Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Exactly this many — page 14

Seven friezes, seventeen wallpapers, fourteen lattices, two hundred and thirty space groups. Each number is a theorem with a finite proof, and the word that matters in every one of them is 'exactly'.
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. Into space

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.

 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.

 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.

 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.

 Where an antiferromagnet's reflections land. Two cubic antiferromagnets, with their nuclear reflections as dots and their magnetic reflections as triangles, in units of the conventional reciprocal lattice. On the left, a simple cubic lattice with every neighbour opposite: the magnetic reflections sit at half-integer indices, in a layer the nuclear pattern never visits. On the right, a body-centred lattice with the corners up and the centres down: the magnetic reflections sit at integer indices with an odd sum — exactly the reflections the body centring had extinguished, lit up again by the moments. In both, no magnetic reflection falls on a nuclear one. What symmetry decides

Magnetic reflections land where nuclear ones cannot

An antiferromagnet scatters neutrons into places its chemical structure leaves dark, and never into a place it lights. The reason is one line of algebra: the magnetic pattern is the nuclear pattern shifted by half a reciprocal-lattice vector. Each of a lattice's seven halvings is one such shift, and how many of them its symmetry keeps decides how many kinds of domain the magnet must form.

 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 group a symmetric motif can lift a plane group to. The seventeen plane groups arranged by lattice, with an arrow from each group to every group a motif of higher symmetry, placed at some position, turns it into, on the generic lattice of the first. Dashed arrows mark the lifts that also halve the cell, two of which return the same type on the smaller cell. p1 lifts only to p2, the rectangular groups to pmm, pmg and cmm, the square ones to p4g and p4m, and p3 to four different groups. Three groups, outlined, cannot be lifted at all without a special metric: p2, pmm and p6m. The classification

Where a symmetric motif lifts the group

A motif with more symmetry than its site asks for can raise the group of the whole pattern — or not, and which one is decided by the site, not the motif. The criterion is one line: a symmetry of the motif survives into the pattern exactly when it normalises the group. Run over every Wyckoff position of the seventeen, it finds twenty-nine of seventy-two where a motif can lift the group, general positions only in the four groups with a direction to slide along, two sites where the same motif gives different groups depending on how it is turned, and seven lifts between groups that halve the cell.

 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 seven edges a sheet can have. The seven frieze groups drawn as the profile of a cut edge: along the strip is along the edge, up the strip is the sheet's normal, and the half-sheet itself stretches away from the viewer. Each is labelled with what its operations are in the sheet — a mirror standing across the edge, a mirror or glide in the sheet's own plane, a half-turn about the cut's in-plane normal — and with how many of the eighty layers have it as the edge in some direction. Only p1 and p2 have no mirror image among their operations, so only they are chiral edges; every one of the seven occurs. Into space

Cutting a sheet can give it a hand

Cut a periodic sheet along a line and the edge that is left has a symmetry of its own: a frieze group, drawn in the plane across the cut. A chiral sheet can only have chiral edges. An achiral one usually has chiral edges too, in all but a few directions — and eight of the eighty layer groups have no achiral edge at all, however the sheet is cut.

 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.

 Two cuts that give back the mirror a single cut takes away. A sheet whose only symmetry beyond translation is a centre of inversion: each flag has a copy turned half round and moved to the underside, drawn open, and the dots are the centres. Every single cut leaves a chiral edge, and the piece on the far side carries its mirror image. The upper ribbon is cut symmetrically about a row of centres, so each centre on its middle line carries the ribbon onto itself with its two edges exchanged, and the ribbon is achiral. The lower ribbon, cut at the same width about a line with no centre on it, keeps nothing but translations and is chiral. Into space

A ribbon can be achiral where its edges are not

Cut an achiral sheet once and the edge is usually chiral, in almost every direction. Cut it twice, into a ribbon, and the ribbon may be achiral even when both its edges are chiral, because an operation can carry the ribbon onto itself by exchanging the edges — but only if the ribbon's middle line sits on a centre of inversion or a mirror parallel to the cut. Seven of the eight sheets with no achiral edge give achiral ribbons, and one achiral sheet gives nothing but chiral strips however it is cut.

 The centrings six four-dimensional lattice groups allow, prime by prime. For each of the six four-dimensional lattices whose symmetry groups are maximal, with the group's order, and each prime from 2 to 7: the subspaces of the lattice modulo that prime the group keeps, sorted by what centring by each one makes. New Bravais types appear only at 2, three of them, all over the cube-and-line lattice, and at 5, where A₄ centred by fifths is A₄*. Both centrings of the hypercubic lattice by halves make D₄, which has three times the symmetry. Lattices

The body-centred hypercube is the checkerboard lattice

In three dimensions the centring letters come out as the subspaces a holohedry keeps modulo a prime, and only two and three make new lattices. In four dimensions the same census, run on the six lattices whose symmetry groups are maximal, finds a prime that the three-dimensional argument never could: five centres A₄ into its dual. It also finds that the most familiar centring does not survive the extra dimension. The body-centred hypercubic lattice is not a centred hypercube at all: its body centre is exactly as far away as an edge, and it is D₄, with three times the symmetry.

 Ninety-six four-colour designs. For each of the seventeen plane groups, the number of its four-colourings, as subgroups of index four, and the number of distinct designs they fall into under the affine normaliser, sorted by the group that permutes the four colours. The 281 colourings make 96 designs: 13 ℤ₄, 39 Klein, 39 D₄, 2 A₄, 3 S₄. The classification

Ninety-six four-colour designs

The seventeen plane groups have 281 four-colourings, counted as subgroups of index four, and many of them are the same design described from a different origin, on different axes or from a neighbouring cell. Identified under every affine map that carries the pattern's group to itself, they make ninety-six designs. The same computation run at two and three colours gives forty-six and twenty-three, and all three numbers match the classical counts of coloured plane groups.

 Two hundred and twelve of the 219 are made of their own operations. The 219 affine space-group types divided by which of their own operations generate them, lattice translations included. 110 are generated both by their screws and glides alone and by their rotations, mirrors and inversions alone; 67 by their screws and glides only; 29 by their point operations only; 6 only by the two kinds together; and 7 by nothing they contain — P1, P2, Pm, Pmm2, P4, P3, P6 — which need a translation given outright. Into space

Seven space groups cannot make their own translations

Every space group contains a lattice of translations, and for all but seven of the 219 types those translations are products of the group's other operations. A screw done twice climbs its axis and a glide done twice slides along its plane, so the non-symmorphic operations make lattices by themselves. The seven that cannot are exactly the seven plane groups with no glide, repeated along a line they all fix, and the reason is a difference between a glide line and a glide plane.

 Two hundred and thirty groups on a hundred and twenty-two diffraction symbols. Every space group placed in a column for its diffraction symbol — Bravais lattice, Laue group and systematic absences — with the columns grouped by lattice. There are 122 symbols. 50 of them name one group; 44 are shared by two, 23 by three, 2 by four and 3 by five. The classification

Every ambiguity in a diffraction pattern, sorted by its cause

A diffraction pattern shows a lattice, the symmetry of its intensities and the reflections that are missing, and the 230 space groups show 122 different combinations of the three. Fifty of those name a single group. For the rest, each pair of groups that share one can be sorted by why they do, and there are five reasons. Four of them are ambiguities some further measurement resolves. The fifth, two groups with the same operations in the same places except for where their axes cross, happens exactly twice, and nothing but a solved structure decides it.

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