Crystallography & Symmetry
patterns derived from their groups
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The theme: The same arithmetic, renamed — page 7

A crystal form is an orbit. A twin law is a coset. The domain states left by a phase transition are the cosets of the low-symmetry group in the high-symmetry one. Four subjects that grew up in different centuries and different departments, doing one piece of arithmetic under four names.
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. Order without repetition

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.

 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. Order without repetition

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.

 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. Operations

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.

 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. Order without repetition

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.

 Three ways to fill a lattice with a wave. Three occupation waves at the same irrational wavevector, each drawn as its profile over one period, the occupancy of a site as a function of where the wave stands there, and as the first thirty sites of the chain it produces, each filled in proportion to its occupancy. A cosine fills every site partly. A crenel of width 0.3 fills three sites in ten completely and leaves the rest empty. A crenel of width 1/φ fills sites exactly where the Fibonacci chain has its long letter, so the third row is that chain written as two species on a lattice. Order without repetition

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.

 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. Order without repetition

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.

 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. Operations

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.

 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. Order without repetition

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.

 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. The classification

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.

 Six groups are made of glides, four of mirrors, eleven of both. For each of the twelve plane groups that contain a reflection, what its mirrors generate on their own, what its glides generate, and what both together generate, counting lattice translations as something the generators must produce. Four groups are generated by their mirrors alone, six by their glides alone — pg, pgg, p4g, p3m1, p31m and p6m — and eleven by the two kinds together; pm is the one they cannot make, because its mirrors are parallel and nothing produces the translation along them. Where a set falls short the cell says how: every coset reached but only part of the lattice, translations in one direction only, or a proper subgroup. Operations

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.

 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. Order without repetition

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.

 Seventy-three classes, seventy-three Smith forms. Every arithmetic crystal class of space, grouped by lattice system, with the cohomology group read off the Smith normal form of the integer conditions its Cayley graph imposes, shaded by the group's exponent. Twelve classes have nothing; fifty-three have exponent two, among them the orthorhombic P holohedry with ℤ2 to the sixth; three have exponent three, three have exponent four, and two — the hexagonal 6 and 622 on the primitive lattice — have exponent six. Nothing larger occurs. The classification

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.

 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. Operations

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.

 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. Lattices

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.

 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. Operations

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.

 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. Order without repetition

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.

 E₈ has the smallest sum at every width. For four eight-dimensional lattices of unit covolume — D₈, its dual D₈, the cubic lattice Z⁸ and two copies of D₄ — the ratio of their Gaussian sums, less the constant term, to E₈'s, at widths from 0.2 to 5, logarithmic. Every ratio exceeds one everywhere: E₈ wins at every width against every rival. The closest approach is at t = 1, where D₈ and its dual are equal, a factor of 1.31 above E₈. D₈ is the better of the pair at wide widths and D₈ at narrow ones, and neither comes close to E₈ at either end. Lattices

In eight dimensions one lattice wins at every width

In space no lattice has the smallest Gaussian sum at every width, because the densest lattice wins when the Gaussian is narrow and its dual wins when it is wide, and fcc is not bcc. The argument fails exactly when the densest lattice is its own dual. In eight dimensions it is: E₈, with 240 shortest vectors, beats D₈, its dual, the cubic lattice and a sum of two four-dimensional lattices at every width tried, by a factor of at least 1.31, and its sum is mirror-symmetric in the logarithm of the width.

 Every vertex of a Penrose patch, by the symmetry of what lies within three edges of it. A region of a Penrose rhomb tiling 15.2 edges across. Each vertex is marked by the local group of the patch of radius 3.05 edges about it: a short line along the mirror where that patch has one (98 vertices), a large dot where it has a five-fold centre (7), and a small dot where it has no symmetry at all (118). The mirrors are common and point in the tiling's ten directions; the five-fold centres are rare; nothing has a half-turn. Order without repetition

Five-fold centres stay few, and mirrors do not

The Fibonacci chain has one palindrome of every even length and two of every odd length, for ever. Run the same census on the patches of a Penrose tiling and the plane gives three different answers at once. Patches with a five-fold centre stop at four, however large they are drawn. Patches with a mirror grow in number with their radius, by about fifty for every edge. And no patch wider than a single tile has a half-turn at all. What decides which answer an operation gets is the size of the set it leaves fixed in internal space.

 Almost every descent ends at D₄. At five widths of the Gaussian, twelve local minimisations of the lattice sum over all four-dimensional lattices of unit covolume, each from a random lattice, each marked by where it ends and labelled with the number of shortest vectors of the lattice it ends at. Fifty-nine of the sixty end at D₄, with twenty-four shortest vectors. One, at the widest Gaussian, ends at A₄*, with ten, a lattice whose sum is higher than D₄'s but which no small change lowers. Lattices

The two lattices where a four-dimensional descent stops short

In four dimensions the argument that rules out a single best lattice in space has nothing to act on, so whether one lattice wins at every width is left to be measured. A search over every four-dimensional lattice finds D₄ from almost anywhere. The exceptions are the finding: two lattices, A₄ and its dual, where a descent can stop short, each at one end of the widths and neither in the middle, for the same reason Voronoi found four dimensions has two perfect forms.

 Every four-colouring of the seventeen, by the group that permutes the colours. The 281 subgroups of index four across the seventeen plane groups, each a colouring of a pattern in four colours that the pattern's symmetries permute transitively, sorted by the permutation group they induce on the colours: ℤ₄ 20, Klein 77, D₄ 164, A₄ 8, S₄ 12. The groups without a three-fold rotation use only the cyclic, Klein and dihedral groups; the five with one use only A₄ and S₄, apart from a single Klein colouring of p6m. The classification

Four colours, five ways to permute them

Two colours can be swapped in one way and three can be permuted in two. Four colours can be permuted transitively by five different groups, and the seventeen plane groups use all five: 281 four-colourings sort into 20 cyclic, 77 Klein, 164 square, 8 tetrahedral and 12 fully symmetric. The last twenty belong to the five groups with a three-fold rotation, and they are the lattice reduced modulo two.

 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. Order without repetition

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.

 The centrings each holohedry allows, prime by prime. For each of the seven holohedries on its conventional lattice and each prime from 2 to 7, how many of the lines and planes of the lattice reduced modulo that prime the holohedry leaves invariant, and which of them give a lattice other than the one being centred. At 2 they give C for the monoclinic system, C, I and F for the orthorhombic, I for the tetragonal and I and F for the cubic; at 3 they give the rhombohedral lattice from the hexagonal one; at 5 and 7 nothing new. Lattices

Only two and three can centre a lattice

The centring letters P, A, B, C, I, F and R are usually a list brought to the Bravais lattices. They can be derived instead: a centring is a subspace of the lattice reduced modulo a prime, and a centring that keeps a crystal system's symmetry is a subspace its holohedry leaves invariant. Enumerating every one for the seven holohedries and the primes up to seven gives exactly the fourteen, with new lattices only at two and three. The reason is the crystallographic restriction, one step removed: every holohedry has an order made of twos and threes.

 The seventy-five rod groups by what fixes the axis and what moves along it. Every rod group sorted by its kernel, the plane point group of operations that leave every point of the axis in place, and by the group of the line it maps onto: translations only, or translations and reversals. Each chip counts the rods with that kernel, that line group and a given number of steps to the repeat. 29 never reverse the axis and 46 do, seventy-five in all, derived from the kernel and the rule that joins it to the line and agreeing cell by cell with the rods counted class by class. The classification

Seventy-five is twenty-nine and forty-six

The eighty layer groups came out as a plane group with a sign on each operation. The seventy-five rod groups have an account of the same shape turned round: a finite group of the plane across the rod, which is what leaves the axis alone, extended by a group of the line, which is what moves along it. Twenty-nine rods never turn their axis over and forty-six do, and the count comes out cell by cell as the rods counted class by class. The screw of a rod is the turn its smallest step gives the plane, and the kernel decides its subscript.

 Heavier atoms spoil the line, and then mend it. Above, the scatter of Sayre's ratio about its fitted line, averaged over four random cells, as a quarter of the atoms is made heavier, for cells of eight and of twelve atoms. Below, one minus the square of ρ, the correlation between the density's transform and its square's, computed from the composition alone. The spread of atomic numbers grows without limit along this axis, but ρ does not keep falling: it is lowest near a weight of three and climbs back towards one as the heavy atoms come to dominate, because a cell ruled by a few equal heavy atoms is nearly a cell of equal atoms. The scatter rises and falls with it. At equal weight what scatter remains comes from atoms that overlap, more of them in the larger cell. How it is known

Unequal atoms break the equality, and very unequal ones mend it

Sayre's identity is exact for a cell of equal atoms. Give a quarter of them more weight and the line it predicts scatters, reflection by reflection, by exactly the difference between weighting atoms by Z and by Z squared. Make them heavier still and the scatter shrinks again, because a cell ruled by a few heavy atoms is nearly a cell of equal ones. The scatter measures the cell, but only if the phases are already known.

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