Seventy-five ways to be a thread

A layer is not a wallpaper counts the symmetry groups of an object that repeats in two directions and lives in three: a monolayer, a surface, a slab, a sheet. There are eighty of them, and they are not the seventeen.

The same question with one direction of repeat has its own answer. A chain of atoms, a helix of protein, a nanotube, a fibre, a single column of a crystal — each repeats along a line and is finite across it, and its symmetry group is a rod group. There are seventy-five.

Both numbers were quoted here rather than derived. The layer-group essay says so in as many words — eighty and seventy-five are “numbers from the literature, as two hundred and thirty is”. This essay removes the second of them from that sentence.

The two questions are not the same question with a number changed. A layer group’s operations either keep the layer’s two faces where they are or exchange them, which is one bit per operation and makes the count a colour-symmetry problem. A rod group’s operations either preserve the direction of the axis or reverse it, which is also one bit per operation — and the bit does something completely different, because the axis carries the translations and the layer’s faces do not.

What a rod group is made of #

The translations of a rod group are a single direction. Put it along z and call it the axis.

Every operation must carry the axis onto itself — otherwise it would carry the repeat somewhere else — so its linear part sends z to plus or minus z. The linear parts are therefore drawn from the crystal classes that single out a direction at all, and this collection already counts those: twenty-seven of the thirty-two, the five cubic ones failing because a class with four three-fold axes has no direction to single out.

That is the point-group half. The other half is the translations.

Why the count is not twenty-seven #

The natural first guess is that a rod group is an axial point group with a repeat attached, so that there are twenty-seven of them. That is wrong for exactly the reason there are two hundred and thirty space groups rather than seventy-three: an operation’s translation part splits into a piece that belongs to the operation and a piece that only records where somebody put the origin, and the first piece is a real distinction.

On a rod the split is unusually clean, because the lattice has one direction and the axis is that direction. An operation either preserves the axis’s direction or reverses it, and those two cases behave in opposite ways under a change of origin. That single dichotomy accounts for the whole difference between twenty-seven and seventy-five, and it can be stated in one line, which the three-dimensional version cannot.

Three facts make the search small #

Only the component along the axis matters. An operation’s translation could in principle point anywhere. A component across the axis is either removed by moving the origin off the axis, or would make the operation’s square a translation across the axis — and a one-dimensional lattice has no such translation. So every translation part is a fraction of the repeat, along the repeat.

The fraction has a denominator. Apply an operation of order n that many times and the result must be a lattice translation, so its translation part is k/n of the repeat for some whole k. A four-fold axis admits 0, ¼, ½ and ¾ and nothing else.

And an origin shift removes some of them. Move the origin along the axis by s, and an operation’s translation changes by (M − I)s — which is zero for an operation preserving the axis’s direction and −2s for one reversing it. So a screw’s pitch is real and no shift touches it, while a two-fold across the axis, the mirror across it, and an inversion all have their translations shifted away.

That single asymmetry is where the count comes from. Twenty-seven classes would give twenty-seven groups if translations added nothing; the screws along the axis are what they add, and the operations across it are what they do not.

A worked case makes the three facts concrete. Class 4 is a four-fold axis and nothing else. Its translation parts along the axis must be k/4 of the repeat, so there are four candidates; none of the four operations reverses the axis, so no origin shift touches any of them; and the four candidates are four rod groups. Class 4/m is the same axis with the mirror across it added. Now the mirror reverses the axis, and its own translation is shifted away. It also does something to the screws. Conjugating by the mirror carries a screw of pitch k to one of pitch −k, so a group containing 4₁ contains 4₃ on the same axis as well, and with them their quotient, a translation of half a repeat, which the lattice does not have. So 4₁ and 4₃ are not merged, they are excluded, and the plain axis and 4₂ remain. Four becomes two.

That is the whole mechanism, and the table below is it applied twenty-seven times.

The enumeration #

For each class: find every way it can sit on the rod, run every combination of translation parts on a grid of twelfths — fine enough for the halves, thirds, quarters and sixths an axis of order 1, 2, 3, 4 or 6 can need — keep the combinations whose closure has the class’s own order, and count two groups as one when a shift of the origin along the rod, or a turn about it, carries one onto the other.

Both halves of that sentence are load-bearing, and both were missing from a first version of this count that still came to seventy-five. A class can sit on a rod in more than one way, and a turn about the rod is a legitimate change of description, because nothing across a rod repeats and so nothing across it has to be kept on a lattice. Why leaving either out does not show in the total is the subject of a section below.

The pattern in the table is legible once the rule above is in hand. A class with nothing but its axis gives exactly n groups — one for each pitch — so class 4 gives four and class 6 gives six. A class with a centre usually gives fewer, because the centre reverses the axis and takes pitches with it: 6 gives six and 6/m two.

Usually, and twice not. Class 2 gives three and 2/m gives four; 222 gives two and mmm gives three. The centre comes with a mirror perpendicular to each two-fold, and when a two-fold lies across the rod its perpendicular mirror contains the rod, which means it can slide along the rod as a glide. The class without the centre has no operation that could carry that slide, so the centre creates a distinction instead of removing one. What the centre does to a count depends on where the class’s axes lie relative to the rod, not on the centre.

The seventy-five split into families in a way worth reading off the table. The five classes that are nothing but an axis contribute seventeen: 1 + 2 + 3 + 4 + 6 = sixteen with the axis along the rod, one per pitch, and one more from class 2 with its two-fold across the rod. The same five with a centre added contribute ten: 1̄ gives one, 2/m four, 3̄ one, 4/m two and 6/m two. And the remaining seventeen classes contribute forty-eight.

The centred row is worth pausing on, because no single rule about centres predicts it. Along a principal axis the centre removes pitches, and by amounts that are not monotone in the order: the three-fold’s three pitches — 0, ⅓ and ⅔ — all collapse to one group, while the four-fold’s four and the six-fold’s six each collapse to two. And in 2/m the centre does the opposite, lifting class 2’s three groups to four, because the mirror it brings can run along the rod as a glide. A count that has to be measured rather than predicted is the kind worth a figure, which is what the figure below is.

The largest contributors are classes 6 and 622, at six groups each. The classes carrying a centre and a principal axis contribute one or two each however large they are — 6/mmm has twenty-four operations and gives three rod groups, while class 6 has six operations and gives six.

Symmetry costs here too, and it is the same observation the packing argument makes about which plane groups a molecule can use: more operations mean more of them reverse the axis, and an operation that reverses the axis takes a distinction away.

One more feature of the table is worth naming because the obvious explanation of it is wrong. Class 6mm gives three rod groups and class 6 gives six, although mirrors containing the axis do not reverse it. A mirror containing the axis carries a screw of pitch k to one of pitch −k, and the natural reading is that it merges them. It cannot: a group holding both 6₁ and its mirror image 6₅ on one axis holds their quotient, a translation of two thirds of the repeat, and the repeat has no such translation. So the mirror does not merge the screws 6₁, 6₂, 6₄ and 6₅, it forbids them. What survives is the plain axis and 6₃, whose pitch of a half is its own mirror image, and the mirror adds one distinction back: it may be a true mirror or a glide sliding half a repeat, which is the difference between p6mm and p6cc. Three groups: p6mm, p6cc and p6₃mc.

Two errors that give the same seventy-five #

The count was first run with one orientation per class and origin shifts as the only identification. It came to seventy-five, the number the Tables record, and it was wrong in nine of its twenty-seven rows. How that happens is worth setting out, because a count agreeing with a book is the ordinary evidence that a count is right.

The first error loses six groups. A crystal class is a set of matrices, and nothing in a set of matrices says which direction somebody means to make periodic. Most classes settle the question themselves, because they single out one direction and the rod must be it. Four do not. A monoclinic two-fold can lie along the rod, where its translation part is a pitch and there are two groups, p112 and p112₁; or across the rod, where a shift of the origin removes any slide it carries and there is one, p211. A mirror can contain the rod, as a mirror or as a glide sliding along it (pm11, pc11), or cut it (p11m). 2/m has two groups each way, and mm2 has three with its two-fold along the rod (pmm2, pcc2, pmc2₁) and two with it across (p2mm, p2cm). Choosing one orientation per class, even the natural one that puts as many operations as possible on the rod, finds half of each of those classes. That is six groups.

The second error doubles six. Identifying groups by origin shifts alone treats pmc2₁ and pcm2₁ as different groups: in one the glide lies in the plane perpendicular to x, in the other perpendicular to y. They are the same group turned a quarter-turn about the rod, and a turn about the rod is as legitimate a change of description as a shift along it. Nothing across a rod repeats, so any turn is allowed, including the eighth of a turn that carries p4₂cm onto p4₂mc and the twelfth that carries p6₃cm onto p6₃mc, which on a hexagonal lattice would not be allowed. The Tables print each of these pairs as one entry with two symbols. There are six such pairs, in mm2, mmm, 4mm, 4/mmm, 6mm and 6/mmm.

Six missing and six doubled cancel exactly, and the total is seventy-five either way. The coincidence has no arithmetic behind it: the missing groups are all monoclinic or orthorhombic, the doubled ones mostly tetragonal and hexagonal, and they are unrelated in every respect except their number. Nothing downstream of the count noticed either. Every group the wrong count produced was a genuine rod group and closed correctly, and a census of which of them are chiral found thirty-one where there are thirty-two, because the one it lacked, p211, is chiral.

That is why the check against the Tables is now made row by row. Twenty-seven small numbers catch what one large number cannot, and they catch both errors separately: a missing role pulls its row down, a doubled pair pushes its row up, and neither can hide behind the other when the rows are compared one at a time.

Two more checks stand behind the rows. The first is arithmetic: an axis of order n laid along the rod must give exactly n groups, and it does for all five values. The second is how the Tables are used, which is the way a number from a book is used everywhere here: the enumeration runs first, the Tables are read afterwards, and a disagreement is a failure of the enumeration rather than a reason to adjust it. The first version of this count met that rule and still passed, because the comparison was with one number. That is the practical lesson of the section. A count of seventy-five that has been checked against a published seventy-five has been checked against one number, and two errors of equal size in opposite directions pass that check.

Where the rod groups sit #

There are four classifications in this family and they differ by two numbers each: how many directions the object repeats in, and how many dimensions it lives in.

• Seven friezes: one direction of repeat, two dimensions.
• Seventy-five rod groups: one direction, three dimensions.
• Seventeen wallpapers: two directions, two dimensions.
• Eighty layer groups: two directions, three dimensions.
• Two hundred and thirty space groups: three and three.

The jump from seven to seventy-five is where most of the interest is, and it is the same jump from seventeen to two hundred and thirty in miniature: the extra dimension brings screws, and screws multiply.

A helix is a rod group, which is the observation that makes the classification matter outside crystallography. A protein α-helix has about 3.6 residues per turn and is not crystallographic at all — its rotation is through an angle that no lattice permits — but a helix whose turn is a rational fraction is one of these seventy-five, and the ones with 2, 3, 4 and 6 residues per repeat are common enough to have names.

What a helix with an irrational turn has instead is a continuous group. Let the angle per residue be anything at all and the operations that carry the helix onto itself no longer form a discrete set: the classification stops, and what takes its place is one of the seven groups a uniform field can have — the turning cylinder, which has every rotation about the axis and no reflection containing it. That is the honest boundary of this rung. The seventy-five are what a lattice permits along one direction, and an object with no lattice along that direction is not badly classified by them; it is outside them.

And a rod group is also what a space group leaves along a line. Take a three-dimensional group, keep the one direction of repeat and discard the other two, and what survives is a rod group — the screws along the kept direction survive intact, since their translations are along it, while everything whose translation lay in the discarded directions loses it. Not every rod group arises this way, and the ones that do are the reason a column of a crystal has a symmetry worth naming separately from the crystal’s own.

The helices that are not on the list #

The classification’s most conspicuous application is to helices, and the helices biology is made of are not among the seventy-five. Saying why is the clearest statement of what “crystallographic” is doing in the phrase.

A rod group’s screw has a rotation of order 1, 2, 3, 4 or 6 and a pitch that is a rational fraction of the repeat. An α-helix has about 3.6 residues per turn, and a strand of B-form DNA about ten base pairs in a turn that is not a lattice repeat of anything. Neither number is one of the five, and the first is not even rational as measured.

So a protein helix has a symmetry group that is a rod group in the general sense — one direction of repeat, operations preserving the axis — and not one of the seventy-five. Its rotation is through an angle the lattice does not permit, and the object is periodic only in the sense that it repeats after however many turns brings the angle back to a multiple of a full turn, if it ever exactly does.

The consequence is an experimental one and it is why fibre diffraction looks the way it does. A helix’s transform is not a set of spots on a lattice; it is a set of layer lines whose amplitudes are Bessel functions, with the order of the Bessel function on each line fixed by the helix’s own turn — which is Cochran, Crick and Vand’s result of 1952 and the calculation Franklin’s photographs were read against. The cross of a helical diffraction pattern is the classification failing, in the specific sense that the object’s symmetry is not one a lattice can hold.

What the seventy-five do cover is the fibre once it is packed into a crystal, since the crystal’s own periodicity then forces the helix onto a rational pitch — which is why a helical molecule crystallises with a slightly distorted turn, and why the distortion is measurable.

Seventy-five against eighty, and where the difference comes from #

The two subperiodic counts are close and the reasons for their sizes are different, which is worth an accounting because the near-equality suggests a symmetry between them that does not exist.

Both draw their point groups from the same twenty-seven axial classes. What differs is what the translations offer.

A layer has two directions of repeat, so its lattice is one of the five plane lattices, and the extra count comes from lattice variety — a centred rectangular arrangement giving a group its primitive counterpart does not have. A rod has one direction of repeat, so there is one lattice and nothing to vary, and its extra count comes entirely from the screw pitches.

So the two numbers are near each other for unrelated reasons: eighty is twenty-seven classes plus the lattice types and the sides, and seventy-five is twenty-seven classes plus the pitches. A near-coincidence, and the arithmetic behind the two is not the same arithmetic at all — which is the sort of thing worth checking rather than assuming whenever two classifications come out at similar sizes.

What the seventy-five do not include #

A rod group is not a frieze with depth. The seven friezes are groups of motions of the plane, and a rod group is a group of motions of space whose translations happen to lie on a line — the same distinction the layer-group essay makes about wallpapers. A frieze has four kinds of operation to work with and a rod has ten, and the extra six are what turn seven into seventy-five. Reading a rod group as a frieze that has been thickened loses every screw in the classification.

They are the crystallographic ones. A rod may turn through any angle at all, and only the angles a lattice permits — halves, thirds, quarters and sixths of a turn — give one of the seventy-five. A helix with 3.6 residues per turn is a perfectly good object with a perfectly good symmetry group, and that group is not on this list; it is an incommensurate structure in one dimension, and needs two integers where the seventy-five need one.

They are groups of a rod, not of a crystal containing one. A fibre in a crystal has whatever symmetry the crystal’s space group leaves it, which is a rod group — but the crystal’s own group is not determined by the rod’s, and several space groups leave the same rod symmetry behind.

Nor do they say which rod a crystal contains. Every column of every crystal has a rod group, and reading it off the space group is a computation of the same kind as what a cleave leaves — the operations of the space group that carry a chosen line onto itself. What this essay counts is the list those answers are drawn from, not the answer for any particular line.

The seventy-five are counted up to a stated equivalence: two groups are one when a map that keeps the rod on itself and does not reverse handedness carries one onto the other. That covers shifts along the rod, turns about it and a half-turn across it, which reverses the rod’s direction and leaves a screw’s hand alone. It does not cover a mirror. Merge the eight enantiomorphic pairs as well, taking a left-handed and a right-handed screw of the same pitch as one, and the count is sixty-seven, in exactly the way two hundred and thirty becomes two hundred and nineteen. Seventy-five is stated with the pairs kept apart.

Nothing here is drawn from a table of rod groups. The twenty-seven classes come from the enumeration of the thirty-two; the translations come from a grid; the closure and the origin quotient are the same routines the space-group count uses. What the literature supplies is twenty-seven integers at the end, one per class, and their job is to be disagreed with if the enumeration is wrong. A single integer did not do that job.

And the count is of groups, not of shapes. Two chemically unrelated objects with the same rod group are one entry in this classification, exactly as two wallpapers with the same plane group are one of the seventeen. What the seventy-five count is the ways a thing can be symmetric along a line, and the things themselves are somebody else’s subject.