Crystallography & Symmetry
patterns derived from their groups
The collection

Every essay — page 13

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

Symmetry at work

Crystal forms, twins, domain walls and grain boundaries. Mineralogy, metallurgy and ferroelectrics each worked these out separately, and every one of them turns out to be an orbit or a coset of a group already built here.

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.

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.

7 figures 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.

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.

8 figures 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.

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.

7 figures 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.

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.

6 figures 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.

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.

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Operations

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

Every 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.

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.

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Symmetry at work

Crystal forms, twins, domain walls and grain boundaries. Mineralogy, metallurgy and ferroelectrics each worked these out separately, and every one of them turns out to be an orbit or a coset of a group already built here.

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.

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.

6 figures 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.

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.

6 figures 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.

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.

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Order without repetition

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

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.

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.

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Symmetry at work

Crystal forms, twins, domain walls and grain boundaries. Mineralogy, metallurgy and ferroelectrics each worked these out separately, and every one of them turns out to be an orbit or a coset of a group already built here.

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.

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.

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Operations

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

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.

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.

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Order without repetition

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

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.

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.

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Symmetry at work

Crystal forms, twins, domain walls and grain boundaries. Mineralogy, metallurgy and ferroelectrics each worked these out separately, and every one of them turns out to be an orbit or a coset of a group already built here.

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.

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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What a lattice forbids

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

The sphere fixes a count; the torus fixes only a difference. Euler's relation for a trivalent net gives Σ (6 − n) pₙ = 6χ, so the surface fixes one linear combination of the face counts and nothing else. On a sphere that combination is twelve, which with no face smaller than a pentagon forces exactly twelve pentagons. On a torus it is zero, which permits any number of pentagons provided as many heptagons pay for them — and permits none at all, which is the plain hexagonal net. On a surface of two holes it is minus twelve, so heptagons become compulsory instead.

As many heptagons as pentagons

A trivalent net on a sphere must have exactly twelve pentagons. The same three lines of arithmetic on a torus give zero — which does not forbid pentagons, it makes them pay: every pentagon has to be balanced by a heptagon, and the counts are otherwise free. One rotated bond in a wrapped honeycomb makes two of each and changes nothing else.

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Operations

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

The domain is a polygon, and its edges are elements. The Dirichlet domain of a point whose stabiliser is trivial: the set of points at least as close to it as to any other point of its orbit. It is a convex polygon, it is a fundamental domain, and each of its edges lies on the bisector of the base point and one image of it — so each edge already carries the element that produced it, with no search. Edges are drawn by kind: paired with another edge, fixed pointwise by a reflection, or folded in half by a half turn.

The relations a polygon dictates

Poincaré's theorem has two halves. The walls of a fundamental domain name the generators, which is the half this collection already computes; walking round its corners names the relations, which needs a domain with corners rather than a domain made of pixels. Building the Dirichlet polygon exactly gives a presentation of each of the seventeen — and coset enumeration says every one of them is right.

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Symmetry at work

Crystal forms, twins, domain walls and grain boundaries. Mineralogy, metallurgy and ferroelectrics each worked these out separately, and every one of them turns out to be an orbit or a coset of a group already built here.

No two cubic grains are more than sixty-three degrees apart. For each proper class: how many rotations describe one misorientation, the largest disorientation there is, and the mean over uniformly random orientations. The maximum is found by sampling and then climbing locally, so it is a lower bound that has stopped moving rather than a solved value — and it lands on the numbers the literature records.

The angle two grains differ by

A crystal's axes are not labelled, so a relative orientation between two grains has as many descriptions as the symmetry allows — five hundred and seventy-six of them for a cubic crystal — and their rotation angles run from a few degrees to more than a hundred and seventy. The honest answer is the smallest, and its largest possible value is a number: no two cubic grains are more than sixty-three degrees apart, whatever anybody does to them.

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The classification

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

The parity argument loses 36 pairs it had won alone. The argument that refutes ten of the twenty-one species walks round a polygon of odd size: the ring of polygons about it is a closed walk of odd length in a graph the species decides, and a bipartite graph has no such walk. With two species at a vertex the flanking pairs come from the union of two graphs, and a union of bipartite graphs need not be bipartite — so the walk stops being constrained. The fourth row is the cost: pairs whose members the argument kills on their own and which it cannot kill together.

The argument that closes eleven

Twenty-one vertex species satisfy the angle equation; a parity argument kills ten before anything is drawn, and the eleven survivors are all built. Asking the same question of tilings with two kinds of vertex, the parity argument evaporates — it constrains a walk in a graph one species decides, and two species decide the union of two graphs, which need not be bipartite. What is left is a search, and a search cannot close a count.

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What a lattice forbids

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

Four angles, and the integer that picks them. Two roots at angle θ have Cartan integers whose product is 4cos²θ. Both are whole numbers and the product is below four, so it is nought, one, two or three — and each value fixes the angle between the two roots, and with it the angle between the mirrors perpendicular to them. The shaded wedge is the region the pair of mirrors folds the plane onto; the smaller it is, the larger the group they generate.

Four root systems, and the same four rotations

Two mirrors meeting at an angle generate a group. Ask that the group be finite and that a certain pairing between the mirrors come out a whole number, and the angle has only four possible values — from which the rotations that survive are of order two, three, four and six. The crystallographic restriction arrives with no lattice anywhere in the argument.

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The classification

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

The cross, from the selection rule alone. The layer lines of a helix with the first maximum of each marked on both sides. Nothing here is a picture of a photograph: each mark is at the radius where the Bessel function of the lowest order the selection rule permits on that layer line first peaks, and that radius is proportional to the order. The order rises by one per layer line until the middle of the repeat, so the maxima lie on two straight lines through the origin — the X — and the larger marks are the layer lines that reach the axis.

What a thread scatters

A helix with ten subunits in a turn is not a screw axis a crystal may have, and nothing about its diffraction pattern is lawless. The pattern lies on layer lines, and on each one only certain angular orders may contribute — a selection rule as hard as any extinction condition. The lowest permitted order rises by one per layer line, a Bessel function of order n does nothing until its argument is about n, and the maxima therefore lie on two straight lines through the origin.

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Lattices

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

One change of basis turns a Gram into its own adjugate. For each Gram matrix: the matrix after the basis change by a right-angle rotation, and the adjugate. They are equal, always — and the adjugate is the determinant times the inverse, which is the dual lattice's Gram. So the dual is the same lattice on a rotated basis, scaled by one over the determinant. Five rows are the named plane lattice types and the rest have entries picked at random, because the claim is an identity in integers and not a property of the five.

Every plane lattice is its own dual

The dual of a lattice has the inverse Gram matrix, and in two dimensions the inverse is the adjugate over the determinant — which is what one particular change of basis does to a Gram. So a plane lattice's dual is the lattice itself, turned through a right angle and scaled, for every lattice with no exception. In three dimensions it is a condition, and the face-centred and body-centred cubic lattices are duals of each other rather than of themselves.

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The classification

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

One group refuses two colours and three refuse three. The two counts side by side, with the rows that refuse a number of colours marked. p3 is the only group with no two-colouring; p4, p4m and p4g are the only ones with no three-colouring. Neither list is a subset of the other and both come from the same arithmetic — a rotation order that divides nothing the symmetric group has.

What a half-turn does to three colours

Ten of the seventeen plane groups have no three-colouring, because a half-turn cannot permute three colours cyclically — that is the first rung of this ladder and it is true. Drop the word cyclically and the answer changes completely: a half-turn permutes three colours perfectly well by swapping two and fixing one, and only the three four-fold groups refuse three colours at all.

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What symmetry decides

Thirty-two classes, enumerated rather than listed — and then the one thing they are for outside crystallography's own bookkeeping: which components of a physical property a crystal is permitted to have before anybody measures it.

12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted.

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

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Lattices

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

Perfection is a rank, and most lattices do not reach it. For each lattice, the rank of the matrices vvᵀ built from its shortest vectors, against the dimension of the space of symmetric matrices those live in. Reaching it means the shortest vectors pin the form down completely: no deformation keeps every one of them at its length. Falling short means there is a direction left to move in, and the lattice is not a local maximum of density.

One perfect form in space

Which lattice packs spheres most densely is a question about a maximum over a continuum, and Voronoi turned it into a rank calculation and a sign check. A lattice is a local maximum exactly when its shortest vectors pin its shape down completely and its inverse can be written over them with positive coefficients. Searching every reduced integer form of minimum two finds one such lattice in the plane and one in space.

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