Crystallography & Symmetry
patterns derived from their groups
Concept

Quasicrystal — where it appears

A solid with sharp diffraction peaks and a rotational symmetry no lattice permits, so it has long-range order and no repeat. Sharpness meant order and tenfold meant no lattice, and the two had been believed inseparable until 1982.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

Assuming a 5-fold rotation. The shortest lattice vector, its rotated copies, and the combination of them that is itself a lattice vector. Where that combination comes out shorter than the vector assumed shortest, the assumed rotation cannot exist.

Why five-fold is impossible

A second proof, geometric rather than algebraic: assume a five-fold centre, and out of it construct a lattice vector shorter than the shortest one there is.

restriction · Restriction A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call.

Order is not periodicity

For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.

aperiodic · Aperiodic A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call.

What Shechtman measured

A diffraction pattern with sharp spots and tenfold symmetry, in April 1982. Sharpness meant order and tenfold meant no lattice, and the two had been believed inseparable — so the interesting question is what the observation had to rule out before it could mean anything.

aperiodic · Quasicrystals The icosahedral group, counted. The sixty rotations of an icosahedron, found by trying every map that sends one adjacent pair of vertices to another and keeping those that carry the whole vertex set onto itself. They fall into 6 axes of order 5, 10 axes of order 3, 15 axes of order 2 — and the fivefold axes are the reason this group cannot be the point group of any crystal, since no three-dimensional lattice admits a rotation of order five. Quasicrystals have it anyway, which is what made 1982 an argument rather than a measurement.

Icosahedral symmetry

Sixty rotations, six fivefold axes, and no lattice in three dimensions that can hold any of them. It is the point group a crystal is forbidden, and the one the first quasicrystal turned out to have.

aperiodic · Quasicrystals A quasilattice down a 5-fold axis: 10-fold, from an axis of order 5. 153 points of a three-dimensional quasilattice, made by keeping the points of Z⁶ whose perpendicular image lies inside a window and projecting them into ordinary space, then viewed along one of its 5-fold axes. There is no lattice here and no unit cell, and the symmetry is nevertheless exact: all sixty rotations of the icosahedral group carry the set onto itself, on 153 points of its core, measured by applying them rather than assumed from the construction. Seen down this axis the set comes back to itself under a turn of a 10th and no finer turn, measured over every turn up to a twelfth — twice the order of the axis, and the factor of two is a centre of symmetry: this set equals its own negation, which is what a window centred on the origin produces, and that is checked here rather than assumed. A diffraction experiment would record the same 10 whatever: the measured intensity acquires a centre whether or not the structure has one, so the tenfold photograph of 1982 does not by itself distinguish a structure with a centre from one without.

Six integers, and the lattice that holds them

A fivefold rotation is not an integer matrix in three dimensions and is one in six. The icosahedral group permutes its own six fivefold axes, so in coordinates along those axes every one of its sixty rotations is a signed permutation — and Z⁶ is a lattice it maps onto itself.

aperiodic · Quasicrystals 66 squares and 106 rhombs. The Ammann–Beenker tiling, built by keeping the points of a four-dimensional lattice whose companion image falls inside an octagon and projecting them into the plane. Every tile has the same edge length; the squares and the forty-five degree rhombs are told apart by their diagonals. Nothing was placed — the faces were found among the projected points.

Eight-fold, with the golden ratio taken out

The familiar quasicrystals are all built on five: Penrose's rhombs, the Fibonacci chain, the ten-fold pattern Shechtman measured. A method that works only on the golden ratio is a method tuned to its answer — so here is the same construction run on eight, where the irrational is √2 and nothing else changes.

aperiodic · Quasicrystals The window of a three-letter chain. The cut-and-project window of the tribonacci chain: every prefix of the chain, projected onto the plane spanned by the two complex roots of x³ = x² + x + 1, and coloured by the letter that follows it. 223,317 points. It is a bounded region in three pieces whose areas are 0.542, 0.296, 0.162 of the whole, which are the frequencies of the three letters; it fills 71.5 per cent of its bounding box, and a straight cut across it meets up to 6 separate pieces. A window for a two-letter chain is an interval.

A window that is not an interval

The usual cut-and-project construction takes a strip through a lattice and keeps the points falling within an interval. Add a third letter and the window stops being an interval: the tribonacci chain's window is a fractal in three pieces, and a straight cut across it meets up to six.

aperiodic · Aperiodic Three rotations of order twelve, and one twelve-fold. Three integer 4 × 4 matrices of order twelve, one from each rational class of that order on a four-dimensional lattice. A four-dimensional rotation turns two perpendicular planes, and each plane is drawn as a dial: a point, its images, and the path the operation takes round them. Φ₁₂ turns its planes by one twelfth and five twelfths of a turn, so a point runs round twelve positions. Φ₃ ⊕ Φ₄ turns one plane by a third and the other by a quarter, and Φ₄ ⊕ Φ₆ by a quarter and a sixth; each has order twelve because twelve is the least common multiple, and neither has a plane with twelve positions in it.

Three rotations of order twelve

A four-dimensional lattice can be carried onto itself by three different integer matrices of order twelve, and only one of them turns anything by a twelfth of a turn. The totient that is usually said to decide which orders a dimension permits decides something else: which angles. The orders are decided by a sum over blocks, the two lists part company at six dimensions, and space already has an operation whose order is not its angle.

restriction · Restriction Patches of an icosahedral quasicrystal grow with the dimension their symmetry fixes. The number of distinct patches about a vertex of the canonical icosahedral quasicrystal that keep a given symmetry, against the patch's radius in edges on logarithmic axes, counted per axis or per mirror plane: along a five-fold, a three-fold and a two-fold axis, and across a mirror plane. The axes' counts grow in proportion to the radius and the mirror's as its square, the slopes given are the last four points'. Along the bottom, the patches with an inversion centre: one up to a radius of 1.9 edges, and none from 2.15 on.

In space, a five-fold axis is a line

In a Penrose tiling the patches with a five-fold centre stop at four, however large they are drawn, because a five-fold rotation fixes only a point of the tiling's internal space. Slice a quasicrystal from six dimensions instead of four and the internal space has three dimensions, where a rotation fixes a whole line. The five-fold patches of the icosahedral quasicrystal then grow in number with their radius, from five to forty-six per axis, while the patches with a centre of inversion, which fixes only a point, vanish past two edges. Counting them takes internal space: a sample of vertices cannot.

aperiodic · Complexity

Named alongside it

The objects these essays reach for when they reach for this one.

Cut-and-projectHigher-dimensional latticeCrystallographic restrictionWindowGolden ratioIcosahedral groupIcosahedral symmetryIndexingPerpendicular spaceRotation orderRotoinversionAmmann beenker

All concepts