Crystallography & Symmetry
patterns derived from their groups
Concept

Rod group — where it appears

The symmetry group of an object periodic in one direction and finite across it: a chain, a helix, a nanotube, a column of a crystal. There are seventy-five, enumerated over the translation parts an axis admits and quotiented by shifts of the origin along it.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

Six classifications, and which are enumerated here. The families of symmetry groups by how many directions they repeat in and how many they live in. The thirty-two crystal classes, the seven friezes and the seventeen plane groups are each built from their own operations and counted. So are both subperiodic families: the seventy-five rod groups class by class, and the eighty layer groups as seventeen one-sided, seventeen mirrored and the forty-six two-colour classes of the plane group underneath. The two hundred and thirty space groups remain a number from the literature, marked as such wherever it appears: reaching it needs the translation extensions of seventy-three arithmetic classes in full.

A layer is not a wallpaper

A sheet repeats in two directions and lives in three, and its symmetry group is not one of the seventeen. There are eighty of them, the difference between one and another is a single sign per operation, and the arithmetic that supplies those signs is the arithmetic of a two-coloured pattern.

classification · Subperiodic 75 rod groups over 27 axial classes. Each axial crystal class, with the number of rod groups it carries: every consistent choice of translation along the axis, in every way the class can sit on the rod, with two groups counted as one when a shift of the origin along the rod or a turn about it carries one onto the other. The total is 75, and every row as well as the total agrees with the International Tables, which are compared with this enumeration rather than used to produce it.

Seventy-five ways to be a thread

Eighty layer groups and seventy-five rod groups are usually quoted, both as numbers from the literature. The second is derived here, class by class — and the total alone turned out to be no check at all, because two errors of six groups each give seventy-five as well.

classification · Subperiodic 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.

classification · Subperiodic 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.

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.

restriction · Finite groups Whether a rolled sheet ever comes back round. Three plane lattices, each with the same rolling vector C = 3a₁ + a₂ drawn from the origin and the line through the origin perpendicular to it. A translation of the rolled pattern straight up the tube, with no turn, is a lattice vector on that line. The square lattice has one, marked T, and the tube repeats every 10 turns. The general rectangular lattice has none in this direction — only along its cell edges — and the general oblique lattice has none in any direction at all, so its rolled pattern climbs forever without returning to the same angle.

Most sheets roll into a tube that never repeats

Rolling the honeycomb along a lattice vector always gives a tube with a repeat, and that is a property of the honeycomb rather than of rolling. Over the seventeen plane groups, 567 of 1,008 rolling directions give a tube with no translation along its axis at all — and every direction of an oblique pattern is one of them.

restriction · Finite groups A thread's two signs, and the four kinds of operation. Every operation of a rod group carries the axis to itself, so it does two independent things: it keeps or reverses the direction along the thread, by a sign σ, and it keeps or reverses the handedness of the plane across the thread, by the determinant of a 2 × 2 matrix. The determinant in space is the product, so the shaded cells are the proper operations — a turn or screw about the axis, and a half-turn crossing it — and the unshaded ones are the improper. A rod group is chiral when all of its operations sit on the shaded diagonal, and polar along its axis when all of them sit on the top row. The two conditions pick out different diagonals of the same square, which is why neither implies the other.

A thread's hand is not a choice

A sheet's handedness in space depends on a sign that the plane pattern does not fix, so one plane group carries several sheets and exactly one of them is chiral. A thread has no such freedom: 32 of the 75 rod groups are chiral, they sit over 9 of the 27 axial classes, and which they are is settled before any structure is drawn. Only its direction depends on how the class lies along it.

space-groups · Chirality Fifty-four of the seventy-five rod groups are a rolled plane pattern. Every rod group, one dot each, grouped by its crystal class. A dot is filled when some plane pattern rolled along some lattice vector has exactly that group, and the 106 crystallographic rollings of the seventeen plane groups fill 54 of them. The eighteen improper classes are full: every one of the 43 achiral rod groups is reached. The nine proper classes are not, and the twenty-one groups named on the right are what is missing — the sixteen whose screw is one of a left- and right-handed pair, and the five bare axes with no climb at all, p1, p112, p3, p4 and p6.

A rolled sheet is never one of a pair

Rolled up along every lattice vector that gives a crystallographic tube, the seventeen plane groups reach fifty-four of the seventy-five rod groups: every achiral one and eleven of the chiral. Not one of the sixteen screws that come in left- and right-handed pairs is among them, and the reason is a single fact about how far a rolled lattice can climb.

restriction · Finite groups 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.

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.

restriction · Finite groups 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.

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.

restriction · Finite groups 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.

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.

restriction · Finite groups One sheet with two faces rolled into two scrolls. The cross-section of a sheet with different atoms on its two faces, rolled three turns into an Archimedean spiral, once with face A outward and once with face B outward. Unlike a tube, a scroll has an inner edge and an outer edge and a radius that grows with every turn, so an isometry of space can neither turn it about its axis nor reverse the direction along the spiral: it can only slide it along the axis, reflect it across the axis, or reflect the spiral itself.

A scroll cannot be turned end for end

A sheet with two different faces rolls into two tubes, face A out and face B out, and for forty-six of the eighty layer groups the two tubes are one object whichever way the sheet is rolled. Roll the same sheets into scrolls, spirals with an inner edge and an outer one, and only seventeen keep that. A spiral cannot be turned about its axis or run backwards, so the half-turns and screws that joined the tubes have nothing to act on. What survives is a mirror lying in the sheet, and a two-fold axis lying along the roll.

restriction · Finite groups The seventy-five rod groups by what fixes the axis and what moves along it. Every rod group sorted by its kernel, the plane point group of operations that leave every point of the axis in place, and by the group of the line it maps onto: translations only, or translations and reversals. Each chip counts the rods with that kernel, that line group and a given number of steps to the repeat. 29 never reverse the axis and 46 do, seventy-five in all, derived from the kernel and the rule that joins it to the line and agreeing cell by cell with the rods counted class by class.

Seventy-five is twenty-nine and forty-six

The eighty layer groups came out as a plane group with a sign on each operation. The seventy-five rod groups have an account of the same shape turned round: a finite group of the plane across the rod, which is what leaves the axis alone, extended by a group of the line, which is what moves along it. Twenty-nine rods never turn their axis over and forty-six do, and the count comes out cell by cell as the rods counted class by class. The screw of a rod is the turn its smallest step gives the plane, and the kernel decides its subscript.

classification · Subperiodic The two forty-sixes over the ten plane point groups. The forty-six two-colour plane groups by the point group of the uncoloured group, and the forty-six rods that turn their axis over by the plane point group their operations show end on. The colourings pile onto the rectangular mirror groups — seventeen on 2mm, ten on m, none on 3 — and the rods onto the hexagonal and square groups, eleven on 6mm and nine on 4mm. Seventeen of the forty-six would have to change point group for the two lists to match.

Two forty-sixes that do not match

The layer groups contain forty-six two-coloured plane patterns, and the rod groups contain forty-six threads that turn their axis over. Both lists are lists of a group with a half of index two, and both halves have a point group in the plane, so they can be laid side by side. They do not line up on any invariant tried: the plane spends its forty-six on rectangular mirrors, which halve a lattice easily, and the rods spend theirs on high rotations, which carry the most kinds of screw. The equal totals are two sums that happen to meet.

classification · Subperiodic Where the four-fold screws of one crystal cross a plane. A space group of the body-centred class I4/m that has a centre of symmetry, seen along the direction of its four-fold axes: the points where the special lines along that direction cross a plane, over four cells of the projection. A rod cut along the large filled lines is p4₃, a four-fold screw of one hand; along the other large lines, p4₁, its mirror image. The crystal's centre of symmetry carries each kind onto the other, so both are there and neither hand is favoured.

A crystal with no hand holds rods of both hands

Rolling a flat pattern into a tube reaches fifty-four of the seventy-five rod groups and never a screw that comes in a pair of hands. Cutting a column out of a crystal reaches all seventy-five. Sixteen of them are handed screws, and half of those can be cut from a crystal that has a centre of symmetry, from a line that happens to be a screw of one hand, with its mirror image a short distance away. A handed thread does not need a handed crystal, only a handed place in one.

space-groups · Chirality

Named alongside it

The objects these essays reach for when they reach for this one.

Screw axisHelixChiralityEnantiomorphLayer groupCrystallographic restrictionEnumerationPlane groupPoint groupSubperiodic groupAxial classDeterminant

All concepts