Eleven ways to turn while climbing

Two constraints, applied in order, and the answer is eleven.

The first constraint is the crystallographic restriction: a rotation that maps a lattice to itself is an integer matrix in the lattice basis, so its trace is an integer, and the trace of a rotation by 2π/n is 2cos(2π/n), which lies between −2 and 2. Five integers live in that range and each corresponds to exactly one order: 1, 2, 3, 4 and 6.

The second is the one the previous rung established: an n-fold screw applied n times gives a pure translation along its axis, that translation has to be a lattice vector, so the rise is m/n for some whole number m.

The sum #

m runs from 1 to n − 1. It cannot be 0, because that is the rotation. It cannot be n, because that is the rotation composed with a lattice translation, which is the same operation modulo the lattice. And every value between is a distinct operation, because two screws with different rises differ in a quantity no origin can change.

So each order of n contributes n − 1 screws, and the total is

(1 − 1) + (2 − 1) + (3 − 1) + (4 − 1) + (6 − 1) = 0 + 1 + 2 + 3 + 5 = 11

The one-fold axis contributes nothing, which is right: an operation that turns by nothing and climbs is a translation, and translations are already accounted for.

The sum is short enough that it can look like a trick, so it is worth saying what it depends on. It depends on the restriction, entirely. If seven-fold rotation were available a lattice would gain six more screws; if five-fold were, four more. The eleven is downstream of the five, and the five is downstream of the trace being an integer.

The same shape as two other counts on this site #

This enumeration has the structure of two others in this collection, and putting the three side by side says something about the subject that none of them says alone.

The seven friezes are counted by generating sixteen candidate subsets of four extra operations and closing each one, after which nine collapse. The seventeen wallpaper groups are counted by a longer version of the same argument, branch by branch on the rotation order. And here eleven screws are counted by a sum over the permitted orders.

What is common is that every one of them is a generate-then-eliminate argument: produce a finite set of candidates by a rule that is obviously exhaustive, then remove the ones that turn out not to be new. What differs is where the elimination happens. The friezes eliminate by closure — a candidate that generates more than it started with is not a new group. The wallpaper groups eliminate by classification. This one eliminates almost nothing: m runs from 1 to n − 1 and every value is a distinct screw, with no collapse at all.

That last point is the reason this is the easiest of the three counts and the least interesting on its own. The interest is entirely in the constraint — in the two lines that force the rise to be m/n and the order to be one of five — and once those are in place the counting is a sum a child could do. A subject where the hard part is stating the constraint correctly and the arithmetic is trivial is a subject in good shape.

Which of them are mirror images of which #

Wind a helix the other way and its rise goes from m/n to (n − m)/n. So the screws pair off: 3₁ with 3₂, 4₁ with 4₃, 6₁ with 6₅, 6₂ with 6₄.

A screw is its own mirror image when m = n − m, which happens when m is exactly half of n, which is possible only for even n. The three cases are 2₁, 4₂ and 6₃.

That pairing is the entire source of the eleven enantiomorphic space-group pairs. A space group that is one of a pair must contain a chiral screw, because that is the only kind of crystallographic operation with a handedness — a rotation has none, a mirror reverses one, and a glide reverses one. Eight of the eleven screws are chiral, and the groups built from them without anything improper to spoil the effect are the twenty-two groups that make 230 rather than 219.

The coincidence of the two elevens is worth naming as a coincidence. Eleven screws and eleven enantiomorphic pairs are different counts arrived at by different arguments, and neither causes the other. The screws are counted by the sum above; the pairs are counted by working out which space groups built on chiral screws survive the reflection as different groups, and eight chiral screws do not give eleven pairs by any arithmetic — several chiral screws appear in more than one pair, and several appear in groups that are not pairs at all.

Where the common factor matters #

The screws where m and n share a factor behave differently from the ones where they do not, and it is the difference between 4₁ and 4₂ that makes it visible.

Apply a 4₁ four times and the point is one cell up, having visited three intermediate positions none of which was on the axis. Apply a 4₂ twice and the point has turned half a turn and climbed a whole cell — a lattice translation — so after two applications it is already back on a translate of its start.

The rule is that the number of turns before a point lands on a lattice translate of itself is n/gcd(m, n). For the coprime cases — 2₁, 3₁, 3₂, 4₁, 4₃, 6₁, 6₅ — that is n, and the orbit under the screw alone has n points per cell. For 4₂ it is 2; for 6₂ and 6₄ it is 3; for 6₃ it is 2.

That has a consequence for what an axis carries. A 6₃ axis contains, besides itself, a threefold rotation — its square, which turns a third and climbs one whole cell, so the climb is a lattice vector and the rotation part is all that is left. And it contains a 2₁ — its cube. One line in space, three operations of three different orders, and the Tables draw one mark for it.

Which orders carry which screws #

Laying the eleven out by order rather than by rise gives a different reading of the same table, and it is the reading a crystallographer uses.

Two-fold: one screw. 2₁, and it is everywhere. It is the only screw available in the monoclinic and orthorhombic systems, which between them account for the majority of published structures, so 2₁ is by a very long way the commonest screw in the literature.

Three-fold: two screws. 3₁ and 3₂, a chiral pair, available in the trigonal and hexagonal systems. These are quartz’s, and they are the reason quartz comes in two forms.

Four-fold: three screws. 4₁ and 4₃ chiral, 4₂ self-paired. Tetragonal only.

Six-fold: five screws. 6₁ with 6₅ and 6₂ with 6₄ chiral, 6₃ self-paired. Hexagonal only, and the largest single contribution to the eleven.

The distribution is uneven for a reason with nothing to do with symmetry: the systems that permit the most screws are the ones fewest crystals adopt. Hexagonal offers five and is comparatively rare; monoclinic offers one and is the most populated system there is. So the available screws and the encountered screws are almost inversely distributed, which is a fact about molecules rather than about groups.

Why the notation puts the number where it does #

4₁ and 4₃ differ in one character and it is the only character that could differ. The order is the size of the polygon in the diagram and the subscript is the numerator of the rise: everything else about the two operations is identical.

The convention has one wrinkle worth knowing. There is no 4₀; a rise of zero is a rotation and is written as a bare 4. And there is no 4₄, because a rise of a whole cell is a rotation composed with a lattice translation and is the same operation. So the subscripts run 1 to n − 1 and the plain symbol covers zero, which is exactly the enumeration above written as notation.

The three that a lattice cannot have #

The enumeration is a statement about what survives, and it is worth naming what does not, because the absences are the interesting part.

No 5₁, 5₂, 5₃ or 5₄. The restriction kills the fivefold rotation and takes its screws with it. That is the same argument that makes fivefold symmetry impossible in the plane, and it is dimension-independent — the three-dimensional version gives the same five orders as the two-dimensional one, which is not obvious and is the reason that essay exists.

No 7-fold, 8-fold or anything above 6. Same argument, same source.

No irrational rise. A screw whose climb is not a rational fraction of the cell never lands on a lattice point, so applying it repeatedly generates infinitely many distinct heights and the structure is not periodic along the axis. That is what a helix in a molecule does — the α-helix’s 3.6 residues per turn is not a fraction with a small denominator — and it is exactly why a protein’s own helix is never a crystallographic screw of the crystal it sits in.

The last one is the most interesting absence, because the object it excludes is common and physically important. Molecular helices with non-crystallographic screws are everywhere in biology, and the way a crystal accommodates them is to have a different symmetry from its contents: the molecule’s tenfold screw is not a symmetry of the crystal, and the crystal’s symmetry relates whole molecules to each other rather than parts of one molecule to other parts. That distinction is easy to blur and worth keeping sharp, because a structure paper’s space group describes the packing and says nothing at all about the helix inside.

What the eleven look like in a real group #

Screws rarely arrive alone, and seeing where they sit in a group is what turns the enumeration into something usable.

Reading a diagram like that against the enumeration is the exercise this rung is for. The 6₃ appears in the list above as one of the eleven; its square appears as a plain threefold, which is not in the list because it is a rotation; its cube appears as a 2₁, which is in the list. So one entry of the eleven has produced marks of three different orders in one cell.

That is why a group’s symbol names one operation per direction and not all of them. P6₃/mmc names the 6₃ because it generates the others, and a symbol listing every axis in the cell would be unreadable and would say nothing extra.

What the count is not #

Two things this eleven does not do.

It is not a count of space groups. Eleven is the number of screw operations available; the number of space groups containing at least one screw is much larger, because a group can contain several and the same screw appears in many groups. The relationship between the two counts runs through the whole classification and is not a multiplication.

It says nothing about how often each appears. 2₁ is by a wide margin the commonest screw in real structures, because it is the one available in the monoclinic and orthorhombic systems that most molecules crystallise in. 6₂ and 6₄ are rare. That distribution is about what molecules are shaped like and how they pack, and no amount of symmetry arithmetic reaches it.

Who wrote them down #

The screw axis as a crystallographic operation arrives with Sohncke in 1879, and it arrives as a correction.

Before him, the accepted list of ways a periodic structure could be symmetric was the list of point symmetries applied at every lattice point — sixty-five arrangements, which Sohncke enumerated, and which are the sixty-five chiral space groups that still carry his name. What Sohncke’s list did not include was any operation that reverses handedness, because his concern was with arrangements of identical points and he took those to be superposable.

Adding the improper operations is what took the count to 230, and it was done by Fedorov and Schoenflies within a couple of years of each other. But the screws were Sohncke’s, and the observation that a rotation could be combined with a translation along its own axis and still be a symmetry of something periodic is the whole reason his list had sixty-five entries rather than the eleven it would have had otherwise.

The naming came later and from a different direction. The subscript notation is Hermann’s and Mauguin’s, from the 1930s, and it was chosen precisely so that the rise appears in the symbol as a number rather than as a name — which is why 4₁ and 4₃ are legible as related and Sohncke's tetragonal enantiomorphic pair is not.

The glide planes are the other family of operations with an intrinsic translation, and their enumeration has the same shape and one extra case — a fifth glide that only exists in a centred lattice, for a reason the enumeration gives rather than states.

What each of the eleven does to a diffraction pattern #

A screw axis has no fixed point, so nothing on a crystal’s outside reveals it and no measurement of the crystal’s shape can find it. It is nevertheless one of the easiest symmetries to detect, because it removes reflections from the diffraction pattern in a pattern of its own.

The rule follows from the turn count established above. A screw $n_m$ along an axis carries a point to a lattice translate of itself after $n/\gcd(m,n)$ turns, and each turn has raised it by $m/n$ of a cell. The reflections along that axis therefore survive only when their index is a multiple of $n/\gcd(m,n)$, and every other one is systematically absent — not weak, not sometimes weak, but zero for every crystal in that group whatever it is made of. The absences a symmetry produces is where the argument is set out; what matters here is that each of the eleven has its own signature.

Reading the eleven through that rule sorts them by how much they give away. A $2_1$ leaves every second reflection along its axis and removes the odd ones — the commonest absence in structural crystallography, and the one a beginner learns first. A $4_1$ leaves every fourth. A $4_2$, whose indices share a factor, leaves every second, so it announces itself exactly as a $2_1$ does along that direction. A $6_3$ likewise reduces to the halving case.

Two screws with the same $n/\gcd(m,n)$ are indistinguishable by absences, and that is not a defect of the technique. The pairs it cannot separate are precisely the mirror-image pairs of the previous section: $4_1$ and $4_3$ both leave every fourth reflection, $3_1$ and $3_2$ both leave every third, $6_1$ and $6_5$ both leave every sixth. A diffraction pattern that has been reduced to its intensities has already lost handedness — that is Friedel’s law — and the absences inherit the loss.

So the eleven fall into seven signatures. Four screws are self-paired and unambiguous, and the other seven form the three enantiomorphic pairs and are read as pairs. Deciding which member of a pair is present is exactly the experiment that decides absolute configuration, and it needs the anomalous signal rather than the absences.

The ones that occur, and where #

A count of possibilities says nothing about frequencies, and the frequencies here are worth knowing because they are lopsided in a way that has a structural reason rather than a statistical one.

The $2_1$ is overwhelmingly the commonest, and the reason is that it is the only screw compatible with the lowest symmetry a chiral molecule can crystallise in beyond triclinic. A molecule with a handedness cannot sit in a group containing a mirror, a glide or an inversion, since those operations would produce its opposite hand and the crystal contains only one. What remains is rotations and screws — and among those the monoclinic group with one $2_1$ axis accounts for a very large share of all molecular structures ever solved, because it is the least demanding way for an object with no symmetry of its own to pack.

The threefold screws are where the classic mineral example sits. Quartz crystallises in $P3_121$ or $P3_221$ depending on which way its helices of linked tetrahedra wind, and the two are the same structure reflected. The mineral occurs in both, sometimes as twins of one crystal, and the handedness is visible in the crystal’s outward form and in its rotation of polarised light — a rare case where an operation with no fixed point reaches the outside after all, by way of the property rather than the shape.

The higher screws demand a helix that closes. A $6_1$ requires six units per turn arriving back where they started one cell up, which is a strong constraint on what can be built and is met most often by chain structures — a polymer, a helical biological fibre, a metal in one of its hexagonal packings. Structures of this kind are why crystallographers of fibres think in screws before they think in point groups.

And the arithmetic of the previous section is what a model builder uses. The number of units per repeat is $n/\gcd(m,n)$ and the rise per unit is $m/n$ of the cell, so a helix with a measured rise and a measured number of units per turn names its own screw immediately. The eleven are not a table to be consulted; they are the complete list of ways a chain can be periodic and turning at the same time, and every fibre pattern ever indexed found one of them.