The collection

Every essay — page 18

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

What a lattice forbids

Only two-, three-, four- and six-fold rotations are compatible with periodicity. The proof takes one line and the exception took a Nobel Prize.

Pentagon contacts as pieces of the icosahedron. The icosahedron drawn with one vertex at the centre, two rings of five, and the twelfth vertex as the outer circle; each vertex is a pentagon and each edge a possible contact. Left: the dodecahedron, whose twelve pentagons all touch five others, uses every edge. Centre: the commonest contact graph among the 5,770 cages to sixty atoms, 6 contacts in clusters of 3 + 3 + 2 + 2 + 1 + 1, carried by 179 cages, placed on the icosahedron with every contact on an edge. Right: the one cage with no contacts at all. All but two of the 340 contact graphs can be placed this way.

Three hundred and forty ways for twelve pentagons to touch

Every cage of pentagons and hexagons has twelve pentagons, and which of them share a bond is a graph on twelve points. Over all 5,770 cages from twenty atoms to sixty there are only 340 such graphs, and the graph almost never tells one cage from another: the commonest is shared by 179 cages and one recurs at thirteen different sizes. And every one of them except two, both from small cages, fits inside a single graph, the icosahedron's, which is what the dodecahedron's twelve pentagons make when every one touches five others.

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Order without repetition

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

Every arrangement on two tori, by how it winds. The ice arrangements on tori eight and nine vertices across, each placed at the pair of numbers it winds by — how much the height on the faces gains going once round the torus across and once round it down — with a disc whose area grows with the number of arrangements there. On the even torus every winding is even and the arrangements crowd the centre; on the odd torus every winding is odd and there is no centre to crowd, so they sit at plus and minus one. Solid discs are the sectors whose windings are both multiples of three, which are exactly the arrangements that are also three-colourings. On the even torus that is exactly a third of everything; on the odd torus it is the four sectors at plus and minus three, and they hold about one arrangement in eighty.

A winding does not dilute

Three-colourings of an odd torus are a small and shrinking fraction of the ice arrangements on it, and the natural guess is that the fraction goes to nothing. It does not. It settles near one in twenty-eight, because the windings that decide it are a property of the whole surface rather than of its edge — and on even tori the same windings put exactly a third of the arrangements in one class, which in the limit is Jacobi's theta identity.

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What a lattice forbids

Only two-, three-, four- and six-fold rotations are compatible with periodicity. The proof takes one line and the exception took a Nobel Prize.

Twelve kinds of operation six dimensions adds, and the lattices that carry them. The twelve rational kinds of operation whose orders first appear in five or six dimensions — 7, 9, 14, 15, 18, 20 (two kinds), 24 (two) and 30 (three) — against six lattices whose symmetry groups were computed element by element. Every kind is carried by at least one. All three kinds of order thirty are carried by A₄ ⊕ A₂, forty-eight symmetries each; the single-block kinds Φ₇ and Φ₁₄ only by A₆, and Φ₉ and Φ₁₈ by E₆ and by three hexagonal planes.

The thirties share a lattice

Four dimensions has three kinds of operation of order twelve and no lattice carries all three. Six dimensions has three kinds of order thirty, and one lattice carries every one of them, forty-eight times each. The difference is not the order. One of the twelves is a single block that cannot be split, and no thirty is: thirty needs eight dimensions to be irreducible, so in six it is always a five-fold piece beside a three-fold one, and the lattice that is both pieces holds every combination.

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Order without repetition

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

$How many locked phases an experiment of a given resolution can see. For three convex repulsions, the number of steps of the staircase wider than a given share of the chemical-potential axis, as that share is made smaller by factors of ten. A resolution of one part in a million shows 199 steps for the short exponential, 829 for the long one and 41 for the inverse sixth power. Every count is exactly the number of fractions with denominator up to some bound, because every step with one denominator has one width. The short exponential and the inverse power cross: the power law shows fewer steps at every practical resolution and more at one part in a hundred thousand million.$

An experiment sees a Farey sequence

A devil's staircase has a step at every fraction, and no experiment sees them all. Which ones it sees is not a matter of luck: every step with the same denominator has the same width, so a finite resolution shows every fraction up to one denominator and nothing beyond it — a Farey sequence, whose length is a sum of Euler's totient. How that denominator grows with the resolution is the one thing the interaction decides, and what is left unseen is steps, never an incommensurate phase.

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Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

A structure that is its own negative, and one that is not. Two arrangements of two species on a four-by-four block with p4m symmetry, each with eight sites of each species, beside the arrangement that exchanging the species produces. On the left the exchanged arrangement is the original one moved by an operation of the group, so the two are one structure: it is self-complementary, and its balance of species is forced by a symmetry. On the right the exchanged arrangement is a genuinely different structure, and the balance is a coincidence of count. Counting structures with the species interchangeable merges the pair on the right into one and leaves the one on the left as it was, which is why the count does not halve.

Swapping the species does not halve the count

Let the two species on a lattice be interchangeable and every structure pairs with its negative, so the count should halve. It does not, because some structures are their own negative: an operation of the group carries them onto the arrangement with the species exchanged. Those are counted by the operations whose every cycle is even, they all sit at exactly half composition, and read as spins they are the arrangements whose zero magnetisation a symmetry enforces.

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Order without repetition

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

Mirrors everywhere in a chain with none. Thirty-four tiles of the Fibonacci chain, long tiles L and short tiles S, with an arc over every window of at least 5 tiles that reads the same backwards, drawn from its own centre, and a tick at the centre where its mirror stands. The chain as a whole has no mirror, since no palindrome in it is infinite, yet mirrors of every size stand all along it, the long ones spaced further apart than the short ones, and every one of them a local symmetry the chain lacks globally.

A chain with no mirror has mirrors everywhere

The Fibonacci chain has no mirror: nowhere can it be reflected onto itself. Yet it holds exactly one window of every even length that reads the same backwards and exactly two of every odd length, each recurring all along the chain, and no aperiodic chain that is not of its kind manages that count. Every letter of it creates a palindrome never seen before, and a mirror of any size is never further away than a fixed multiple of that size.

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Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

Three rules for 'square' are three shapes, and the cell falls between them. A measured cell placed by how far it is from each of the two conditions a square cell needs — equal edges across, a right angle up — each in its own standard uncertainties, so the square cells are the origin. Three rules for accepting the cell as square at the five per cent level are three regions. The two-condition χ² accepts inside the larger circle, of radius 2.45; testing each condition separately accepts inside the square of half-side 1.96; reading the distance as if it were one condition accepts inside the smaller circle, of radius 1.96. The measured cell, 2.22 and 0.75 from the two conditions, is inside the first region and outside the other two.

Two conditions are not one condition twice

Whether a plane structure has a higher-symmetry supergroup is decided by its measured cell, and the obvious measure — how many standard uncertainties the cell sits from a right angle, or from equal edges — cannot rank the candidates on its own. A square needs two conditions and a rectangle one, so a single distance means different things on each; counted properly, a square can score above the equal-edged lattices that contain it, and walking up the diagram one condition at a time is the reading that keeps the ranking coherent, at a price that has to be paid in its error rate.

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Order without repetition

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

An occupation wave is its own Fourier series

A modulated crystal whose atoms stay put while a wave decides how much of an atom sits on each site diffracts to satellites, and the satellite of each order is simply the matching Fourier coefficient of the wave's profile. A cosine stops at first order and can never exceed a quarter of a site; a step makes satellites of every order, skips exactly the orders its width divides, and makes the strongest first-order satellite any wave can. And a step as wide as its own wavevector is the Fibonacci chain.

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Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

Three mirrors of a 30–30–120 triangle, closed. A triangle with angles of thirty, thirty and a hundred and twenty degrees, shaded, and every image of it under the group its three side-mirrors generate, within a window. The images alternate between the triangle and its mirror image, and they tile the plane — so the closure is discrete. But the group it produces has a mirror along the triangle's own altitude, drawn dashed, that is none of the three given: the triangle is two copies of the thirty–sixty–ninety triangle, and the group is that triangle's, p6m. The closure found a mirror the generators did not name.

Four groups are made of mirrors

Close the plane from located mirrors instead of rotation centres and the closure is discrete only for four triangles of mirrors — and one of those four manufactures a mirror along its own altitude. Of the ten plane groups that contain a reflection, exactly four are generated by their mirrors; three hold a kaleidoscope at index two, three and four, which are the numbers in front of the star in their orbifold symbols; and three have mirrors in one direction only, which generate nothing along themselves.

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Order without repetition

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

One invariant for every invertible chain. The quantity x² + y² + z² − 2xyz − 1, built from the half-traces of the transfer matrices of the two words a substitution produces at each level and of their product, for six substitutions at one energy, logarithmic. For the golden, silver and bronze chains and for the golden rule reversed it is V²/4 = 0.09 at every level — the same number for all four, computed in exact integers. For period-doubling and Thue–Morse it wanders by orders of magnitude from one level to the next.

One invariant for every chain that can be undone

The Fibonacci chain's transfer matrices obey a recursion with a quantity it cannot change, and the natural guess is that every quasiperiodic chain has its own. It does not. The conserved quantity is the trace of the commutator of the two tiles' matrices, so every chain a reversible substitution builds from the same two tiles — golden, silver, bronze — conserves the same number, exactly, at every length and every energy. Period-doubling and Thue–Morse, whose rules cannot be undone, conserve nothing.

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Into space

Seventeen becomes two hundred and thirty. Screw axes, glide planes, and the translations that no choice of origin can remove — the operations a flat surface has no room for.

How many reflection operations a plane of each lattice can carry. Every mirror of each of the fourteen Bravais lattices' holohedries, sorted by how many classes of lattice vector lie in it modulo the conventional cell's own translations — which is how many reflection operations any plane of that orientation can carry in any space group on that lattice. Primitive lattices carry one on every plane. Centred ones carry two on the planes their centring vectors lie in: one plane of the C-centred orthorhombic lattice, the diagonal planes of the body-centred tetragonal and cubic ones, and every mirror plane of the face-centred lattices. The rhombohedral lattice, on the hexagonal axes the Tables use, carries three on each of its three mirror planes. None carries four.

Three on a plane, and never four

A reflection plane in a space group carries one operation for each class of lattice vector lying in it, and a face-centred lattice has planes holding two independent centring vectors — enough for four. None of those planes is ever a mirror. The fourteen lattices decide the question without the two hundred and thirty: one, two or three operations on a plane, and never four — with the three on every mirror plane of the rhombohedral lattice, where the Tables' own hexagonal axes put them and the letters have no name for them.

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Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

Every square-lattice superstructure, by the size of its cell. The number of distinct arrangements of two species on the square lattice whose primitive cell holds exactly n sites, for n from one to twelve, logarithmic, counted over every cell shape at once: every sublattice of index n up to the lattice's own symmetry, every colouring of it up to translations and the sublattice's stabiliser, and only those whose true period is that sublattice. Odd prime sizes are low, because only a few cell shapes exist; sizes with many divisors are high.

Every superstructure counted once

The 805 arrangements of two species on a four-by-four cell were never the structures with sixteen atoms: forty-nine of them repeat in a smaller cell, and every sixteen-atom structure whose cell is not a four-by-four square is missing. Counting structures rather than arrangements means counting over every shape of cell at once, and discarding every arrangement that repeats sooner. On the square lattice that gives 2, 2, 4, 11, 16, 40, 48, 148 structures for cells of one to eight atoms, and 22,413 with sixteen.

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Order without repetition

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

Two waves that are each symmetric enough make a lopsided pair. The first-order satellite amplitudes at h − q and h + q around the main reflections h = 1 to 4, for three chains: an occupation wave on the lattice, whose pairs are equal; the strain wave alone, the displacement the size effect produces with every site equally full, whose satellites are weak and grow with h; and both together. Together the pairs are unequal, the low side stronger, by an amount that grows from one reflection to the next — an imbalance neither wave has on its own, from a displacement of less than a hundredth of a spacing.

The strain wave that tips the satellites

An occupation wave puts equal satellites on both sides of every main reflection, and a strain wave a hundredth of a lattice spacing high is enough to tip them. Real composition waves drag such a strain with them, because atoms of different sizes want different spacings, and the tilt that results is not a nuisance. To first order it does not depend on how strong the occupation wave is at all. It reads the size mismatch directly, and its sign says which species is the larger.

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The classification

Exactly seventeen ways to fill the plane with a repeating pattern. Not a fact to memorise: a theorem, with a finite argument and no eighteenth case.

Eighty layer groups, over the seventeen plane groups. Each of the seventeen plane groups with a block for every layer group that projects onto it. Every plane group carries one layer whose two faces differ and nothing turns it over, and one with a mirror in its own plane. The rest are turned over by some of their operations, and there is one for each class of two-colouring of the plane group: the pale-dark blocks turn the layer over only with a rotation, a mirror line or a glide line of the pattern, and the dark blocks turn it over on a translation — a glide plane lying in the layer. The totals are 17, 17, 26 and 20, which is eighty.

Eighty is seventeen, seventeen and forty-six

The eighty layer groups are usually quoted, and an earlier count here reached sixty-three and blamed the translations. The eighty follow in three lines from things already counted: a layer group is a plane group with a sign on each operation, and the signs are either all plus, all doubled by a mirror in the layer, or a two-colouring of the plane group — seventeen, seventeen and forty-six. The sixty-three had missed the twenty colourings that turn a layer over on a translation, which are glide planes lying in the layer itself.

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Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

Six groups are made of glides

Four plane groups are generated by their mirrors. Six are generated by their glides, and they are mostly different ones, because a glide done twice is a translation where a mirror done twice is nothing. Closing the plane from located glide lines with their slides reaches six of the seventeen; mirrors and glides together reach eleven, every group with a reflection except pm. The classification does not fall out of the closure, and where it stops says what each kind of reflection can and cannot make.

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Order without repetition

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

Second-order satellites that only the strain makes. Satellite intensities on a logarithmic scale around the first four main reflections of a chain whose occupation is a pure cosine wave, with no strain and with the strain a size factor of 0.1 brings. Without strain the second-order satellites are absent, down at the chain's numerical floor; with it they appear at between a ten-thousandth and a hundredth of the first order's intensity, larger on the outer side of each reflection and growing with its order.

The second order reads the strain again

An occupation wave with a pure cosine profile has no second harmonic, so on its own it makes no second-order satellites. Strain the chain by the size difference of its atoms and they appear. Their strength measures the square of the displacement and cannot see its phase, where the tilt of the first-order pair measures the displacement times the cosine of its phase. So the two give the size factor twice, and the two readings agree only if the strain sits where a size effect puts it.

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Into space

Seventeen becomes two hundred and thirty. Screw axes, glide planes, and the translations that no choice of origin can remove — the operations a flat surface has no room for.

Four inversion centres, and a band's charge on each. A square crystal with four atoms in each cell, at the points a quarter of the way along both diagonals, and the four points of the cell that an inversion maps onto themselves: the corner, the midpoints of the two edges, and the centre. None of the atoms sits on one. For each of four versions of the crystal, differing only in which bonds are stronger, the ring marks where the lowest band's charge is centred — computed from the band's Berry phase and predicted from four parities. Each version puts it on a different inversion centre, and in every case on a point with no atom.

Four signs say where the charge sits

In a plane crystal whose only symmetry beyond its lattice is a centre, a band's charge must sit on one of the cell's four inversion centres — and which one is decided by four signs: whether the band is even or odd under inversion at the four wavevectors inversion leaves alone. The centre can be a point no atom occupies, and when the four signs multiply to minus one there is no centre at all, only a charge that winds round the cell.

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The classification

Exactly seventeen ways to fill the plane with a repeating pattern. Not a fact to memorise: a theorem, with a finite argument and no eighteenth case.

Seventy-three Smith forms

The cohomology group that counts a crystal class's intrinsic translations has a closed form only when the point group is cyclic. For every other class it is still one integer matrix away: the conditions the Cayley graph's cycles impose, reduced to Smith normal form, leave the group on the diagonal. Run on all seventy-three classes of space, the diagonal says the largest exponent is six, that a group of order forty-eight needs nothing finer than halves, and that no class needs a denominator its operations do not already have.

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Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

$How many superstructures a diffraction pattern tells apart. For two-species superstructures of the square lattice with one to twelve sites in the primitive cell: the number of structures, and the number of classes that the positions of their superlattice spots separate them into, the positions together with the composition, and every intensity. At twelve sites 2,012 structures give 167 sets of spot positions, 504 with the composition and 1,815 sets of intensities. At five, seven and eleven sites the positions distinguish only the cells; the intensities distinguish everything. The bar shows the share resolved by positions, dark, and by intensities, light.$

The spots say the cell and little else

Of the 2,012 two-species superstructures of the square lattice with twelve sites in the cell, more than half light every superlattice spot their cell allows, so the positions of the spots say which cell and nothing more. Positions separate the 2,012 into 167 classes, the composition takes that to 504, and every intensity to 1,815. At a prime number of sites no spot is ever missing, because a sum of distinct p-th roots of unity cannot vanish.

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Order without repetition

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

The h = 1 anomaly moves with the length of the chain. For a size-coupled chain with ε = 0.03, the ratio of the second-order pair's tilt, divided by k², to the first-order tilt at h = 1, from plain sums over the sites and from sums with a smooth window, for 22 chain lengths from 10,000 to 480,000 sites. The plain ratio swings between -0.08 and 5.60; at 30,000 and 120,000 sites, the two lengths first tried, it sits near 0.31 both times. The windowed ratio is one half at every length.

The satellite beside the zero-order reflection

An earlier essay found that the second-order satellites of a modulated chain tilt half as far as the first-order ones at every reflection but the first, where they tilted a third as far, the same on chains of 30,000 and 120,000 sites. It left the departure unexplained and proposed a higher order of the expansion. The expansion was never wrong. The weak satellite at h = 1 sits a quarter of a reciprocal unit from the strongest reflection a chain has, and a plain sum over a finite chain carries that reflection's tail, which does not shrink steadily as the chain grows. It swings, and the two lengths tried happened to swing to the same place.

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Lattices

The repeating scaffold underneath every crystal. Five in the plane, fourteen in space, and no others — a classification with a short proof.

Face- and body-centred cubic change places at t = 1. The Gaussian sum of the face-centred cubic lattice minus that of the body-centred cubic lattice, both at unit volume, across widths from 0.35 to 2.8. Below t = 1 the difference is positive, so the body-centred lattice has the smaller sum; above, it is negative and the face-centred lattice does. At t = 1 the two are equal to the last digit a computer carries, and not by accident: at unit volume each lattice's sum at width t is the other's at width 1/t times a known factor, and at t = 1 the factor is one. Neither lattice is best at every width, and no other lattice can be either.

No lattice in space wins at every width

Spread a Gaussian over every point of a lattice and add them up. In the plane one lattice gives the smallest sum at every width — the hexagonal one — and that is a theorem. In space no lattice does, and the proof is two lines: at large widths the winner must be the densest lattice, at small widths it must be the densest lattice's dual, and in space those are face- and body-centred cubic. They change places at exactly the width where each is the other's dual.

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The classification

Exactly seventeen ways to fill the plane with a repeating pattern. Not a fact to memorise: a theorem, with a finite argument and no eighteenth case.

I222 and I2₁2₁2₁: the same axes, apart. The two-fold rotation axes of the two space groups on the body-centred class of 222, drawn in one conventional cell seen down its third edge: axes along that edge as diamonds, axes along the first edge as solid lines and along the second as dashed lines, with the height of each written beside it. Both groups have pure two-fold rotation axes in all three directions. In the symmorphic group some axes of different directions cross, at the origin among other places; in the other none does, because the heights of the axes in each direction are a quarter out of step with the positions of the axes in the others.

Two groups no single operation reveals

A space group is non-symmorphic when no choice of origin removes all of its translations at once, and every textbook example has a screw or a glide to show for it. Two of the two hundred and thirty do not need one. Every operation of I2₁2₁2₁, taken on its own, is also a plain rotation about some axis; so is every operation of I2₁3. What makes them non-symmorphic is that their rotation axes never meet — and no diffraction extinction can tell them from I222 and I23.

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Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

When a three-species superstructure's spots can go dark. Every structure of three species on the square lattice with three to eight sites in its primitive cell, using all three species, and every superlattice spot of each, sorted by whether the spot is dark for every choice of scattering contrasts, dark at exactly one ratio of the two contrasts, or never dark; with the number of distinct ratios and their degree over the rationals. At the primes 3, 5 and 7 no spot is dark for all contrasts and no dark ratio is rational: none at three sites, quadratic ones at five and cubic at seven. At four and six sites every dark ratio is rational, at eight some are and some are not, and at six and eight some spots are dark for every contrast.

Three species hide a spot at the golden ratio

With two species and a prime number of sites in the cell, no superlattice spot can ever be missing, because a sum of distinct roots of unity of prime order vanishes only when it is all of them. With three species the structure factor weighs two sums against each other with two scattering contrasts, and the argument changes shape without breaking. At a prime size no spot is dark for every contrast and none at a rational ratio of them, but a spot can be dark at one irrational ratio: at five sites, a power of the golden ratio. It happens only in structures with a centre of symmetry.

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Order without repetition

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

The exact third on every even rectangle. Square ice on twelve tori with both sides even, eight of them rectangles that are not squares, up to 6 × 12 and 8 × 8: the number of ice arrangements, the number whose two windings are both divisible by three, and their ratio. It is one third exactly, as a ratio of integers, on every one.

The third survives a rectangle

On every even square torus, exactly a third of the square-ice arrangements have both windings divisible by three, and the explanation offered was a theta function that is its own dual. A test was proposed: on a torus twice as long as it is wide the two windings have different stiffnesses, the explanation gives a different number, and the exact third should fail. It does not fail on any of eight rectangles. But the explanation, worked through properly for a rectangle, never predicted that it would. The two directions of a rectangle are each other's duals, and their factors cancel to a third at every shape.

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