Six integers, and the lattice that holds them

Where five-fold becomes legal asks how many dimensions a single five-fold rotation needs, and answers four: the cyclotomic polynomial of order five has degree φ(5) = 4, and an integer matrix of order five exists in four dimensions and in no fewer.

That is a question about one rotation. A quasicrystal has a group — the full icosahedral group, order 120 with its inversion — and holding all of it costs more.

Six. The icosahedral group has six five-fold axes; it permutes them, reversing some; so in coordinates along those axes every operation is a matrix with exactly one ±1 in each row and column. Every entry is an integer, and Z⁶ is a lattice the group maps onto itself.

Why the six axes are the right coordinates #

An icosahedron has six axes of order five, running through opposite pairs of its twelve vertices, and no three of them lie in a plane. So they span three-dimensional space six times over — any three of them are a basis, and there are relations among the six that involve the golden ratio.

Take those six directions as the basis of an abstract six-dimensional space instead. Then:

Every rotation of the icosahedron is a signed permutation of them. A rotation carries five-fold axes to five-fold axes, since it preserves the icosahedron; it may reverse an axis, since an axis is a direction up to sign. There is nothing else it can do.

A signed permutation matrix is an integer matrix, with a single ±1 in each row and column and zeros elsewhere. Its determinant is ±1, so it is in GL(6,ℤ), and it maps Z⁶ onto itself.

And the check that this is a group and not a coincidence of shapes is a homomorphism test: the matrix of a product must be the product of the matrices. That is verified on all 3,600 pairs, and the sixty matrices are distinct, so the representation is faithful. The five-fold rotation forbidden in three dimensions is an ordinary lattice symmetry in six.

The two three-spaces, and why they are conjugate #

Six-dimensional space splits into two three-dimensional pieces that the group preserves, and the split is the whole construction rather than a technicality.

Parallel space is the map sending the i-th basis vector to the i-th icosahedral axis in ordinary space. Perpendicular space is the same map with the golden ratio τ replaced by its algebraic conjugate σ = −1/τ — the other root of x² = x + 1.

That substitution is what makes the split exist. τ and σ are the two roots of one integer polynomial, so an integer matrix cannot tell them apart, and a matrix that acts as a rotation R in one embedding acts as the conjugate rotation in the other. The split is verified rather than fitted: for every one of the sixty matrices M, E‖M = R E‖ and E⊥M = R′E⊥, to 4 × 10⁻¹⁶.

An eigenvalue decomposition would find the same two subspaces and would not say why they are there. They are there because the group’s matrix entries are integers and its geometry needs an irrational number, and the only way to have both is to have two copies of the geometry, conjugate over the rationals. Everything the cut-and-project construction does is a consequence of that.

Cutting the lattice, and detecting the symmetry back #

The construction is cut and project with six dimensions instead of two.

Take the points of Z⁶. Each has a parallel image and a perpendicular one. Keep the points whose perpendicular image lies inside a bounded window, and record the parallel images. What comes out is a point set in ordinary three-dimensional space with no lattice at all, dense in structure and never repeating.

Then forget how it was made. Apply each of the sixty rotations to the point set and ask whether the set comes back to itself. It does — exactly, on every point of the set’s core, with positions compared at six decimals.

That is the round trip this site runs on every plane pattern, in three dimensions and on an aperiodic set. It is the only exact symmetry claim on this site about a point set with no lattice, and it is available for a reason worth stating: the window is icosahedrally symmetric too. Symmetry of the result is symmetry of the construction, and both halves have to have it.

The refusal, and what it says about windows #

Change the window to a box and the same test fails: 48 of the 60 rotations move the point set off itself.

That is the check doing its job rather than a defect of the box. A box window is a perfectly good window — it produces a point set with long-range order and sharp diffraction, and such structures exist — but it is not icosahedrally symmetric, so neither is what it produces. A test that passed for any window would be testing nothing.

It also marks where this construction departs from the standard one. The window used here is a ball, and the standard window for an icosahedral quasicrystal is the projection of the six-dimensional unit cube — a rhombic triacontahedron. A ball is icosahedrally symmetric, which is what the symmetry claim needs; it is not the standard window, so the point set is not the standard icosahedral quasilattice and its diffraction is not the standard one. That is stated here rather than glossed, because a construction that is nearly the textbook one is exactly the kind of thing a reader will assume is the textbook one.

Six integers, and the experiment that needs them #

The six-dimensional lattice is not a formal convenience. It is what indexes the diffraction pattern.

A crystal’s reflections are indexed by three integers, because its reciprocal lattice is three-dimensional. An icosahedral quasicrystal’s reflections need six, and that is a measurement rather than a modelling choice: the observed peak positions are integer combinations of six basis vectors and are not integer combinations of any three. Shechtman’s 1984 pattern was indexed that way within a year of publication, and the indexing is the reason the six-dimensional description won.

The dense set of peaks is the other half of the same fact. Six integers mapping into three-dimensional reciprocal space means the set of possible reflection positions is dense — arbitrarily close to any point there is a peak with some indices. What makes a diffraction pattern look discrete is that intensity falls away rapidly with the perpendicular index, so only peaks with small perpendicular components are visible. The window’s Fourier transform supplies that falloff, which is why the window is a physical object rather than a bookkeeping device.

What the projection looks like along each kind of axis #

The projected point set is three-dimensional, and looking down its axes is how its symmetry is read in a laboratory.

Down a five-fold axis the pattern of points has ten-fold appearance, because the set is centrosymmetric — projecting a five-fold arrangement along its own axis gives ten directions once the inversion is included. That is the same doubling the self-rotation function reports for autocorrelations, arriving here from geometry rather than from a transform.

Down a three-fold axis the appearance is six-fold, for the same reason, and down a two-fold axis it is two-fold. So an electron diffraction study of an icosahedral phase reports patterns with ten-fold, six-fold and two-fold symmetry at particular angles to each other, and the angles are the identification: 37.38° between a five-fold and its neighbouring three-fold, 31.72° between a five-fold and a two-fold. Those numbers come out of the icosahedral group and out of nothing else, and they are what Shechtman’s tilt series matched.

Where the exactness stops #

The group in six dimensions is exact. Sixty integer matrices, a homomorphism verified on 3,600 products, and a faithful representation.

The splitting is exact to machine precision. E‖M = RE‖ holds at 4 × 10⁻¹⁶, which is the floating-point floor rather than a residual, and it holds because the algebra is exact and only the arithmetic is not.

The symmetry of the projected set is exact in the sense that matters: the set maps to itself under all sixty rotations, with no tolerance beyond the rounding of coordinates for comparison. What is not exact is the set’s extent — it is cut from a finite box of six-dimensional integers, so its outer edge is an artefact of the box, and the test is run on the core where that edge cannot reach.

And there is no detector here. The plane patterns on this site are verified by a detector that enumerates every operation a lattice permits and keeps the ones that work; that machinery needs a lattice and there is none. So what is checked is that a stated group of sixty rotations maps the set to itself — a strong claim, and a weaker one than “these sixty and no others”. A construction with accidental extra symmetry would not be caught here, which is the one direction of this site’s usual round trip that the aperiodic case cannot supply.

What a lattice in six dimensions is not #

Two misreadings of this construction are common enough to be worth refusing directly, and both come from treating the extra dimensions as physical.

The atoms are not in six dimensions. The structure is a set of points in ordinary space, and every one of them is where it is. The six-dimensional lattice is a description: a bookkeeping device in which the aperiodic set becomes a periodic one, so that the tools built for periodic structures — reflections indexed by integers, systematic absences, structure factors — can be used on it. Nothing is being claimed about space having extra dimensions, and no measurement of this construction would differ from a measurement of the point set.

And the extra dimensions are not free parameters. The perpendicular space is fixed by the group, not chosen: it is the conjugate embedding, and there is exactly one, because τ has exactly one algebraic conjugate. A construction with a different perpendicular space would not commute with the group and would not produce an icosahedrally symmetric set.

What the description buys is precisely the thing an aperiodic structure otherwise lacks: a finite list of parameters. A periodic crystal is described by a cell and the contents of one cell. A quasicrystal described in three dimensions has no repeating unit and no finite parameter list; described in six it has a cell and the contents of one cell again, and the contents are the window. That is why refinement of quasicrystal structures is done in six dimensions and reported in three.

Three lattices, not one #

The six-dimensional description has the same structure as the three-dimensional one, and that includes centring.

Just as three-dimensional space has primitive, body-centred and face-centred cubic lattices, six-dimensional space has three icosahedral Bravais classes: P, F and I. They give quasicrystals with different systematic absences and different indexing conventions, and real materials are found in each — the aluminium–manganese phases Shechtman measured are face-centred, and the icosahedral phases in aluminium–copper–iron are primitive.

That is the point at which the six-dimensional picture stops being an analogy and starts being ordinary crystallography done in a different number of dimensions. The reflections are absent in patterns the centring predicts, exactly as they are in three dimensions, and an experimenter reads the class off the absences by the same reasoning reading a space group from its absences uses.

The perpendicular component is not observable and its size is #

One consequence of the six-dimensional description is measurable in the first minute of an experiment, and it is worth naming because it is what a diffraction pattern of a quasicrystal looks like.

Every reflection carries six integers, and those six split into a parallel part — where the peak sits in ordinary reciprocal space — and a perpendicular part, which is not a position at all and cannot be observed directly. What the perpendicular part decides is the intensity. A peak’s structure factor is the transform of the window evaluated at the perpendicular component, and a window of finite extent has a transform that falls away from the origin. So a reflection with a large perpendicular component is weak, whatever its parallel component is.

That produces the pattern’s characteristic appearance. The peaks are dense, as the arithmetic requires, and they are not all visible: a few are strong, more are weak, and the rest are below any background. The visible pattern is a sparse selection from a dense set, and which reflections are selected is decided by a quantity that exists only in the description. Two peaks at nearly the same place can differ in intensity by orders of magnitude, and nothing about their positions says why.

It also gives the practical indexing check. An assignment of six integers to a set of observed peaks is testable: the strong peaks must be the ones with small perpendicular components, and an indexing that puts a large perpendicular component on the strongest reflection in the pattern is wrong even though every position it predicts is right.

Who found it, and when #

The six-dimensional description arrived quickly and from several directions at once, because the mathematics was already in place when the measurement appeared.

De Bruijn gave the projection construction for Penrose tilings in 1981, three years before Shechtman’s publication, and showed that a Penrose tiling is a slice of a five-dimensional lattice. Kramer and Neri published an icosahedral tiling from a six-dimensional projection in 1984, independently of the experiment. Within two years of Shechtman’s paper, Duneau and Katz, Elser, and Bak had the six-dimensional formalism in the form it still has.

The reason it settled so fast is the indexing. A description that says “these peaks are integer combinations of six vectors” makes a prediction about every peak in the pattern, and the prediction was checked and held. The competing explanation — Pauling’s multiple twinning of a large cubic cell — made a different prediction about peak positions, and it did not hold.

The same idea carries a second family of structures, and it is worth naming because the two are usually met separately. An incommensurately modulated crystal has a periodic average structure with a displacement wave running through it whose period is not a rational multiple of the cell — and it is described by restoring periodicity in one extra dimension, so that a four-dimensional lattice cut at an irrational angle produces the three-dimensional structure with its modulation. The construction is this one with three fewer added coordinates. That is the whole difference between the two: a modulation needs one extra dimension, a decagonal quasicrystal five, and an icosahedral one six, and the number is decided by how many independent irrational periods the structure carries rather than by anything about its chemistry.

Why six and not five #

One number deserves a paragraph, because the neighbouring case is different and the difference is instructive.

A Penrose tiling needs five dimensions, and its symmetry group is the ten-fold dihedral group acting in the plane. Five is φ(5) + 1: the five-fold rotation needs four dimensions as a single matrix, and the fifth coordinate carries the direction the projection drops. De Bruijn’s construction uses exactly five, one per family of grid lines.

The icosahedral group needs six, and six is not φ of anything relevant. It is the number of five-fold axes, and the reason the group’s matrices are integral in those coordinates is that the group permutes the axes. Two different arguments, giving two different numbers, for what looks like the same phenomenon.

The general statement behind both is that the smallest integral representation of a group is decided by its character theory rather than by any one of its elements. A single rotation of order n needs φ(n) dimensions; a group containing it may need more, and how many more is a fact about the group. The gap between four and six is the clearest example this site has of the difference between an element and the group it belongs to — which is the same distinction thirteen ways to hold a lattice makes about integer matrices in two dimensions.

The construction’s status is worth restating once more against its neighbour. Cut and project supplies the method and says nothing about which group is being realised; the icosahedral group supplies the group and says nothing about a lattice. Six is where the two meet.

Where the ladder goes next #

This rung supplies the lattice. Two rungs sit above it.

The window as a physical object. Its shape decides the diffraction intensities, its size decides the density, and sliding it is a phason — the degree of freedom the freedom a crystal has not is about. The standard triacontahedral window is the projection of the unit hypercube, and computing with it rather than with a ball is the next piece of machinery this ground needs.

Six-dimensional space groups. Once there is a lattice and a point group there are space groups, with screws and glides in six dimensions, and the icosahedral quasicrystals are classified by them. The counts are large and the arithmetic is the arithmetic of a space group is an extension, which is the last thing this site would need before its three-dimensional machinery had a genuine six-dimensional counterpart rather than a construction with a stated symmetry.