A crystal with no hand holds rods of both hands

Every band group is a ribbon cut strips out of sheets and named every strip’s symmetry group. All thirty-one band groups turned out to be the group of some ribbon, and twenty-four of them only of a ribbon, never of a single edge. It ended on the other kind of strip. A band keeps a plane, but a rod keeps only a line, and its group is one of the seventy-five rod groups that seventy-five ways to be a thread derived class by class. A rod’s operations can turn it about its axis by any angle its line allows, so its group is not a matter of three signs as a band’s was. Cutting a rod out of a three-dimensional crystal is the question with the same structure. Which of the seventy-five are reached, how many only by cutting along a special axis, and how many of those are chiral?

There is a second reason to ask. A rolled sheet is never one of a pair made rods the other way, by rolling the seventeen plane groups into tubes. It found fifty-four of the seventy-five, every achiral one and eleven of the chiral, and not one of the sixteen screws that come in left- and right-handed pairs. The reason was a fact about how far a rolled lattice can climb per turn. A cut rod has no such restriction.

Cut from the 230 space groups, the rods reach all seventy-five, including every handed screw, and eight of the sixteen handed screws can be cut from a crystal that has no hand at all.

A rod, cut #

A rod cut from a crystal is a cylinder about a line $L$ along a lattice direction $d$. Its symmetry group is every operation of the crystal’s space group that carries $L$ onto itself. Such an operation must send $d$ to $d$ or to $-d$, since the line has to stay a line in the same direction. It must also, after adding a lattice translation, fix the point where $L$ crosses a plane at right angles to $d$. The operations that send $d$ to $\pm d$ act on that plane as a plane group, the projection of the crystal along $d$. A line in general position keeps only the operations that act on the plane trivially: translations along $d$, and any mirror at right angles to $d$, which keeps every line along $d$ where it is. A line through a special position of the projected plane group keeps more. So the rods a crystal can give along one direction are one for each special position of its projection, and one more for everything else.

The picture at the head of this essay is one such projection. The crystal belongs to the body-centred class I4/m, and its space group has a centre of symmetry, so it has no hand. Along its four-fold direction some special lines are four-fold screw axes that turn one way as they advance, and others are screw axes that turn the other way. A rod cut along one of the first kind is a four-fold screw of one hand, the rod group $p4_3$. A rod cut along one of the second is its mirror image, $p4_1$. The centre of symmetry carries each kind of line onto the other, so the crystal is not handed, but no single line in it is centrosymmetric.

The census #

The census runs over the 219 affine space-group types, each with its operations written on its primitive lattice. They are the 230 with each of the eleven mirror-image pairs counted once, the distinction two hundred and thirty or two hundred and nineteen draws, and the hands are restored below. For each, it takes every lattice direction whose coordinates are $-2$ to $2$, forty-nine up to sign, keeping only those along which some operation other than a translation sends $d$ to $\pm d$. For each such direction it finds every special position of the projection by solving for the fixed points of every operation, and for the mirror lines, taking a generic point on each. The operations keeping each line are collected and closed into a group.

The one delicate step is measuring slides along the rod. A lattice basis adapted to $d$ generally has its other two vectors at an angle to $d$, and a slide measured along the basis vector $d$ would then pick up a share of the displacement across the rod. The slides are therefore measured along $d$ itself, with a metric the crystal’s point group preserves. Where the rod has an operation that reverses its direction, the origin along the rod is put at the point that operation fixes, so that no irrational offset from the metric is left in any slide. Each rod’s group is then named by rodKey in the derivation of the seventy-five. That is the same canonical form the list was built with, so a cut rod and a derived rod are compared by one rule and not two.

Every rod the census cuts is one of the seventy-five, and every one of the seventy-five is cut. No rod fails to be named, which is the first test of the construction: a slide measured wrongly would give an operation that is not in any rod group, and the key would find nothing.

The picture sets the seventy-five out by their twenty-seven axial classes. Every cell is filled. The shading varies over three orders of magnitude, from the groups with translations alone or a single mirror, which almost every crystal gives in almost every direction, to the six-fold screws, which one crystal type each gives along one direction.

Special lines do the work #

A line in general position gives one of only two rod groups. Either it keeps translations and nothing else, $p1$, or the crystal has a mirror at right angles to the direction and the rod keeps that too. The other seventy-three rod groups need the line to be special: on a rotation or screw axis, in a mirror or glide plane containing the direction, or through an inversion centre or a two-fold axis crossing it. Seventy-three of seventy-five rods exist only because the cut was placed. That is the rod’s version of the ribbon’s finding that exchanging operations need a special centre line. A ribbon had two edges to place and a rod has one axis, but the axis has more ways to be special, since the operations fixing a line through space include every rotation about it.

The rod groups cut most often are the ones with an inversion centre on the axis, $p\bar1$, and the ones with a two-fold axis crossing the rod. Both are special in the weakest way, through a single point of the line, and a crystal with such points has them on many lines in many directions.

The handed rods #

Thirty-two of the seventy-five rod groups are chiral, the count a thread’s hand is not a choice made from the rods’ own operations, and sixteen of those are one of a mirror-image pair. A reflection in a plane containing the axis turns each into the other, and no motion of space does. They are the rods with a three-, four- or six-fold screw that climbs by something other than a half turn: $p3_1$ and $p3_2$, $p4_1$ and $p4_3$, $p4_122$ and $p4_322$, $p3_112$ and $p3_212$, and the four six-fold pairs, $p6_1$ and $p6_5$, $p6_2$ and $p6_4$, $p6_122$ and $p6_522$, and $p6_222$ and $p6_422$. Rolling never produces any of them.

Cutting produces all sixteen. That is expected in one sense: the crystal $P4_1$ has only $4_1$ screws along its four-fold direction, and a rod cut along one of them is $p4_1$. The surprise is where else they come from. Four of the eight pairs are also cut from crystals that have a centre of symmetry or a mirror, and so no hand of their own. The three-fold screws $p3_1$ and $p3_2$ come from rhombohedral groups of the classes $R\bar3$ and $R3m$ and from cubic groups of six classes, the four-fold screws $p4_1$ and $p4_3$ from body-centred tetragonal groups of the classes $I4/m$ and $I4mm$, the $p4_122$ pair from $I4/mmm$ and from body-centred and face-centred cubic groups, and the $p3_112$ pair from $R\bar3m$ and three cubic classes.

The mechanism is the one the picture at the head of this essay shows. A crystal with an improper operation contains, for every screw axis of one hand, a screw axis of the other, placed where the improper operation sends it. Each individual line is handed. The crystal is not, because it holds both hands in equal number. It is the reverse of quartz, which a hand made of pieces that have none described as a handed crystal built from unhanded tetrahedra. Here an unhanded crystal is built from handed columns. The diamond structure is the familiar case: its space group, in the face-centred cubic class $Fm\bar3m$, has four-fold screw axes of both hands along each cube edge, and a rod cut along one of them through the diamond lattice is a four-fold screw with two-fold axes across it. Diamond has no hand, and every such column cut from it does.

The six-fold screws are the exception. Every cut that gives one of the four six-fold pairs comes from a crystal that is itself handed, one of $P6_1$, $P6_5$, $P6_2$, $P6_4$, $P6_122$ and their partners. The reason is arithmetic, and it explains the other cases too. Suppose a crystal has a screw axis of one hand and, through an improper operation, a parallel screw axis of the other. Take an operation of each that turns by the same angle. One climbs by $k$ parts in $n$ of the axial repeat and the other by $n - k$, so the first followed by the inverse of the second is a pure translation that climbs by $2k/n$ of the repeat. It must be a lattice vector. So an unhanded crystal can hold a $3_1$ and $3_2$ pair only if some lattice vector climbs a third of the repeat along the axis, a $4_1$ and $4_3$ pair only if one climbs a half, and a six-fold pair only if one climbs a third or two thirds.

A rhombohedral lattice has lattice vectors that climb a third of its axis, and so does a cubic lattice along a body diagonal, since each cube edge projects onto the diagonal at a third of its length. A body-centred lattice has one that climbs half its four-fold axis. The hexagonal lattice, the only one with a six-fold axis, climbs its axis only in whole repeats. Three-fold and four-fold screws can hide in pairs inside an unhanded crystal, and six-fold screws cannot, because no lattice that allows a six-fold axis allows the translation a hidden pair would need. The census found exactly the classes this predicts: rhombohedral and cubic for the three-folds, body-centred tetragonal and cubic for the four-folds, and none at all for the six-folds.

What rolling misses and cutting reaches #

The twenty-one rod groups rolling cannot reach split into two kinds. The sixteen handed screws are missed because a rolled lattice climbs by nothing or by half a repeat per turn, and those are the two climbs a mirror leaves alone. The other five are $p1$, $p112$, $p3$, $p4$ and $p6$, bare axes whose screws do not climb at all. The earlier census found why rolling misses them: a tube with such a group needs a rolling direction of primitive index one, which only square and rectangular lattices have, and every plane group on those lattices brings a half-turn or a reflection to every such rolling. A crystal has no such obstacle. Cut along a pure rotation axis of a crystal whose other operations all miss that line, and the rod keeps the rotation and nothing else.

So the two ways of making a rod are not two routes to one list. Rolling produces the rods whose climb per turn the sheet’s own lattice fixes, and those rods are never handed. Cutting inherits whatever screw axes the crystal has, of either hand, and gets every rod group there is. The tube has a screw no lattice allows found rolled tubes with screws of order fourteen and ninety-eight, which no crystal has. Rolling, then, gives screws of orders a crystal cannot have and never a handed screw of an order a crystal can have. Cutting gives every crystallographic screw and nothing beyond.

What the census does not decide #

Only directions with coordinates up to two are cut. That covers every direction along which a crystal’s rotation and screw axes lie, since those are the axes of its point group and point along the crystal’s short lattice vectors, and it covers every rod group. A direction further out has fewer operations sending it to $\pm d$ and gives rods in the classes $1$, $\bar1$, $m$ and $2/m$ more often, which changes the counts of cuts but not the list.

The counts of cuts are counts of the census’s own enumeration. A crystal type, a direction and a special position make one cut, however many symmetry-equivalent lines that position stands for. So the shading in the census picture says how many kinds of place give a rod group, not how often a random line through a crystal does.

Hands are counted through the 219 affine types. A rod cut from one of the eleven enantiomorphic crystal types has a mirror image cut from the other hand of that type, and the census adds those mirror images rather than cutting the eleven again. The claim that a handed rod can come from an unhanded crystal does not depend on this, since those crystals are among the 219 as they stand.

And a rod is not a crystal. A column cut from diamond along a screw axis is handed as an ideal set of atoms, and whether a real column of finite width, with surfaces, shows that hand in any measurement is a question about the surfaces. Chiral in the plane is not chiral in the room found how much the setting of an object in space decides its hand. A rod is the most restricted setting there is.

Where the pieces come from #

The rod groups were enumerated in the 1930s and 1940s and are tabulated in the International Tables for Crystallography, which also list, for each space group, the site symmetries of its special positions. The rods reached by cutting are, in effect, the line symmetries of those positions extended along a direction, and the correspondence is standard in the theory of subperiodic groups. That diamond’s four-fold screw axes come in both hands is a textbook fact about $Fd\bar3m$. The census above, and the count of which handed rods need a handed crystal, are made on this page.

What the census must refuse #

The refused claim is the one the phrase “chiral crystal” invites. A rod with a handed screw is chiral, and it seems it should come from a chiral crystal. The census finds eight of the sixteen handed rod groups among the rods of crystals with a centre or a mirror. The test that a rod cut from a crystal with no improper operation has none is the check in the other direction. It cannot fail if the construction is right, since a rod’s operations are a subset of the crystal’s, and it would fail at once if the slide or the axis were measured wrongly and an operation appeared that the crystal does not have.

Still open: the rods of the crystals that pack chiral molecules #

A protein, or any single-handed molecule, can only crystallise in one of the sixty-five Sohncke groups, the space groups with no improper operation, for the reason the groups one hand may sit in gives. Every rod cut from such a crystal is chiral, and the census says which chiral rod groups each Sohncke group gives. The columns that such a crystal is made of, the stacks of molecules along a screw axis, are exactly these rods, and the rod group of a column constrains how the molecules in it can touch the next column. Which Sohncke groups give only rods with screw axes, which give rods with pure rotation axes that a single-handed molecule would have to sit on, and whether the groups proteins actually crystallise in are the ones whose columns are all screws, is a comparison the census makes possible and has not made.