Crystallography & Symmetry
patterns derived from their groups
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The theme: The same arithmetic, renamed — page 5

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.
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. Lattices

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.

 How many similar sublattices the cubic lattice has at each scale. Every integer matrix satisfying MᵀM = α²I, counted up to the lattice's own point group by marking orbits rather than dividing. The even scales are drawn apart because they are the ones that give nothing new: a factor of two in the scale never produces a shape the smaller scale did not already have. Lattices

The shapes a lattice in space can thin to

In the plane, which indices admit a sublattice of the same shape is a question about which integers a quadratic form represents, and Fermat answered it. In space the question collapses: taking determinants shows the index is always a perfect cube, so there is nothing to represent. What is left is how many there are at each cube — and for a hexagonal lattice, whether there are any at all depends on one number.

 Which indices a screw axis contains itself at. The fifteen kinds of axis a space group may have, against the index of the sublattice taken along the axis. A filled cell is an index at which the axis contains a copy of its own kind; the darker cells are the indices at which what comes back is the mirror image instead. A pure rotation axis is filled everywhere and a screw is not, and which indices a screw loses is decided by one congruence rather than by any geometry. Into space

A screw that contains its own mirror image

No operation of a crystal turns a right-handed screw axis into a left-handed one — that is what makes the eleven enantiomorphic pairs pairs. And yet a 4₁ axis contains copies of 4₃ as subgroups, at every index congruent to three modulo four. One congruence decides both which indices are possible and which hand comes back, and it is the same congruence for all fifteen kinds of axis.

 Both sides of the transformation, on five lattices. A Gaussian of width set by t on every point of a lattice, summed; and the same sum over the dual lattice with the width inverted and the covolume divided out. The two agree to the last bit a double carries, at every t and on lattices with no symmetry in them, so nothing here is a coincidence of parameters. The identity is exact and the reason to have it is that the two sides do not cost the same. Lattices

The sum that turns a lattice into its dual

Put a Gaussian on every point of a lattice and add them up. The answer equals the same sum over the dual lattice with the width inverted and the covolume divided out — exactly, to the last bit a double carries, on lattices with no symmetry in them. The identity is free and the reason to have it is that the two sides do not cost the same: at one end of the range the direct sum needs forty thousand terms and the dual sum needs a hundred and twenty-five.

 One curve falls and the other does not. The boundary's share of a ball, against the radius, for a plane group and for the free group on two generators. The plane group's falls like one over the radius and goes to zero; the free group's rises to two thirds and stays. A group with no sequence of regions whose boundary becomes negligible has no shape-independent average, and that is not a difficulty in the analysis — it is a property of the group. Operations

The boundary a growing region forgets

Quoting a density assumes the region it was averaged over does not matter, and that assumption is a property of the group of translations rather than of the crystal. A ball in a plane group grows like R² and its boundary like R, so the edge becomes negligible — and where that fails, the average genuinely moves. The free group on two generators keeps two thirds of itself on the boundary forever, and a slab seven layers deep is wrong by exactly one seventh however wide it is made.

 Cubic means the Sylow 3-subgroup is not normal. The thirty-two sorted two ways at once: by crystal system and by whether the Sylow 3-subgroup is normal. Two of the four boxes are empty, so the two properties coincide exactly. That gives 'cubic' a definition with no geometry in it — a class is cubic when its threefold subgroups are conjugate to each other rather than unique — and it explains why a cubic class has no principal axis: a group cannot single out one member of a conjugate family. What symmetry decides

How many axes there are is a Sylow count

Sylow's theorems say that the subgroups of prime-power order in a finite group are all conjugate and that how many there are is congruent to one modulo the prime. Applied to the thirty-two crystal classes that arithmetic counts axes: the number of Sylow 3-subgroups is the number of equivalent threefold directions, it is four in exactly five classes, and those five are the cubic ones. So cubic has a definition with no geometry in it.

 Three lattices no congruence can separate. The three reduced forms of discriminant minus twenty-three, with the integers each represents. The principal form represents one and the others do not; the others represent two and it does not. So they are genuinely different lattices — and they represent exactly the same residues modulo twenty-three, so they are in one genus and no congruence condition of any kind distinguishes them. Lattices

Lattices that agree at every prime

Counting the plane lattices with a given metric determinant is a class number. Above it sits a coarser count — the genus, which is what congruences can see — and for most small determinants the two agree. At discriminant minus twenty-three they part: three lattices representing exactly the same residues modulo everything, and different integers. No argument modulo any number can tell them apart, and they are not the same lattice.

 The same accounting, at every coordination number. One row per number of edges at a vertex. The bill a sphere charges is 2dχ; the face worth nothing is 2d/(d − 2), which is a whole number at three, four and six and is 10/3 at five; the faces that can pay are those with fewer sides than that; and the last column is every way of paying the whole bill with faces of a single size. At three edges a vertex there are three such ways and twelve pentagons is one of them. At six there are none, which is the statement that six-fold coordination belongs to the plane and to no closed surface at all. What a lattice forbids

The twelve belongs to the vertex

Twelve pentagons is read as a fact about closing a surface. It is not: it is a fact about three edges meeting at a point. Let four edges meet instead and the sphere charges eight triangles; let five meet and it charges twenty; let six meet and it cannot be paid at all.

 Every closed surface, and the two that charge nothing. The same accounting indexed by Euler characteristic rather than by genus. An orientable surface has χ = 2 − 2g, so it only ever occupies an even row; a non-orientable one has χ = 2 − k and occupies every row from one downwards. The odd rows therefore belong to surfaces that cannot be oriented and to nothing else — and the first of them, the projective plane, charges six. Six pentagons is a bill no orientable surface presents. What a lattice forbids

The surfaces a count by genus skips

A count indexed by genus steps in twelves and lands only on even numbers. A closed surface can have any characteristic at or below two, and the odd ones belong to the surfaces that cannot be oriented — where the projective plane charges six pentagons, a bill no orientable surface ever presents.

 19 site symmetries, and the counts each can impose. Every distinct site symmetry across the space groups this site builds, named by the multiset of its operation types, with the orientation counts a disordered molecule there may take. The counts are the indices of the site group's subgroups, computed by closing subsets under multiplication rather than looked up. Nearly every row offers every divisor of its order. One does not: a site of order twelve whose group is the tetrahedral rotation group refuses an orientation count of two, because that group has no subgroup of order six. What a lattice forbids

How many orientations a disorder needs

A molecule at a site with more symmetry than it has resolves the contradiction by occupying several orientations at once. How many is not fitted: it is the index of the molecule's symmetry in the site's, so the occupancy is the reciprocal of a whole number — and one site in the census refuses a divisor of its own order.

 Two kinds of atom, and more ambiguity rather than less. Exhaustive searches on rings of four sizes. The third column counts homometric groups when every atom is identical; the fourth counts them when each atom may be one of two kinds. The fourth is larger at every size, and at nine and ten sites the third is nothing at all — there is no pair of arrangements of four identical atoms that a diffraction experiment cannot separate, and there are six and four once the atoms may differ. Distinguishing the atoms adds information to the structure and adds ambiguity to the measurement. How it is known

When the atoms are not all the same

Every homometric pair found so far is a pair of point sets, where an atom is a point and counts once. Give the atoms different scattering powers and the ambiguity does not go away — it grows. On a ring of nine there is no pair of four identical atoms that diffraction cannot separate, and there are six once two kinds of atom are allowed.

 Four groups whose description count a metric can raise. Every plane group, the number of ways of writing one arrangement down on the lattice the group requires, and the number on the most symmetric lattice it may sit on. Eight groups already occupy the most symmetric lattice available to them and have nowhere to go. Four have a metric that raises the count, by two and in one case by six. The starred rows belong to groups whose normaliser has a free direction, where the quantity is a count of grid points rather than an index and cannot be compared. Operations

The normaliser is not a function of the group

How many ways there are of writing one structure down is computed from the group and printed in a table beside its name. It is not a property of the group. Draw a p2 pattern on a hexagonal cell rather than an oblique one and the number goes from four to twenty-four, with nothing done to the group at all.

 Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds. What a lattice forbids

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

 The seven friezes rolled into cylinders are the seven axial families. Each of the seven frieze groups drawn on a strip 3 cells long, beside the same strip rolled into a cylinder so that its ends meet. A translation by one cell becomes a rotation by a 3th of a turn about the axis, a mirror across the strip a mirror containing the axis, the centre line a mirror perpendicular to it, a half-turn in the strip a half-turn about a horizontal axis, and a glide a rotation by half a cell's angle combined with that perpendicular mirror. Each cylinder's symmetry group was built from the rolled strip and again from the family's own generators, and the two agree. At n = 3 the orders are 3, 6, 6, 6, 6, 12, 12, and the last column names the crystal class each member is, coloured by whether it is proper, contains the centre, or is neither. What a lattice forbids

Seven friezes round a cylinder

A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.

 Where the sphere and the projective plane have no net. The number of different closed nets with three bonds at every atom and faces that are pentagons and hexagons only. On the sphere, with twelve pentagons and k hexagons for k up to 12, every count has at least one net except k = 1. On the projective plane, with six pentagons and h hexagons, each count sits under the sphere count it lifts to, since every hexagon of a projective net becomes two on the sphere. The projective counts for h = 0 to 6 are 1, 0, 0, 1, 1, 3, 3, so the projective plane has no net at h = 1 or 2: two gaps where the sphere has one. Every sphere count was found by enumeration and agrees with the published one. What a lattice forbids

A gap the sphere does not have

A net of pentagons and hexagons on the projective plane must have six pentagons, and the count permits any number of hexagons. Not every number happens. Lifting each net to the sphere turns the question into one about which cages have a centre — and the answer leaves two gaps where the sphere has one.

 Disorder models against occupancies, site by site. Every distinct site symmetry in the space groups built here — 19 of them — with its order, its number of subgroups, the number of distinct disorder models, which are the subgroups up to conjugacy by the operations of the site symmetry, and the number of different occupancies those models can have. The last column is the largest number of models that share one occupancy. In all, 162 models share far fewer occupancies; the most crowded is 4/mmm, where 11 different models all give an occupancy of 1/4. At every site, the classes of operation a model keeps separate it from every other model with the same occupancy. What a lattice forbids

The occupancy does not name the disorder

A molecule disordered on a special position takes a number of orientations fixed by a group index, and its occupancy is the reciprocal. Many different disorders share one occupancy — eleven at a single kind of tetragonal site — and what separates them is which of the site's operations the molecule keeps, which the averaged structure records and the occupancy does not.

 One hat patch laid out as hats, as equilateral tiles and as turtles. A patch of 36 hats found by exact cover on the kite grid, 4 of them reflected and drawn in the second colour, laid out three times. Every edge keeps its direction; short edges and long edges are given their own lengths. At short 1 and long √3 the tiles are hats, at equal lengths they are the equilateral member of the family, and at short √3 and long 1 they are turtles. In all three the same tiles touch the same neighbours along the same edges, and each layout was checked to be a tiling: 0 gaps and 0 overlaps, 0 gaps and 0 overlaps, 0 gaps and 0 overlaps among 1500 sample points, and every interior vertex surrounded by a full turn. Order without repetition

The hat and the turtle are one tiling

The hat has short sides and long sides; the turtle has the same turns with the two lengths exchanged, and looks nothing like it. Take a patch of hats, keep every edge pointing the way it points, stretch the short edges and shrink the long ones, and the patch becomes a patch of turtles — every tile touching the same neighbours along the same edges.

 Seventeen plane groups, and one chiral sheet over each. The seventeen plane groups, whether each is chiral as a pattern in the plane, how many sheets can be built over it by giving each operation a sign on the sheet's normal — 63 in all — and which of those sheets is chiral in space. There is always exactly one. For the five groups chiral in the plane it is the sheet whose two faces differ and nothing turns it over. For the twelve achiral in the plane it is the sheet turned over by exactly the operations that reverse orientation in the plane, so that every mirror line becomes a half-turn axis lying in the sheet. Into space

Chiral in the plane is not chiral in the room

A pattern with mirrors all over it can be a sheet with a hand, and a pattern with no mirror can be a sheet without one. Whether a layer is chiral depends on what each of its operations does to the side of the sheet, and over every one of the seventeen plane groups exactly one sheet is chiral in space.

 Seventy-two positions and the sets they fall into. Every plane group with the number of its Wyckoff positions, and the number of sets those positions fall into when the positions a normaliser exchanges are counted once: on the cell the group requires, on the most symmetric cell it may sit on, and under every change of basis carrying the group onto itself. 72 positions become 53 sets on the required cells and 51 on the best ones, and the last column never goes lower. The two groups a special cell changes are pmm and cmm. Operations

The same site under two names

A structure report puts each atom on a Wyckoff position, and two correct reports of one crystal can name different positions. The positions that can trade places are exactly the ones the normaliser exchanges — and which those are depends on the cell as measured, not only on the group.

 Two turns and their undoing leave a slide. A turn g by 90° about the point c and a turn h by 60° about d. The marked point p is carried back 60° about d, back 90° about c, forward 60° about d and forward 90° about c, and does not return: it arrives displaced by a vector of length 2.371, which is 4·sin 45°·sin 30°·|c − d|. Two other points put through the same four motions move by the same vector, drawn beside them, because the commutator g h g⁻¹ h⁻¹ of two rotations of the plane is a translation — (I − A)(I − B)(c − d) exactly — whatever the angles and the centres. What a lattice forbids

What forces a lattice

Every enumeration here starts from a lattice of translations, and the lattice is usually taken as given. It need not be. A group of motions that is discrete, and leaves no point far from an orbit, has to contain one — in the plane by an argument four lines long, each line a picture, and in space by an inequality whose threshold turns out to be the six-fold rotation.

 The rectangle a (4, 2) tube is rolled from. A patch of honeycomb turned so that the rolling vector C = 4a₁ + 2a₂ lies along the page. C has length √28 ≈ 5.292; the shortest lattice vector perpendicular to it, T, has length 4.583; and the rectangle on the two holds 28 hexagons and 56 atoms. Rolling the rectangle so that its left and right edges meet makes one repeat of the tube. The two lines through the corner are the sheet's mirror directions nearest C: the zigzag direction along a₁ and the armchair direction thirty degrees from it. C makes an angle of 19.11° with the first and lies on neither. What a lattice forbids

The tube has a screw no lattice allows

Roll a honeycomb along one of its lattice vectors and the tube turns and climbs with a screw of order 14, 98 or 794 — orders the flat sheet could never have. The rolling keeps the sheet's translations and spends them on turns, and it keeps the sheet's mirrors only along two directions, which is why almost every carbon nanotube comes in two hands.

 Square ice scatters a pinch at the origin. The intensity scattered by the horizontal arrows of square ice, averaged over 100 configurations on a 32 by 32 torus, over the whole Brillouin zone with the origin at the centre; darker is more intense. Along the horizontal axis through the origin the intensity falls to zero — 2.9e-32 at the smallest wavevector — while along the vertical axis it stays near 1.75, so the two meet at the origin in a pinch. Order without repetition

The ice rule is a conservation law

Two arrows in and two out at every vertex is a statement that nothing flows in or out anywhere. That makes one half of the arrow field vanish identically, in every arrangement and not merely on average, and what is left scatters with a pinch at the origin: an intensity that approaches different values from different directions. Break the rule now and then and the pinch acquires a width Debye and Hückel predicted for a salt solution.

 What lies between a group and its copies at index 25. For each group, the lattices carried to themselves by its point group that contain a copy of the group at index 25, arranged by index, with lines for containment; each lattice is labelled by the Gaussian or Eisenstein integer that generates it. Copies are the bottom row, drawn large when nothing lies between them and the whole lattice. p4: 3 copies at index 25, 0 maximal, with invariant lattices between of index 5; p4m: 1 copies at index 25, 1 maximal, with invariant lattices between of index none. Into space

The primes a cell can grow by

A plane group contains copies of itself in bigger cells, and the International Tables list the ones that are maximal — the copies nothing else sits between. For p4 they come at 2, at 5, 13, 17 and 29 twice each, and at 9 and 49 once, and the list is the list of primes of the Gaussian integers. Put mirrors on the pattern and the copies at 5 and 13 vanish while 25 becomes maximal: a mirror cannot keep one factor of a prime without the other.

 Subgroups, the classes a group sorts them into, and the sets its normaliser does. For every plane group, the number of subgroups of index two and of index three, the number of conjugacy classes those fall into under the group's own operations, and the number of sets they fall into under its Euclidean normaliser. Over the seventeen there are 74 subgroups of index two in 74 classes and 56 sets, and 82 of index three in 36 classes and 32 sets. 9 of the thirty-four rows have fewer sets than classes, which is where the tables' "equivalent" entries come from. Counts of subgroups and of classes agree with an independent count from transitive actions on n points. Operations

Three of them, and they are equivalent

The subgroup tables print a count and sometimes a word beside it. Three subgroups of one type may be three copies the group itself shuffles, or three the group holds firmly apart and only a change of description exchanges. p3 has three copies of itself at index three, no operation of p3 moves any of them, and one shift by a third of a cell exchanges all three.

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