picture, turned twice as fast, and it is usually called perpendicular space.

The reason to keep it is that the two projections together lose nothing. A lattice point has four coordinates; the two projections have two each; and the map from the lattice to the pair of shadows is a bijection. So a point in physical space carries a hidden position in perpendicular space, and the rule “keep only the points whose perpendicular position falls inside a window” is a rule that can be stated and applied.

That is precisely what turns the dense shadow into a pattern. The window in perpendicular space selects a discrete subset of the dense projection, and the shape of the window determines which quasicrystal comes out. A square window on a two-dimensional lattice gives the Fibonacci chain; a pentagonal window on a five-dimensional lattice gives a Penrose tiling.

So the answer to “what does the extra dimension buy” is: a place to keep the information that decides which points are in the pattern. In the plane there is nowhere to put it, and a five-fold set is either dense or impossible.

What is observed, against what is permitted #

The list of orders four dimensions permits — 5, 8, 10 and 12 on top of the crystallographic five — is short enough to compare against what has been found in real materials, and the comparison is unusually clean.

Icosahedral quasicrystals have five-fold axes and need six dimensions for their full description, since the icosahedral group needs more room than a single five-fold rotation does. The first, Shechtman’s aluminium–manganese, is icosahedral.

Decagonal phases have a ten-fold axis and are periodic along it — quasiperiodic in a plane, periodic perpendicular to it, so they need five dimensions. Aluminium–cobalt–nickel is the standard example.

Octagonal and dodecagonal phases, with eight- and twelve-fold axes, are rarer and are found in chromium–nickel–silicon and in some tantalum–tellurium systems.

And nothing has been reported with seven-fold or eleven-fold symmetry, which is what the table predicts: $\varphi(7) = 6$ and $\varphi(11) = 10$, so those would need six and ten dimensions respectively, and the higher the dimension the less plausible a physical mechanism becomes. The agreement between what arithmetic permits cheaply and what nature produces is close enough to be worth noticing and loose enough not to be a law.

What the plane argument was really saying #

Reading the descent proof again with the totient condition in hand changes what it looks like it was about.

That argument produces a vector of length $2\cos 72°$ times the original — about $0.618$, which is $1/\varphi$ — and the appearance of the golden ratio there is not decorative. It is the same $\mathbb{Q}(\sqrt5)$ showing up that makes $\varphi(5) = 4$, seen from the geometric side. The descent works in the plane because the plane cannot hold the algebra that five-fold symmetry needs, and the shorter vector it constructs is the arithmetic overflowing.

Go up to four dimensions and the construction still produces its vector — but there it is a genuine lattice vector of the four-dimensional lattice, not a contradiction, because the four-dimensional lattice has room for vectors at that length. Nothing about the geometry changed; the container did.

The price of the extra dimension #

Gaining a forbidden symmetry costs something, and the cost is worth stating because it is easy to present four-dimensional crystallography as a free upgrade.

The lattice is not in physical space. The four-dimensional lattice is a bookkeeping object. What exists in a laboratory is its three-dimensional slice, or the two-dimensional one drawn here, and that slice is not periodic. Nothing has been made periodic by going up; the periodicity has been moved somewhere unobservable.

The projection is dense without a window. Selecting which lattice points to project — the “window” or “acceptance domain” — is an extra choice with real content, and different windows give genuinely different patterns from the same lattice. So the higher-dimensional description does not determine the quasicrystal by itself.

The description is not unique. A given quasicrystal can be described as the projection of several different higher-dimensional lattices, and choosing among them is a convention.

Against those costs, the gain is substantial: a quasicrystal’s diffraction pattern needs six indices rather than three, and in the six-dimensional description those six indices are the ordinary Miller indices of an ordinary periodic lattice. Everything that indexing a crystal does becomes available again, which is why the description is standard.

Where the exactness stops #

The matrix and its order are exact integer facts, and everything downstream of them is not.

The projection involves irrational numbers — $\cos 72°$ and its relatives — so every coordinate in the shadow is a floating-point number and every comparison between two of them is a comparison with a tolerance. The assertions on this page reflect that: the integer claims are equalities, and the geometric claims come with a stated distance.

The physical claim is a further step again. That a real alloy is described by a six-dimensional lattice is a model, supported by how well the indexing works and by the sharpness of the diffraction, and it is not decided by any calculation here. What is decided here is that the mathematics permits it, which is a much smaller statement and the only one this site’s machinery supports.

And the standing limit: the integer machinery on this site is two-dimensional and periodic. The matrices on this page come from a separate, small piece of exact arithmetic — cyclotomic polynomials by polynomial division, companion matrices, orders by multiplication — and nothing in it touches the round trip that verifies the plane patterns.

The surprising part #

The restriction is usually taught as a fact about crystals, and the totient condition shows it is a fact about arithmetic.

Here is the connection worth carrying. $\varphi(n)$ is the degree of the field extension $\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}$ — how many rational coordinates are needed to write down a primitive $n$-th root of unity. So “how many dimensions does an $n$-fold rotation need” and “how complicated is the $n$-th root of unity as an algebraic number” are the same question, with the same answer.

That makes the crystallographic restriction a statement in algebraic number theory that happens to have been discovered by mineralogists. The five permitted orders are the $n$ for which $\zeta_n$ is a quadratic irrational or simpler — which is to say, the orders for which the arithmetic of the plane is rich enough. And the appearance of the golden ratio in every five-fold quasicrystal is the same fact from the other side: $\varphi(5) = 4$, the field $\mathbb{Q}(\zeta_5)$ contains $\mathbb{Q}(\sqrt5)$, and $\varphi$ is what lives there.

Who worked it out #

The number-theoretic statement belongs to the study of finite subgroups of $GL_n(\mathbb{Z})$ and to Minkowski, whose bound on the order of such a group dates from the 1880s. The four-dimensional space groups were enumerated in 1978 by Brown, Bülow, Neubüser, Wondratschek and Zassenhaus — 4783 of them, by computer, and the calculation was a substantial undertaking at the time.

Its physical use came from the other direction and later. When Shechtman’s tenfold pattern appeared in 1984, the higher-dimensional description was supplied almost immediately by several groups at once, and it worked so well that it became the standard language within a few years. The International Union of Crystallography redefined “crystal” in 1992 to mean any solid with an essentially discrete diffraction pattern, which puts quasicrystals inside the definition and makes periodicity a special case rather than a requirement.

The mathematics had been sitting there for a century, complete and unapplied, waiting for somebody to measure a diffraction pattern it explained.

Where the ladder goes next #

The two-dimensional prohibition this essay lifts is the crystallographic restriction, and its geometric form is the descent argument.

The dimension where nothing changes is three, and the construction that turns the shadow into a pattern is cut and project.

What the shadow looks like when it is measured rather than computed is what Shechtman measured, and the pattern it produces in the plane is the Penrose tiling.

What the pictures here cannot show. The shadow drawn on this page is a finite sample of a dense set, and density is exactly what a finite drawing cannot exhibit — every drawing of a dense set looks like a drawing of a discrete one with small spacing. That the projection has points arbitrarily close together is a consequence of the projection direction being irrational, argued rather than shown, and the picture would look the same if it were false.