Field

Order without repetition

Penrose tilings and quasicrystals — patterns with long-range order, five-fold symmetry, and no repeating cell at all.

A Penrose tiling, 4 inflations. Two rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself.

Penrose tilings

Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.

One tile becomes 130 in 3 inflations. One tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell. The panel counts are 10, 20, 50, 130, and each is the one before it multiplied by the substitution matrix — an identity in whole numbers, checked every time the figure is drawn rather than quoted.

Inflation, and where the golden ratio comes from

A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio squared.

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

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.

Matching rules, and what actually forces aperiodicity

The two Penrose rhombs are usually said to tile the plane only aperiodically. They tile it periodically without difficulty. What cannot be done periodically is tiling them according to the decoration, and the distinction is the whole result.

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.

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.

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.

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.

$A chain modulated at q = 0.211. The lower row is the lattice: 34 sites, evenly spaced. The upper row is the structure: the same sites displaced by a wave of amplitude 0.12 of a spacing and wavevector 0.211, drawn through them. Because 0.211 is not a ratio of small whole numbers, no cell of any size holds the structure — the displacement pattern never repeats — and yet the atoms are nowhere near random: each one is exactly where a single sine wave says it should be. That is what an incommensurately modulated crystal is, and its diffraction pattern is sharp.$

The satellites that need a second integer

A crystal whose atoms are displaced by a wave of the wrong wavelength has no unit cell at all, and diffracts to sharp spots anyway. Indexing them takes two integers per reflection instead of one — and the intensities of the extra spots are Bessel functions, which is a check the arithmetic can be made to pass.

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.

Which inflation factors a tiling may have. Every distinct inflation factor produced by a two-letter substitution whose matrix has entries up to 4, plotted against its algebraic conjugate. The two grey lines are the unit circle, which in the quadratic case is the pair of values ±1. A factor whose conjugate lies strictly inside is a Pisot number and the chain it grows has sharp Bragg peaks; 39 of the 77 factors here lie outside and cannot. The golden ratio is the smallest of them all, which is the arithmetic reason it turns up in every quasicrystal anybody has drawn.

Which inflation factors exist

A tiling grown by substitution has an inflation factor, and it is an eigenvalue of an integer matrix — so it is an algebraic integer, and sharp diffraction demands that its conjugates be small. That condition is an inequality between two integers, and it explains why the golden ratio turns up in every quasicrystal anybody has drawn.

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.

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

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.

The Fibonacci chain: p(n) = n + 1. The number of distinct windows of each length in the Fibonacci chain, measured by sliding a window along 46,368 tiles. Every count is checked against the same count on half the chain, and only lengths where the two agree are drawn — a factor count on a finite word is otherwise a lower bound wearing the clothes of an answer.

n plus one, and no fewer

Slide a window along a chain and count what it can show. A periodic chain runs out of new views; an aperiodic one never does; and the fewest an aperiodic chain can manage is one more than the window's length — which is exactly what the Fibonacci chain manages.

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.

One patch of radius 2. A Penrose patch of 476 vertices, with the vertices within 2 edge lengths of one of them marked and the circle drawn. That marked set, written in coordinates relative to its centre, is what the census compares: two vertices have the same patch when their marked sets agree. Every vertex of the tiling is the centre of one such patch, and the question is how many different ones there are.

How many patches of each size

A periodic tiling has one kind of neighbourhood however far out you look. Random points have as many kinds as neighbourhoods. A Penrose tiling has a number in between that never stops growing and never catches up — and the count is a measurement rather than a theorem.

625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound.

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

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.

$Order 5: 32,768 arrangements. An Aztec diamond of order 5, with every possible dimer drawn at an opacity equal to the fraction of arrangements it appears in — a probability computed exactly, by counting the arrangements of the region with that dimer's two sites removed, rather than sampled. The four corners come out nearly certain and the middle nearly even, with a circle between them. The most certain dimer here occurs in 0.97 of the arrangements, which is 1 − 2⁻5 exactly, so nothing is frozen at any finite size.$

How many arrangements one rule allows

Every count in this collection so far has been a count of symmetries, or of orbits under one. Here is a different count: the arrangements a purely local rule permits on a fixed lattice, with no symmetry quotient anywhere in it. The answers are enormous, they are exact, and the useful quantity is not the number but its growth per site.

Arrangements per site, falling towards the exact value. The number of configurations of an L × L torus obeying the ice rule, taken to the power of one over the number of sites. The largest computed here is 4,484,823,396 configurations on a 7 × 7 torus. The values fall towards Lieb's exact 1.54 from above, and every one of them is above Pauling's estimate of 1.5 — which undercounts, because it treats the vertices as independent and they are not.

The arrangements a crystal keeps at absolute zero

Ice has a residual entropy, and the number a calorimeter measures is the logarithm of a count of arrangements. Pauling's one-line estimate of that count is out by two and a half per cent; the exact count in two dimensions is available, falls towards its limit from above, and the whole disorder is invisible to a diffraction experiment, which sees only the average.

21 points, 3 gap lengths. The first 21 multiples of 377/610, marked on a circle of circumference one, together with the point at zero. The 22 gaps between neighbours take 3 distinct lengths — 13/610 (1 of them), 21/610 (9 of them), 34/610 (12 of them). The largest is the sum of the other two: 13 + 21 = 34. Every quantity here is a whole number over the denominator, so nothing is measured.

Three gaps, and never four

Mark the points α, 2α, 3α round a circle of circumference one. They look scattered. The gaps between neighbouring points do not: for every angle and every number of points there are at most three distinct gap lengths, and when there are three the largest is the sum of the other two. That is where a chain with exactly two tile lengths comes from.

$two chains, periods 1 and 1.62. Two interpenetrating chains of atoms with periods 1 and 1.62, whose ratio is irrational, so no length is a whole number of both. Each chain is displaced from its own lattice by a wave with the other's period — the short ticks show each atom's displacement from where an unmodulated chain would put it — which is what makes this one crystal rather than two side by side. Nothing here is a unit cell: any length chosen contains a whole number of one chain's atoms and a fractional number of the other's.$

Two lattices, one crystal, and no cell at all

A modulated crystal has a lattice and a wave running through it. A composite has neither host nor guest: two interpenetrating substructures with periods that share no common multiple, each modulating the other. Every reflection needs an index from both, and the unit cell a diffractometer reports belongs to whichever half scattered harder.

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.

The allowed energies of the Fibonacci chain, level by level. The set of energies at which a wave neither grows nor decays, for periodic approximants of the Fibonacci chain of 5, 8, 13, 21, 34 and 55 sites. Each row has exactly one band per site, and each band splits into smaller ones at the next level rather than growing. Nothing in the picture converges to an interval: the gaps opened at one level survive at every level after it, and the limit is a Cantor set — closed, containing no interval at all, and of measure zero, which is a theorem of Sütő's rather than something these six rows prove.

A spectrum that is a Cantor set

A wave in a periodic chain has bands with gaps between them. A wave in the Fibonacci chain has gaps inside the gaps, at every scale — and the traces that decide where they are obey a recursion with a quantity it cannot change.

The average is the same wherever it is taken

A measurement is made on a window somewhere, and the question is whether the answer belongs to the chain or to the window. For the Fibonacci chain the error falls as one over the window's length; for a shuffle of the same letters it falls as one over the square root, and the two exponents are fitted rather than asserted.

One tile, and no period

Every aperiodic pattern in this collection so far needs two shapes. A search over the eight-hundred-and-seventy-three ways of gluing eight kites together, filtered by nothing but whether a shape tiles and whether it repeats, returns exactly one — and it is the shape announced in 2023.

The tile that needs no reflection

One shape tiles the plane and never repeats, and it does it with copies of both hands. Cut the tiles out of card and that is nothing; ask for it in a molecule, where handedness cannot be undone by turning something over, and it is the whole question.

A peak that grows, and not fast enough. The strongest peak of three chains, divided by the square of the number of letters, as each chain is lengthened. A Bragg reflection is a sum of terms in phase, so its intensity grows as the square of the count and this number settles: the Fibonacci chain and the period-doubling chain both do, at exponents of about two. The Thue–Morse chain does neither — its strongest peak grows, so it is not diffuse scattering, and it grows more slowly than the square, so it is not a Bragg peak. The fitted exponents are printed beside each curve and no threshold enters the comparison.

Neither a peak nor a bump

A chain whose strongest reflection grows as the length to the power one and a half. A Bragg peak grows as the square and a diffuse bump grows as the length itself, so this is neither — and the essay that ruled out the first possibility could only say so by quoting a theorem.

How often each patch occurs

That a patch has a frequency at all is the ergodic theorem. What the frequency is turns out to be an eigenvector: the substitution acts on blocks as well as on letters, the block matrix has a Perron vector, and its entries are the frequencies exactly. For the Fibonacci chain those entries take three values at every length, and the three values are the three gaps of a rotation.

21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part.

Superspace groups in the plane

A modulated crystal has no space group, and in a space of one more dimension it has one. Counting them in the plane is the seventeen's own extension arithmetic with a third coordinate on which the point group acts by a sign — and the sign has to be plus or minus exactly, which kills the three-fold, four-fold and six-fold classes before a single extension is counted.

Four of the 980 piles in a three-cube box. A stack of unit cubes in the corner of a box, seen down the body diagonal. Every visible face is one of three rhombi and the picture is a tiling of one fixed hexagon — the same hexagon for every pile, because a pile in an a×b×c box always shows ab+bc+ca faces however it is stacked. The four here are taken at even intervals through the enumeration, from the empty box to the full one.

A facet with no energy in it

Stack cubes into the corner of a box and look down the body diagonal: the pile is a tiling of a hexagon by three rhombi, and the number of piles is a product MacMahon wrote down in 1916. Because the count is exact, so is the average pile — and the average has a flat corner meeting a rounded middle, which is the shape of an equilibrium crystal, arrived at by counting with no surface energy anywhere in the argument.

One hat patch laid out as hats, as equilateral tiles and as turtles. A patch of 36 hats found by exact cover on the kite grid, 4 of them reflected and drawn in the second colour, laid out three times. Every edge keeps its direction; short edges and long edges are given their own lengths. At short 1 and long √3 the tiles are hats, at equal lengths they are the equilateral member of the family, and at short √3 and long 1 they are turtles. In all three the same tiles touch the same neighbours along the same edges, and each layout was checked to be a tiling: 0 gaps and 0 overlaps, 0 gaps and 0 overlaps, 0 gaps and 0 overlaps among 1500 sample points, and every interior vertex surrounded by a full turn.

The hat and the turtle are one tiling

The hat has short sides and long sides; the turtle has the same turns with the two lengths exchanged, and looks nothing like it. Take a patch of hats, keep every edge pointing the way it points, stretch the short edges and shrink the long ones, and the patch becomes a patch of turtles — every tile touching the same neighbours along the same edges.

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

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.

Square ice scatters a pinch at the origin. The intensity scattered by the horizontal arrows of square ice, averaged over 100 configurations on a 32 by 32 torus, over the whole Brillouin zone with the origin at the centre; darker is more intense. Along the horizontal axis through the origin the intensity falls to zero — 2.9e-32 at the smallest wavevector — while along the vertical axis it stays near 1.75, so the two meet at the origin in a pinch.

The ice rule is a conservation law

Two arrows in and two out at every vertex is a statement that nothing flows in or out anywhere. That makes one half of the arrow field vanish identically, in every arrangement and not merely on average, and what is left scatters with a pinch at the origin: an intensity that approaches different values from different directions. Break the rule now and then and the pinch acquires a width Debye and Hückel predicted for a salt solution.

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.

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

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.

Identical layers, each turned by an angle no number of turns undoes. Plan views of 4 layers of a stack. Each layer is the same square lattice with one cell shaded and one direction drawn, and each is turned from the one below through the angle whose cosine is three fifths, about 53.13 degrees. That angle is not a rational part of a full turn, so no number of layers brings the drawn direction back to where it started. A tiling of space with this structure has a symmetry that turns one layer onto the next and climbs one layer, and it has no translation.

Aperiodic is two words in space

A tile is aperiodic when none of its tilings is periodic, and periodic has been read two ways: a tiling with a translation, or a tiling with infinitely many symmetries. In the plane those are one condition, provably. In space they come apart, and a prism found in 1988 sits exactly in the gap.

What a defect costs the count

Each broken vertex relaxes the rule and so adds arrangements — the question left standing was whether each adds a fixed amount or the cloud around it costs some back. The exact count at every defect number at once answers both halves: almost all of the rise is the freedom to choose which vertices break, and with that removed the first defects subtract rather than add.

The count that depends on the edge

A residual entropy is supposed to be a bulk number: so much per vertex, whatever surrounds the lattice. Square ice has two of them. On a torus the count per vertex heads for 1.5396 and inside a domain wall it heads for 1.2990, with the same rule on the same lattice — and the sixteen per cent between them is sitting in the corners.

Three colours on a chessboard

Colour the cells of a board in three colours so that no two sharing an edge agree. The number of ways is the number of ice arrangements on the same board — the same integer, to the last digit, at every even size — so a residual entropy a calorimeter reads is also the answer to a colouring problem with no physics in it at all. At odd sizes the two counts part company, and why they do is a condition on going round.

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