Crystallography & Symmetry
patterns derived from their groups
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The theme: Exactly this many — page 9

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'.
66 reflections that no rotation reaches. Every reflection inside the limiting sphere of an orthorhombic cell, 7 × 11 × 13 Å at 1.4 ångström, plotted by its distance from the rotation axis against its height along it. The ones marked are those a rotation about that axis can never bring into diffracting position: turning the crystal moves a point on a circle at fixed height, so a point too close to the axis can never acquire the component along the beam that the Ewald condition demands. The blind region is a cusp about the axis, it is 4.3 per cent of the sphere here, and nothing but remounting the crystal removes it. How it is known

What one turn of the crystal reaches

Every reflection inside the limiting sphere is measurable by some orientation. A crystal on a spindle has one axis, and a region around it never reaches the Ewald sphere at all — however patiently the crystal is turned.

 Two plane structures with one Patterson. Two arrangements of 4 atoms on a 4 by 4 torus. No translation and no half-turn carries one onto the other, so they are different structures; every interatomic vector occurs the same number of times in both, so no measurement of intensities distinguishes them. The plane case is not the chain case with an extra index: a mirror in the plane sends a vector set to its mirror image rather than to itself, which is the one place the analogy with a cycle breaks. How it is known

Two structures on a torus, and one Patterson

Homometry was settled here on a ring of positions, which is a crystal in one dimension. Moving the same exhaustive search to a torus asks whether the coincidence is commoner or rarer when the vectors have a plane to land in — and the honest answer is that dimension is not what decides it.

 the kagome net: 34 of 144 wavevectors carry a mechanism. The zone of the kagome net, with a mark at every wavevector whose rigidity matrix drops rank — which is to say at every wavevector that carries a motion of the bars. There are few of them and they are isolated, so enlarging the cell adds mechanisms slowly. The ranks at the half-integer wavevectors are exact; the others are computed with a stated tolerance, because the matrix there has genuinely complex entries. Symmetry at work

A mechanism that is a wave

The framework essays found the kagome net's mechanism count growing with the cell it was looked for in, and recorded it as a finding without an explanation. Here is the explanation: the motions lie along lines in reciprocal space, and a larger cell samples a line at more places.

 One vertex, two edges: one net. Three edges: no answer at all. Every net with one vertex and the stated number of edges, counted inside boxes of voltages of three sizes, up to change of basis and the sign of an edge. Two edges give one net whatever the box, and the reason is a sentence: two voltages that generate the translations are a basis of ℤ², and every basis is carried to every other. Three edges give more nets in every larger box, and that is not a failure of the search — normalise two of the voltages to a basis and the third is a free pair of integers, so the family is infinite. An enumeration inside a bound reports which of those two situations it is in rather than reporting the count it happened to reach. Symmetry at work

Every net with one vertex, counted

A net is a few vertices, a few edges and a pair of integers on each, so a census is available: fix the numbers, bound the integers, enumerate. Two edges give exactly one net at every bound. Three give three, then nineteen, then a hundred and forty-three — and the question changes.

 How many reflections the centre test needs. The error rate of two tests for a centre of symmetry against the number of reflections used, measured on 60 centrosymmetric and 60 non-centrosymmetric structures at each point. The moment test — the one in every textbook, comparing ⟨|E|² − 1⟩ to its two theoretical values — reaches one error in twenty at 160 reflections and one in a hundred at 320. A likelihood ratio, which uses each reflection's own value instead of one average, reaches the same at 40 and 80. The gap is the price of summarising a distribution by its mean, and it is about a factor of four. How it is known

How many reflections it takes to know there is a centre

The test for a centre of symmetry compares one average of the intensities against two theoretical values a quarter apart. Whether that is a measurement depends on how many reflections went into the average, and the only honest way to find out is to run the test on structures whose answer is already known and count the mistakes.

 6°: a boundary with a dislocation every 9.5 cells. Two crystals of the same lattice, each turned by half of 6 degrees in opposite senses, meeting on the dashed line. Almost everywhere along it the atoms of one side face the atoms of the other at very nearly the right distance — the boundary is good crystal — and at the marked places the misfit has accumulated to a whole lattice vector and an extra half-plane has to be inserted. Those are the edge dislocations, and they are 9.5 cells apart against the 9.6 that Frank's formula gives. Symmetry at work

A small angle is a row of dislocations

Turn one crystal a degree against another and the coincidence arithmetic says they share almost nothing. The boundary between them is nevertheless nearly perfect crystal, and both statements are true: the misfit stays small for a long way and then, all at once, needs an extra half-plane.

 Which descents change the shape of the cell, and into how many shapes. Each descent the modes produced, with the number of independent strain components the parent class permits and the number the child permits. A transition is ferroelastic exactly when the second is larger — the child leaves alone a distortion the parent moves — and the difference is a spontaneous strain the crystal acquires without being pushed. The count of distinct shapes is the orbit of that strain under the parent, which can be smaller than the number of domains: two domains may differ in something a change of shape cannot show. Every count here is a rank of an averaged set of quadratic forms, computed twice — once by averaging, once from a character. Symmetry at work

The strain that arrives with the transition

A crystal that loses symmetry usually changes shape, and whether it does is a subtraction: how many strain components the child permits, minus how many the parent did. The difference is a distortion nobody applied, and it is what makes a domain visible in a microscope.

 3m → 1: the two directions a wall between domains may take. The difference between the strains of two domains, sampled around a circle of directions: the first colour where that direction is stretched, the second where it is compressed. The two solid lines are the directions where it is neither, and those are the only orientations a straight wall between the two domains can take without straining itself — Sapriel's condition, one dimension down from the planes it is usually written for. There are exactly two, and that is not luck: the two domains are images of one another under the parent group, so their strains have the same area change and their difference changes no area at all. A form that changes no area takes both signs, and its zero set is a pair of directions. Symmetry at work

The walls a strain permits

Two domains of different shape can only meet along a line neither of them stretches. That condition is a quadratic in a direction, so a pair of domains has exactly two permissible walls — and the reason there are always two rather than sometimes none is that their strains differ by no area at all.

 Σ5: three lattices in one picture. Two copies of the square lattice turned by 36.87 degrees against one another — one drawn pale, one drawn in the second colour — with the points they share ringed. The fine dots are the lattice generated by both together, the DSC lattice, which contains each crystal with index 5 exactly as the coincidences sit inside each crystal with index 5. Three lattices nested at the same index, and the middle one is the crystal. Symmetry at work

The dislocations a boundary allows

Two crystals meeting at a coincidence angle share one lattice and generate another. The second is where a boundary's own defects live, its shortest vector is one over the square root of the index, and a dislocation's energy is the square of that.

 Every wall is a frieze. Each ferroelastic descent, with the frieze group of each of the two walls its domains permit. A wall is periodic along its length and bounded across it, so its symmetry group is one of the seven — the classification this collection derived early as the same argument on a strip, arriving here as a fact about interfaces. The last column counts the operations in the wall's group that exchange the two domains rather than fixing them: a wall is unchanged by having its sides swapped, so those belong to it, and they are why a wall is often more symmetric than either domain. Symmetry at work

The wall has a group of its own

A boundary between two domains is periodic along its length and bounded across it, so its symmetry is a frieze. The seven, derived here early on as an exercise on a strip, turn out to be the classification of interfaces.

 A hundred and thirteen orbit types, merged into shapes. The three counts, and what stands between them. 113 is the number of kinds of form this site publishes: one for every stabiliser a face can have, in every class. Allowing a stratum to change shape along its own family raises it to 164. Merging entries that are the same solid with the same symmetry, wherever they occur, brings it down to 48 — 30 that enclose a volume and 18 that do not, which is the count every mineralogy text prints, with the dome and the sphenoid kept apart rather than merged. The last line is the warning: throwing away the symmetry of the solid and keeping only its combinatorial type leaves 35, because a rhombic dipyramid, a tetragonal dipyramid and an octahedron are one and the same arrangement of eight triangles. Symmetry at work

A hundred and thirteen orbits, and forty-eight shapes

This collection reports 113 kinds of crystal form and every mineralogy text reports 47. That difference was explained here in a paragraph and never computed, which means nobody had checked it. Computing it needs a definition of *shape* a program can decide, and the definition turns out to be the interesting part.

 The figure of merit a supercell always beats. One line list, indexed on the true cell and on five multiples of it. The mean discrepancy is not merely similar down the column, it is identical to every digit: the supercell's grid of allowed Q values contains the true cell's grid exactly, so each line lands on precisely the same place and misses by precisely the same amount. Any figure of merit built on agreement alone therefore returns one number for the whole family, and cannot prefer the true cell. What falls is the last column, and the only thing in it that the fit does not already contain is the count of lines the cell says should have been seen. Symmetry at work

The figure of merit a supercell always beats

Indexing a powder pattern returns a ranked list rather than an answer, and the ranking needs a number. The obvious number — how well the cell accounts for the lines — is exactly the number a supercell cannot lose on, because the supercell's grid contains the true cell's grid and the discrepancies are identical to every digit. What has to be paid for is the lines nobody saw.

 How densely each shape packs, by translation alone. The densest lattice packing of each shape, as its area over the critical determinant of its difference body. The two that tile the plane by translation reach one and must, which is a check on the search rather than a result of it. The triangle reaches exactly two thirds because its difference body is a hexagon. The many-sided approximation to a circle reaches π/√12, which this collection computes a completely different way. And the pentagon is the worst of them, which is where the search is doing work nobody could do by inspection. Symmetry at work

The densest packing of a shape that is not a disc

Which lattice packs equal discs most densely has a proof that finishes. Replace the disc with a pentagon and the same question has no closed form, but it does have a reduction: translates overlap exactly when the difference of their positions lies inside the shape minus itself, so the question becomes the smallest determinant a lattice can have while avoiding one convex body — and that is a search with a resolution attached.

 One net, six descriptions, four different answers about its symmetry. The honeycomb written against six bases of ℤ², all of them the same net. The detector tests each lattice type's holohedry in standard position, so a symmetry written against another basis is a matrix that is not in the list and is never tried — and the answer comes back as p6m, or an unnamed group of order four, or p2, or cmm, depending on how the voltages were typed. The metric column is the form the net's own edges make, inverted; the reduced column is that form after Lagrange–Gauss reduction, and it is the same in every row, which is what makes the last column a property of the net. Symmetry at work

The symmetry a net was written with

A net has no coordinates, so its symmetry is whatever its best drawing has. This collection measured that by handing the drawing to a detector — and the detector tests a fixed list of matrices, so the answer depended on which pair of translations the voltages had been written against. The honeycomb came back as p6m, or p2, or cmm, or nothing, one net and four answers.

 Two vertices and three edges: two nets, at every box size tried. Every net with two quotient vertices and the stated number of edges, counted inside boxes of voltages of several sizes. One cross voltage is set to zero by the gauge — the freedom that moving one vertex into another cell gives — and the rest are drawn from the box. Each entry is the count of nets whose placement separates their vertices, plus the count of those whose does not: the first has a canonical description and stops growing, and the second does not have one and therefore keeps rising with the box. The reducible column is the descriptions thrown away for a reason the one-vertex census never had — cycles generating the whole of ℤ² and a net whose own cell holds one vertex rather than two — and it is empty at every odd edge count, because the swap that would reduce a description pairs its edges and an odd number cannot pair. Symmetry at work

Every net with two vertices, counted

The one-vertex census could not contain the honeycomb, because the honeycomb has two vertices in its cell. Adding the second one closes a family at two nets, removes the floor of p2 entirely, makes a third of the members undrawable, and forces the census to refuse a kind of description the first one never met: an honest quotient graph written on twice the cell it needs.

 Every subgroup of index two is normal; at index three most are not. For each of the seventeen plane groups, its abelianisation and the number of normal subgroups of each small index against the number of subgroups of that index. The index-two column is complete every time, because the left and right cosets of a subgroup of index two are the same pair of sets. At index three and four the two numbers part, and the gap is what normality costs: a subgroup that is carried to a different subgroup by some operation of the group it sits in. Operations

The quotient each normal subgroup leaves

Two hundred and eighty-one subgroups of index four across the seventeen plane groups, and ninety-seven of them normal. Which ones, and what is left when they are divided out, needs no enumeration at all: below order six every group is abelian, so a normal subgroup of small index is a subgroup of the abelianisation and its quotient is decided by a product of greatest common divisors.

 One framework has a count of zero, one mechanism and one self-stress. Every net this collection has a placement for, as a periodic bar-and-joint framework in a fixed cell: its point group, the joints and bars of one cell, the scalar Maxwell count 2n − e − 2, and the mechanisms and self-stresses found exactly from the rank of the rigidity matrix. The scalar count is always the difference of the last two, which is Maxwell's identity — and the bathroom net is the row that shows what the identity costs: nought equals one minus one, and a framework that reads isostatic moves. Symmetry at work

The mechanisms a count cannot see

Maxwell's count subtracts constraints from freedoms, and a mechanism and a state of self-stress cancel in the subtraction — so a framework with one of each reports the same number as a rigid one. The bathroom net reports nought and moves. Doing the same subtraction with representations instead of numbers separates them, because a mechanism and a self-stress cancel only when they belong to the same representation.

 Cube, octahedron, rhombic dodecahedron — from connectivity alone. Every form of index two or less, classified by how many chains lie inside it: two or more and the face is flat, exactly one and it is stepped, none and it is kinked. The number beside each flat form is how many chains it contains, which is the rule's own tie-break — a face with three chains is flatter than one with two. Nothing about interplanar spacing enters, and the three structures are told apart by their bonds. Symmetry at work

Which faces are flat

Bravais ranks a crystal's faces by how far apart their planes lie. Hartman and Perdok classify them by how many uninterrupted chains of bonds run inside them, which uses no spacing at all — and on the three cubic structures the two rules put the same face first every time. Then the second rule's power turns out to live entirely in where the chain list is cut off.

 One local configuration, and a number of structures that doubles. How many kinds of adjacent pair a close-packed stack has, how many kinds of triple, and how many stackings of each period there are. The first column never moves: every pair of layers is congruent to every other, at every period, which is what an order-disorder family is. The last two agree with 2ⁿ + 2(−1)ⁿ, which is the chromatic polynomial of a ring of n layers at three colours, because a stacking of period n is exactly a proper three-colouring of that ring. The gap between the first column and the last is the whole subject. Symmetry at work

A stack with no space group

Every pair of layers in a close-packed stack is congruent to every other pair, and the number of stacks doubles with every layer added. A family whose local configuration is completely determined and whose global structure is not determined at all has no single symmetry group — what it has is a set of operations that compose only when their ends match, which is a groupoid.

 21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part. Order without repetition

Superspace groups in the plane

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

 A 21.79° twist, and the cell its beat has. Two copies of the same lattice, one turned. The coarse pattern a reader sees is the beat between them, and the outlined cell is computed from the two lattices rather than measured off the picture: the moiré reciprocal lattice is the original acted on by (I − R), so the moiré cell is the original scaled by one over twice the sine of half the twist, and turned through a right angle plus half the twist. Symmetry at work

A beat is not a period

Lay one lattice on another and turn it: the coarse pattern that appears has a spacing anyone can compute, a over twice the sine of half the twist, and it exists at every angle whatever. Whether the superposition actually repeats is a different question with a different answer — countably many angles say yes, and at most of those the true cell is larger than the beat by a definite factor. On a square net it always is.

 Four of the seventeen have a centre, and they are the four with no rotation. For each plane group: the order of its point group, how many of its operations are rotations, the lattice vectors every operation of the point group fixes, and the centre those vectors make. A central element must commute with every translation, which forces its linear part to be the identity — so the centre is a group of translations, and a translation is central exactly when the point group leaves it alone. A rotation leaves nothing alone but zero. Operations

The four groups with a centre

An element that commutes with everything has to commute with every translation, and that forces its linear part to be the identity. So the centre of a plane group is a group of translations — the ones its point group leaves alone — and a rotation leaves nothing alone but zero. Four of the seventeen have a centre and thirteen have nothing at all.

 Four of the 980 piles in a three-cube box. A stack of unit cubes in the corner of a box, seen down the body diagonal. Every visible face is one of three rhombi and the picture is a tiling of one fixed hexagon — the same hexagon for every pile, because a pile in an a×b×c box always shows ab+bc+ca faces however it is stacked. The four here are taken at even intervals through the enumeration, from the empty box to the full one. Order without repetition

A facet with no energy in it

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

 The alias accounts for every line and predicts more. The observed lines above, and below them the grid of a supercell that explains all of them. The full ticks are the observed lines, which the alias reproduces exactly; the faint ones are lines the alias predicts and nobody saw. That second set is the only thing that separates the two cells, and it is why an indexing criterion has to charge for unobserved lines rather than measure agreement. Symmetry at work

Every alias is a supercell

A cell that explains every line of a powder pattern is not a near miss and not a coincidence: its reciprocal grid contains the true one, which means its own cell is a superlattice of the true cell. So the ambiguity of indexing is the arithmetic of superlattices, and it can be counted — two cells with one unknown, sixteen with two, sixty-two with three, all of them accounting for the same twenty lines exactly.

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