Crystallography & Symmetry
patterns derived from their groups
Concept

Cut-and-project — where it appears

A construction slicing a higher-dimensional lattice with a strip at an irrational angle and keeping the shadow of what falls inside. The result never repeats and is built from something perfectly periodic that is simply not where anybody was looking.

Named by 12 essays across one field — each of them below, with the objects they name alongside it.

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 Cut and project. A square lattice, a strip along a line of the given slope, and the shadow on that line of every lattice point inside the strip. The shadow has two gap lengths; whether their order repeats depends entirely on whether the slope is rational.

Cut and project

Take a periodic lattice, cut a strip through it at an irrational angle, and keep the shadow of what falls inside. The result never repeats, has exactly two spacings, and is a quasicrystal — built from something perfectly periodic that is simply not where anybody was looking.

aperiodic · Aperiodic The substitution, 6 generations. The rule "every long tile becomes a long and a short, every short tile becomes a long", applied 6 times from a single tile. Each generation is as long as the previous two together, so the tile counts are Fibonacci numbers — 13 long and 8 short at the last row — and their ratio is 1.62500 against the golden ratio's 1.61803. The sequence never repeats and every finite piece of it recurs infinitely often, which is order without periodicity in its smallest form.

The smallest quasicrystal

Two tile lengths on a line, in the golden ratio, in a sequence that never repeats. Three completely different constructions produce it, they are required here to agree, and its diffraction needs two integers per peak where a periodic chain needs one.

aperiodic · Quasicrystals The chain as a cut through a periodic pattern. A periodic pattern in two dimensions: one atomic surface through each lattice point, drawn as the curve x = n + A·sin(2πy). The physical chain is the cut along the line y = qx with q = 0.211, and the atoms are where that line meets the curves — plotted along the bottom. Every cut meets every curve exactly once, so every cut gives a chain with the same 9 atoms and the same lattice, differently displaced. That is the difference from cut-and-project, where the atomic surfaces are intervals with ends and moving the cut adds and removes points: here the extra coordinate is a phase, and shifting it is a symmetry of the material rather than a different material.

The extra dimension that makes it periodic

A structure with no cell in three dimensions can be a slice through one that has a cell in four. The construction is cut-and-project with continuous atomic surfaces instead of intervals — and that single difference is what separates an incommensurate crystal from a quasicrystal.

aperiodic · Modulation Sliding the window catches different points. The periodic lattice that cut-and-project starts from, with the strip drawn at two positions 0.21 apart, which is 15 per cent of the window's width. Most lattice points are caught by both; a few are caught by one and not the other, and those are the whole difference between two quasicrystals. The slope has not changed, so the density, the two tile lengths and the ratio of their frequencies are identical — the offset is a parameter with no energy attached to it, which is what makes a phason a degree of freedom rather than a defect.

The freedom a crystal has not

Slide the window of a cut-and-project construction and the tiling changes — different tiles in different places — while its density, its tile ratio and its diffraction pattern do not. That parameter is a phason, it costs nothing, and no local measurement whatever can determine where it sits.

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 Every window returns within 3.0 n. For each window length, the largest distance between two consecutive occurrences of the same window, measured over 46,368 tiles. The gaps are Fibonacci numbers, and the ratio to the window length stays below 3.00 — the chain is linearly repetitive. That is a strong statement of uniformity: there is no stretch of the chain, however far out, in which a given patch fails to occur within a bounded multiple of its own size.

Every patch comes back

A chain that never repeats still repeats everything in it. Every block of tiles occurs again, and again, within a bounded multiple of its own length — and how large that multiple is turns out to be a fact about the continued fraction of a slope.

aperiodic · Complexity 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

Every quasicrystal on this site has been 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 9 approximants, period 2 to 89. The approximants of the Fibonacci chain: the n-th Fibonacci word taken as a unit cell and repeated. Each is a perfectly ordinary periodic crystal — it has a lattice, a cell and a space group — and each has the composition of the quasicrystal to the accuracy a ratio of Fibonacci numbers can manage, since its long and short tiles are consecutive Fibonacci numbers and their ratio is a convergent of the golden ratio. The last column is where the approximant stops agreeing with the infinite chain letter for letter: always past its own period, because the infinite word begins with every finite Fibonacci word, and never for ever. At order 9 the error in the composition is -3.87e-4, and it falls by a factor of τ² at every step up the sequence.

The crystal you get by rounding τ off

Everything aperiodic about a Fibonacci chain comes from one irrational number in the slope of a cut. Replace it by a fraction and the whole construction survives: the same lattice, the same strip, the same rule, and a chain that is periodic — agreeing with the quasicrystal for a length that grows with the denominator.

aperiodic · Quasicrystals A patch of hats scatters a pattern that repeats. The diffracted intensity of the 1217 points of a patch of 183 tiles laid out as the hat, over 2 by 2 cells of the kite grid's reciprocal lattice, whose edges are the faint lines. Every local maximum above a hundredth of the central peak is a disc with area proportional to its intensity; 12 reach the central peak's full height. 72 maxima are drawn. Adding a reciprocal lattice vector to the scattering vector changes the intensity by at most 1.1e-15 of the central peak, so each cell holds the same pattern.

How much of the hat is a crystal

Put a scatterer on every corner of a patch of hats and the diffraction pattern repeats exactly, because every corner sits on a lattice. Inside each repeat the strongest reflections are those of an ordinary crystal with partly filled sites, and by Parseval's identity they carry sixty-three per cent of what the pattern holds. The aperiodicity the hat is famous for lives in the remaining third, in reflections a hundred times weaker.

aperiodic · Monotile 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 Every rational holds a window, and there is nothing in between. The ground state density of a chain of particles with a convex repulsion, against the chemical potential that sets how many of them there are. Every density with denominator up to 24 is a flat step of positive width — 177 of them — and the steps with the simplest fractions are the widest: a half takes 19 per cent of the whole range on its own. The risers between them are not smooth stretches; they are where the densities with larger denominators sit, and a finer computation fills them with more steps. What is left after every rational has taken its window is the irrational densities, which are the genuinely incommensurate ground states and have no width at all.

Every fraction holds a window

Three essays here name the devil's staircase and none computes one. A chain of particles with any convex repulsion has a ground state at every rational density holding an interval of chemical potential to itself — 709 of them computed, the widest taking 19% of the axis and the narrowest two parts in a million million — and the incommensurate densities are what is left over.

aperiodic · Modulation

Named alongside it

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

Golden ratioWindowThe Fibonacci chainLong-range orderQuasicrystalQuasiperiodicPerpendicular spaceAperiodicityCommensurateContinued fractionDiffractionDiffuse scattering

All concepts