Where the plane had places, space has pitches

Two mirrors a coset cannot tell apart asked when two symmetries of a plane group are the same symmetry, meaning that some operation of the group carries one onto the other. Modulo the lattice every plane group is finite and the answer is a small list. In the infinite group the list grows: p2’s half-turns fall into four classes, one at each kind of centre, where the quotient has one. The whole decision came down to a single lattice, $(I - B)\mathbb{Z}^2$, whose index is four for a half-turn, three for a third of a turn, two for a quarter-turn and one for a sixth. Those four numbers count the families of rotation centre that translations alone leave apart.

The essay ended by pointing at space. There a rotation fixes a line rather than a point. An operation with the same linear part can also slide along that line, and the slide is where screws come from. Every motion of space is a screw established that every proper motion is a rotation about some axis combined with a slide along it. This essay runs the plane’s decision on all 219 affine space-group types and every kind of rotation axis in them. The plane’s four numbers come back unchanged, in every lattice. What they count is different. Where the plane’s families were rotations in different places, a family in space can be a screw of a different pitch on a line that no translation reaches.

An axis, and the lattice it lives modulo #

Write an element of a space group as $(B, b)$, the motion $x \mapsto Bx + b$, in a primitive basis so that the lattice is $\mathbb{Z}^3$. When $B$ is a rotation through $2\pi/n$ about a line, the element is a rotation or a screw about some parallel line. Its slide along that line is the average of $b$ over the powers of $B$, a vector along the axis. The smallest lattice vector along the axis, $a$, gives the natural unit: the slide is $k/n$ of $a$ for a whole number $k$ between $0$ and $n-1$, and $k$ is the subscript in $2_1$, $3_1$, $4_2$ and the rest. Eleven screws and no others counted the possible values.

An element and the element followed by a translation along its own axis act on the same line and differ only in how far along it they reach. So the object worth counting is an axis: an element taken together with its translates along its own line, and with its inverse. For a fixed linear part $B$ every element is $(B, b_0 + \lambda)$ with $\lambda$ in $\mathbb{Z}^3$. Conjugating by a translation $t$ changes $b$ by $(I - B)t$, and moving along the axis changes it by a multiple of $a$. So the axes with linear part $B$, up to translations, are the classes of $b_0 + \mathbb{Z}^3$ modulo

$$L_B = (I - B)\,\mathbb{Z}^3 + \mathbb{Z}\,a .$$

In the plane $(I - B)$ was invertible and $(I - B)\mathbb{Z}^2$ was already a full lattice. In space $(I - B)$ has rank two and misses the axis direction entirely, so the $\mathbb{Z}a$ term is what makes $L_B$ full. The rest of the group then acts on these classes. An operation $(R, r)$ with $RBR^{-1} = B$ sends the axis of $b$ to the axis of $Rb + (I - B)r$. One with $RBR^{-1} = B^{-1}$ sends it to an element of linear part $B^{-1}$, whose inverse is the same axis again, read with $B$. The orbits of that action are the classes of axes of that kind in the group. In the quotient by the lattice there is exactly one element with linear part $B$, so there is exactly one class.

The plane’s numbers, in every lattice #

The index of $L_B$ in $\mathbb{Z}^3$ is the number of axis families that translations alone keep apart. It turns out to be the plane’s number in every lattice, and the reason is short. Dividing $\mathbb{Z}^3$ by the line $\mathbb{Z}a$ leaves a two-dimensional lattice, and $B$ acts on it as a rotation of the plane through the same angle. The index of $L_B$ is the index of $(I - B)$ on that plane lattice, which is $2 - 2\cos\theta$, the number the crystallographic restriction reads off a trace. The census confirms it on every entry: 4 for every two-fold axis, 3 for every three-fold, 2 for every four-fold, 1 for every six-fold, in primitive, side-centred, body-centred, face-centred and rhombohedral lattices alike.

So the count does not change from lattice to lattice. What changes is what a family is, and the figure reads it off. In a primitive lattice the four families of a two-fold axis along a cell edge are four parallel lines of the same kind, all rotations or all $2_1$ screws, exactly as in the plane. In a centred lattice the same four families are two rotations and two $2_1$ screws. All seventy body-centred entries and all twenty-eight face-centred ones are like that, with no exception.

The mechanism is a lattice vector that is not along the axis but has a fractional component along it. Composing a rotation with such a translation keeps the linear part and moves the line, and it also adds that fraction to the slide. In a body-centred lattice the vector $(\tfrac12, \tfrac12, \tfrac12)$ has half a period along every cell edge, so every two-fold coset holds both $2$ and $2_1$. Every four-fold coset holds either $4$ and $4_2$ or $4_1$ and $4_3$, and every three-fold coset along a cube diagonal holds $3$, $3_1$ and $3_2$. A primitive lattice does the same along a direction that is not a cell edge. The two-folds along a face diagonal of a tetragonal cell pick up half a period from $(1, 0, 0)$, which accounts for the sixty-one primitive two-fold entries holding both kinds. The three-folds along a body diagonal of a primitive cube pick up a third of a period from each cell edge.

A cube seen down its diagonal #

That last case is the clearest place to see a family become a pitch.

Down the body diagonal of a primitive cube the lattice projects to a triangular net. The index of $L_B$ is three, so the three-fold axes parallel to that diagonal fall into three families under translation. In the plane group p3 the three families would be three kinds of rotation centre. Here they are one rotation and two screws. The rotation passes through lattice points. The other two lines pass through none, and a rotation about either one, composed with the cell-edge translation that carries the lattice back onto itself, climbs a third of the diagonal per turn, one way or the other. They are the same linear part, in the same coset of the lattice, and their pitches differ.

In P23, whose operations are all proper, the three families are three classes. The rotation cannot become a screw and the two screws cannot become each other. In Pm3̄ the rotation is still its own class, but the $3_1$ and the $3_2$ are one class: the inversion carries a right-hand screw about one line onto a left-hand screw about another. The figure colours them accordingly. Both groups are made from the same lattice and the same three-fold axes, and they differ by one operation that reverses handedness.

A body-centred four-fold, worked #

The body-centred tetragonal groups show every case at once, because their four-fold cosets hold two pitches and their two-fold cosets two more.

In I4, the symmorphic group, the quarter-turns about the cell’s vertical edges are rotations. The centring translation composed with one of them gives a $4_2$ screw about the line through the middle of each face. The index for a quarter-turn is two, so there are exactly two families, and they are the rotation and the $4_2$: two classes, of different pitch. The half-turns, with index four, fall into four families under translation. Two are rotation axes that nothing in I4 exchanges, and two are $2_1$ screws that the quarter-turn carries onto each other, so they make three classes. In the quotient all of this is one element with linear part a quarter-turn and one with linear part a half-turn.

In I4₁ the same two four-fold families are a $4_1$ and a $4_3$, a right-hand and a left-hand screw of the same pitch. I4₁ has no improper operation, so they stay two classes. Its half-turns come out the other way round from I4’s: the two families of $2_1$ are the ones nothing exchanges, and the two rotation families are joined. In I4₁/a, which adds an inversion to I4₁, the $4_1$ and the $4_3$ become one class, and the half-turns fall into two classes of two. The same lattice and the same four-fold linear part give two, two and one classes of four-fold axis in these three groups, and the reason in each case is a single operation that the group has or lacks.

Why a pitch is never changed and a hand only by a reflection #

The whole pattern follows from what conjugation does to a slide. Conjugating $(B, b)$ by $(R, r)$ gives an element whose slide is $R$ applied to the original slide. $R$ is an isometry and maps the axis direction to itself or to its reverse, so the slide keeps its length. The pitch of a screw is a conjugacy invariant. A rotation can never be conjugate to a screw, nor a $4_2$ to a $4_1$, however the group is built, which is the refusal at the end of this essay.

The hand is subtler. A screw’s hand is the sign relating its sense of turning to its direction of travel. A proper $R$ preserves that relation, whether it keeps the axis direction or reverses it. An improper $R$ reverses it: a mirror perpendicular to the axis keeps the turning and reverses the travel, and a mirror containing the axis does the opposite. So conjugation by a motion keeps a screw’s hand, and conjugation by a reflection, an inversion or a rotoinversion reverses it. A right-hand and a left-hand screw of the same pitch can be one class only if the group contains an improper operation.

The census checks the converse as well, and it holds without exception. Screws of both hands share a coset in 54 (type, axis) cases across 49 types. In all 36 cases whose group contains an improper operation the two hands are one class. In all 18 whose group has proper operations only they are two. The proper-only cases are the groups a chiral molecule can crystallise in, and among them are several that carry both hands at once. The body-centred four-fold groups of the $4_1$ kind carry $4_1$ and $4_3$ axes together, and the rhombohedral groups carry $3_1$ and $3_2$. Every primitive cubic group without improper operations carries $3_1$ and $3_2$ along its diagonals.

That refines a sentence written here before. A screw that contains its own mirror image observed that no proper motion turns a right-hand screw axis into a left-hand one. That is what makes P4₁ and P4₃ a pair, and it is exactly true. But a group without reflections can still contain screws of both hands, as the body-centred and cubic cases do. It holds them as different classes, and nothing in the group identifies them. The groups one hand may sit in is about the other side of the same fact: which groups a crystal of one hand can adopt at all.

Counted over all 219 #

Across the 219 affine types there are 542 pairs of a type and a kind of rotation axis, one pair for each class of linear parts up to inversion. Each is one class in the quotient. In the infinite group 471 of them split. The plane’s census had twenty-eight entries growing out of forty, so splitting is the ordinary case in space even more than in the plane. Two-fold kinds, 376 of them, fall into 993 classes, at most four each. Three-fold kinds, 80, fall into 183, at most three. Four-fold kinds, 68, fall into 104, at most two. The eighteen six-fold kinds never split, because the index for a sixth of a turn is one and translations alone reach every six-fold axis, as they reached every six-fold centre in the plane.

The maximum in each order is the index of $L_B$, reached when nothing but translations acts on the families. The kinds that split less than the maximum do so for the plane’s three reasons. Another operation of the group exchanges two families, as pmg’s half-turn exchanged its two mirror lines. A reflection joins two hands. Or, the case that has no plane counterpart, the families are already joined because a centring translation composed with the rotation lands on another family of the same pitch. Of the 542 cosets, 239 hold axes of more than one pitch. No class holds more than one.

The count, made twice #

The classes were computed by reducing vectors modulo $L_B$ with its Hermite normal form, and letting the group act on the residues. That calculation was repeated by a route that never forms $L_B$. Each axis is written as an actual line in space, reduced modulo the lattice by its coordinates across a basis that completes the axis direction, and carried by every operation of the group to another line. The orbits of lines, with their signed screw types, were then counted. The two routes agree on all 542 entries: the same number of families and the same number of classes.

The one thing both share is the list of axes to start from. That list is enumerated once, and the second route checks that it has exactly as many lines as the index of $L_B$ says. What is independent is the identification, which is what is being counted. In the plane the corresponding check was orbits of rotation centres, and here it is orbits of rotation axes, which is what a crystallographer draws.

What this is a table of #

In the plane the conjugacy classes turned out to be the table of special positions: two points share a Wyckoff position when an operation carries one to the other, which is when the rotations fixing them are conjugate. In space only the rotation axes fix points, so only they correspond to positions. The screw axes fix nothing, and their classes are what the International Tables draw rather than list: the symbols in the diagrams of symmetry elements, where a $4_1$ and a $4_3$ on different lines are drawn with different symbols. The diagrams are a table of these classes, with a picture where the list of positions has an entry.

What the count does not settle is what the same site under two names settled for positions in the plane: which of these classes a change of origin or of setting exchanges. A class that the group keeps apart can still be joined by the group’s normaliser, the motions that carry the group onto itself. For a group with only proper operations the normaliser often contains an inversion, which would join the two hands that the group keeps apart. Computing that needs the normalisers of the 219 types, which this census does not use.

What the count has to refuse #

The refused claim is the one the quotient would make if asked. Modulo the lattice, a rotation and a screw with the same linear part are the same element. In 239 of the 542 cosets that element stands for axes of different pitch, and the infinite group keeps every such pair apart. The test would catch any error in the action that let a translation change a slide’s length. The hands test is the sharper one. It predicts, before any group is examined, exactly which of 54 cases are joined, and it could have failed in any of them.

Where this goes: the normaliser, and which hand is kept apart only by the group #

The census answers the question inside each group. The next question is the one the lead of the plane essay pointed at: for the eighteen cases where a proper-only group keeps two hands apart, does its normaliser join them? If the normaliser contains an inversion, the group’s two classes of screw are exchanged by a change of description. Then a structure report could list the $4_1$ axes of one crystal as the $4_3$ axes of another report of the same crystal. Telling two words apart and the same symmetry, somewhere else do this for the plane. In space it needs the normaliser of each type, and the eighteen cases are the place to start.