Crystallography & Symmetry
patterns derived from their groups
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The theme: The same arithmetic, renamed — page 6

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.
The same structure, mapped from intensities and from differences. Left, the ordinary Patterson map of the structure: 14762 interatomic vectors, a continuous field of overlapping peaks, and the two vectors between the anomalous scatterers — circled — nowhere among its strongest. Right, the map from squared Bijvoet differences over the same reflections: two peaks after the origin, and they are those two vectors. The difference map is 7381 times smaller a problem to read. How it is known

A map of the atoms that break the law

Feed a Patterson synthesis the differences between the two halves of each Friedel pair instead of the intensities, and the map that comes back holds the vectors between the anomalous scatterers and nothing else. Two atoms among a hundred and twenty-two: 14,762 vectors become two.

 One change of setting, four rules. The four things a structure report contains and the rule each obeys under a change of setting with basis change P and origin shift p. The cell and the indices are multiplied by P; a coordinate is multiplied by its inverse, after the origin has been subtracted; and an operation is conjugated and then shifted by (W − I)p, a term the other three have no equivalent of. Applied to Pnma with the change below, the operations still close into 8, the orbit maps point for point, and every |F| is unchanged. Into space

One matrix, four rules

Changing the setting of a structure is one matrix and one origin shift — and the cell, the coordinates, the indices and the operations each obey a different rule under it. Three of the four ways of getting it wrong still leave a closed group of the right order, so closure catches none of them.

 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. Order without repetition

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. Order without repetition

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.

 Every vector realised, and not at the same hexagon count. Each row is a set of faces other than hexagons whose charge — the sum of 6 − k over them — comes to twelve, which is what a closed trivalent net on the sphere must pay. Each column is a number of hexagons added to that set, and the entry is how many different solids exist with exactly those faces, found by winding up every arrangement of them into a spiral. A dash means the search found none; a question mark means the planar reader declined the row and it is not evidence either way. Every row has an entry somewhere, which is Eberhard's theorem, and the first one is at 0, 2, 3, 4 hexagons depending on the row — so the charge decides everything except the number of hexagons, and the number of hexagons is not a function of the charge. What a lattice forbids

Everything except the hexagons

Three counts of what a closed net must carry end on the same admission: an arithmetic saying what a net must charge does not say that a net exists. Eberhard's theorem says how close the charge comes to being enough, and the answer has a shape nobody would guess — it fixes every face count except the hexagons, and the hexagons are exactly the entry it cannot see.

 Three conditions, and a near-miss for each. Zassenhaus's characterisation asks a group for a normal subgroup that is free abelian of finite rank, of finite index, and maximal among the group's abelian subgroups. Four groups against those three clauses. The free group on two letters has no non-trivial abelian normal subgroup at all; the discrete Heisenberg group has one that is free abelian of rank two and maximal abelian, and its index is infinite; ℤ² × ℤ/2 has a free abelian normal subgroup of index two, and the maximal one has torsion in it. Each fails a different clause, which is what shows no clause is redundant. The infinite dihedral group passes and is crystallographic in one dimension. What a lattice forbids

Which groups a crystal could have

Bieberbach's theorem is a statement about a group acting: discrete, no point far from an orbit. Zassenhaus turned it round into a statement a group can satisfy on its own — a maximal abelian normal subgroup, free of finite rank, of finite index — and each of those three clauses is kept out of redundancy by a group that fails it and nothing else.

 Two, seventeen, two hundred and thirty, and then. The number of arithmetic crystal classes and the number of crystallographic groups in each of the first six dimensions, with the second divided by the first. The classes multiply by between five and fourteen a dimension; the groups multiply by much more, and the quotient — how many groups an average class carries — goes 1.00, 1.31, 3.15, 6.74, 36.5 and 339. The last column says what is derived on this page and what is quoted: the plane in full, six of the seventy-three classes in space, and nothing at all above three dimensions, where the counts come from machine enumerations of the 1970s onwards. What a lattice forbids

Finitely many is not few

Bieberbach's third theorem says each dimension holds finitely many crystallographic groups and gives no idea how many. The counts are 2, 17, 230, 4783, 222018 and 28927922, and dividing them by the number of arithmetic classes says which of the classification's three steps supplies the explosion — the step that attaches translations, not the one that finds the matrix groups.

 Every arrangement on a torus 4 across, sorted by defects. The transfer matrix that counts ice arrangements chooses, at each vertex, the one horizontal arrow the rule permits. Enumerating both choices instead and carrying a polynomial that records how many vertices end up with three arrows in or three out gives the number of arrangements at every defect count at once. The first column, drawn solid, is the ice count — 2970 arrangements with no defect at all, which is the number the earlier transfer matrix gives and is checked against it. The second column is empty: no arrangement has exactly one defective vertex, because a defect carries a charge and the charges on a closed surface must cancel. The columns together add to two raised to the number of edges, which is every assignment of arrows whatever. Order without repetition

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 the first half outright: almost all of the rise is the freedom to choose which vertices break, and what is left once that is taken out exactly is small, uneven, and lowest at the first count that needs a four.

 The turns a rolled sheet can make are square roots. Every primitive rolling of a square and of a hexagonal sheet with at most 66 cells in a repeat, one mark per distinct turn. Across is the number of cells n₀ in one repeat of the tube; up is the turn, as a fraction of a circle, of the screw that climbs one n₀-th of the repeat. Most columns are empty, and the occupied ones hold two or four marks rather than anything up to n₀: on the square sheet the turns are exactly the square roots of minus one modulo n₀, on the hexagonal sheet the roots of the corresponding congruence for minus three. The pattern is symmetric about half a turn because a root and its negative are the two hands of one tube. What a lattice forbids

A rolled sheet turns by a square root

Roll a square sheet into a tube of thirteen cells and its screw could, on paper, turn by any of twelve fractions of a circle. It turns by five thirteenths or eight, and nothing else, because five squared plus one is a multiple of thirteen. Every rolled tube's screw is a square root of minus the sheet's determinant — which is why handed tubes appear only at the orders Fermat allows, why they always come in both hands, and why the crystallographic ones need an accident.

 How the space groups are spread over their classes. For each number of groups, how many arithmetic classes carry exactly that many space-group types, in space and in the plane, every count derived from the classes' cohomology and the normaliser's action on it. In the plane ten of the thirteen classes carry one group. In space twelve of the seventy-three carry one, thirty-five carry two and fifteen carry four, so the typical class carries two; the long tail is the orthorhombic and tetragonal classes with eight, ten and sixteen. What a lattice forbids

Two hundred and nineteen over seventy-three

The space groups average three to an arithmetic class, and an average of three could be a few classes carrying dozens or every class carrying a handful. Derived class by class from the cohomology and its normaliser, the answer is the second: twelve classes carry one group, the median carries two, and the largest carries sixteen. Sixty-one of the seventy-three admit a non-symmorphic group, and the step that merges relabellings, which the plane barely uses, removes more than a quarter of the list.

 A colouring, and the arrows it writes. A proper three-colouring of the cells of a four-by-four torus — no two cells sharing an edge carry the same colour — with an arrow drawn on each shared edge by the difference of the two colours it separates. The difference is one or two modulo three, never nought, so every edge gets a direction. At each corner four cells meet and their four differences go round a cycle and add to nothing modulo three, which forces two of the arrows in and two out. That is the ice rule, arrived at from a colouring with no arrows in its statement. Order without repetition

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.

 P4₁: the lattices, as an ideal across and a multiple along. Every sublattice the point group of P4₁ carries to itself, indexed by the norm of the ideal it uses across the axis and by the multiple it takes along it. The entry is the space group that sits on it: the parent's own type in one colour, a different type in the other, and a dash where no group with the parent's point group survives at all. A dot marks a lattice that is maximal — one whose step is a single prime, across or along, with nothing between it and the whole. The rows and columns are two divisibility orders and the table is their product, which is the shape the plane's answer predicted. Into space

An ideal across and a prime along

In the plane a copy of a group inside itself grows by a prime ideal, and the maximal indices are the norms of the primes of a ring. In space with one principal axis there are two directions to grow in, and the question the plane left was whether the two constraints multiply. They do not — and the place they fail is an index the plane calls maximal, because the step in between carries the group's mirror image.

 Three rotations of order twelve, and one twelve-fold. Three integer 4 × 4 matrices of order twelve, one from each rational class of that order on a four-dimensional lattice. A four-dimensional rotation turns two perpendicular planes, and each plane is drawn as a dial: a point, its images, and the path the operation takes round them. Φ₁₂ turns its planes by one twelfth and five twelfths of a turn, so a point runs round twelve positions. Φ₃ ⊕ Φ₄ turns one plane by a third and the other by a quarter, and Φ₄ ⊕ Φ₆ by a quarter and a sixth; each has order twelve because twelve is the least common multiple, and neither has a plane with twelve positions in it. What a lattice forbids

Three rotations of order twelve

A four-dimensional lattice can be carried onto itself by three different integer matrices of order twelve, and only one of them turns anything by a twelfth of a turn. The totient that is usually said to decide which orders a dimension permits decides something else: which angles. The orders are decided by a sum over blocks, the two lists part company at six dimensions, and space already has an operation whose order is not its angle.

 A row of coefficients for every group. For eight of the seventeen, the number of sublattices of each index that the group's point group carries to itself. A nought means the group has no copy of itself at that index at all. p1 and p2 preserve every sublattice, so their rows are the counts of sublattices themselves — 1, 3, 4, 7, 6, 12 — which is the sum of the divisors. The rows thin out as the point group grows, and p4m and p6m have almost nothing in them. Every row is a sequence a Dirichlet series can be built on, and the next figure is what that series factors into. Into space

A row written as a product

Every group's copies of itself sit at a row of indices, and every row so far has been read one entry at a time. Counting all of them at once turns a row into a Dirichlet series, and every one of the seventeen rows factors into a product over the primes — which is the statement that a copy is a chain of maximal steps, written as arithmetic. The plainest group of all has the most famous series in mathematics.

 All thirteen families of line groups, reached by rolling. Each of the seventeen plane groups, on its generic lattice, rolled into a tube along every lattice vector that gives a repeat, with a mark under each of the thirteen families of line groups some rolling lands in. The band above each column is the family's translational part: a general screw for families 1 and 5, a half-step screw for 4, 8 and 13, pure translations or glide planes for the rest. Every family is reached, six of them by a single plane group, and the two oblique groups reach nothing because an oblique lattice rolled in any direction never repeats. What a lattice forbids

The centring that turns into a screw

Rolled with no limit on the order, the seventeen plane patterns reach every one of the thirteen families of line groups — six of them through a single pattern each. And the families' translations are decided by one integer: a tube that keeps any reflection holds one or two cells of its sheet in a repeat, never more, and it holds two exactly when the sheet is centred along the rolling direction. The centring vector of the flat lattice is the half-step screw of the tube.

 One lattice, two copies of pm. pm on a lattice doubled across its mirrors. The mirrors of the parent are every vertical line; a copy of pm on the doubled lattice has mirrors every other line, and there are two ways to choose which — the solid set or the dashed set. Both are copies of pm with the same lattice and the same point group, and no translation of the parent carries one onto the other, because the translation that would is exactly the one the doubling removed. So a count of invariant lattices is not a count of subgroups, and the gap is visible in the smallest case there is. Into space

A lattice is not a subgroup

Every count of copies so far has counted lattices, and the International Tables count subgroups. One invariant lattice can carry several copies of a group that nothing in the parent carries onto one another — pm's doubled lattice carries two, with its mirrors on the even lines or the odd ones — and how many is a cohomology computation, a first where the classification of the seventeen used a second.

 Seven indices in 40, and one lattice at each. For every index to 40, how many sublattices of a cubic lattice there are and how many of them the full cubic point group carries to itself. The first number runs into the hundreds; the second is nought at almost every index and one at 1, 2, 4, 8, 16, 27, 32. A cubic point group is forty-eight conditions on a sublattice, and forty-eight conditions leave very little. The richest point group in three dimensions has the poorest arithmetic of copies, and the two are the same fact. Into space

The richest group has the poorest arithmetic

A plane group with no symmetry has seven hundred and sixty-two sublattices to grow by; one with a six-fold axis and mirrors has eight. Take the trend to its end in space and a cubic group has six to index forty — one at every cube, one at twice a cube, one at four times a cube, and nothing anywhere else. Every operation of a point group is a condition, and forty-eight conditions leave almost nothing.

 Six matrices and eight: a monoclinic normaliser modulo two. For P2/m and C2/m, every 3 × 3 matrix over the field of two elements that preserves the reductions of the two eigen-sublattices of the class's two-fold or mirror — six on the primitive lattice and eight on the centred one — each drawn as a tile of nine squares filled where the entry is one. The whole infinite normaliser reduces into this set, and a dot under each tile marks that some normaliser element with entries −1, 0 and 1 reduces to it. Every tile is realised, and the same holds in all six monoclinic classes. What a lattice forbids

The normaliser only acts modulo two

The space groups were counted as orbits of a small, finite set of changes of basis — only those with entries −1, 0 and 1 — out of normalisers that are infinite in the monoclinic classes. That the counts came out right was evidence, not proof. It becomes proof in two steps. A missing change of basis can only leave the count too high, never too low, so the published totals already force every class. And in the monoclinic classes the infinite normaliser acts on the cohomology only through its reduction modulo two, where the small set already reaches everything.

 The eighty layers by what rolling does to their two faces. The eighty layer groups sorted by how the two tubes a sheet makes, one with each face outside, are related. Seventeen one-sided layers make two different tubes in every direction. Forty-six make the same tube in every direction, because they have a mirror or glide in the sheet or a centre of inversion. Sixteen relate the two tubes only along two, four or six special directions, and there as mirror images. One, the layer with a four-fold rotoinversion and nothing else that turns it over, exchanges its faces and still makes two different tubes in every direction. What a lattice forbids

A sheet with two faces rolls into two tubes

Roll a sheet whose two faces differ and every operation that turns it over dies, as expected: a tube's inside is not its outside. But the sheet can be rolled with either face out, and the operations that died are exactly the congruences between the two tubes that result. Over the eighty layer groups that settles, direction by direction, whether the two are one tube, a pair of mirror images or two different objects, and one layer exchanges its faces and still never rolls the same tube twice.

 One group without an axis, and as many as the order with one. For a rotation of each order that an integer matrix can have in a small dimension, the number of space groups its arithmetic class admits — computed from the cohomology rather than enumerated. A rotation acting on the smallest lattice that will hold it fixes no direction and admits exactly one group: the symmorphic one, with no screw. Add a direction it leaves alone and the count becomes the order of the rotation, and the extra groups are its screws. The four-fold with an axis gives four, which are P4, P4₁, P4₂ and P4₃; the five-fold with an axis gives five, in five dimensions, where no published table exists to check it against. The classification

The screw a dimension does not have

The extension count is a machine that runs in any dimension, and the seventeen were the case where every step could be checked against a list arrived at four other ways. Run on a cyclic point group it has a closed form two lines long — and it says a five-fold screw axis does not exist in four dimensions, which is a prediction rather than a check.

 Tight where there is an axis and vacuous where there is not. The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — the bound every account of the subject quotes. What occurs is one over the exponent of the cohomology, which divides the bound. For a rotation with a direction it fixes the two agree exactly: a four-fold screw does need quarters and a six-fold sixths. For a rotation acting with no fixed direction the exponent is one — the cohomology is trivial and no fraction occurs at all — so the bound is slack by the whole order. The same bound is sharp and useless in the same table. The classification

The denominator a group actually needs

The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — a bound every account of the subject quotes. What occurs is one over the exponent, which divides it. The same bound turns out to be attained exactly and to be slack by its whole size, in two rows of one table, and what decides which is whether the rotation fixes a direction.

 Every finite normaliser of a lattice's group, found whole. The fourteen Bravais lattices on primitive bases, with the number of primitive vectors along the axes of their rotations, the order of the lattice's group, and the order of its normaliser in the integer matrices, found by trying every assignment of axis vectors to three independent ones. On the three lattices up to monoclinic the axis vectors lie on one line or none and the normaliser is infinite. On the other eleven it is finite (48, 16, 48, 48, 16, 16, 24, 12, 48, 48, 48), its largest entry is one, and it is exactly the set of matrices with entries −1, 0 and 1 that the space-group census used. What a lattice forbids

A normaliser must permute the axes

The space groups were counted with a small set of changes of basis, and past the monoclinic system nothing had shown that set was the whole normaliser. It is, and the reason is geometric: a change of basis that carries a lattice's group to itself must carry each rotation axis to a rotation axis, so it permutes a finite set of vectors, and past the monoclinic system three of them pin it down. Every such permutation tried, the eleven finite normalisers turn out to be exactly the matrices with entries −1, 0 and 1, and 219 and 230 stand without any table behind them.

 An equality, with a known factor in it. Sayre's identity says the structure factor is proportional to the convolution of the structure factors with themselves — an equality, not a probability — with a factor that depends on the atoms and the resolution and on nothing else. Computed for a structure of Gaussian atoms, the ratio of the two sides falls with resolution exactly as predicted: the logarithm of the ratio is a straight line in the squared index, its slope is minus half the width in the atomic factor, and the ratio is very nearly real and positive, so the identity relates the phases and not only the amplitudes. How it is known

The relation that is an equality

Every relation of this kind so far is a probability — right nine times in ten, useless applied once and decisive applied ten thousand times. One is not. For a structure of equal, resolved atoms the squared density has peaks in the same places, so every structure factor is exactly a convolution of all the others, with a factor that depends only on the atom. It is exact, and on its own it is useless.

 How often a broken vertex carries four. Of all the broken vertices in all the arrangements with a given number of them, the fraction whose four arrows all point the same way — on tori four, five and six vertices across. If the vertices were independent that fraction would be one in five at every count. With only two defects it is far lower, 0.085, 0.075 and 0.066 on the three tori, and falling with the size; as the count rises it climbs, oscillating, to one in five and stays there. The odd counts sit above the even ones because an odd number of defects cannot be made of twos alone — three defects must hold exactly one four, which is a share of one third. Order without repetition

A four is not two twos

A vertex of square ice can break the rule with three arrows one way or with all four, and a census that counts the two apart finds them behaving as charges of two and four should: the fours are rare while defects are few, a pair of fours is squeezed about four times as hard as a pair of twos, and the dip an earlier census reported turns out to be nothing but the first count that cannot be made without a four.

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