Every motion of space is a screw

The plane has four motions and the list is easy to believe, because each of them can be acted out with a sheet of paper. Space is asked the same question and gives a stranger answer: there is one motion, with two numbers in it, and everything else is that motion with one of the numbers set to zero.

A rigid motion of space that preserves handedness is a rotation about some axis together with a slide along that same axis. A rotation is the case where the slide is nothing. A translation is the case where the turn is nothing. Nothing else occurs, and the axis is not put in by hand — it is computed from the motion, along with how far the motion climbs along it.

Splitting the translation, which is where the axis comes from #

The reduction is short enough to state completely.

A proper motion is x ↦ Ax + b, with A a rotation matrix. Rotation matrices in space always fix a direction — the axis — because a three-by-three rotation has an eigenvalue of one, which is why the plane’s list and space’s differ in the first place. Call that direction u.

Now split b in two: the part along u, and the part across it.

The part along u cannot be removed. Moving the origin changes b by (A − I)s, and (A − I)s is always across the axis, never along it — because A leaves u alone. So the component of b along the axis is a property of the motion, not of anybody’s choice of origin, and it is the pitch.

The part across u is a statement about where the axis is. Solving (A − I)p = −b⊥ inside the plane across the axis finds a point the motion moves only along u. The axis is the line through p in the direction u, and it is located rather than assumed.

That split is exactly the one the space groups are built on, applied to one operation instead of to a whole group. What is new here is that nothing is left over: after the split, the motion is the screw with those three data, and multiplying them back out reproduces it.

The two ends of one description #

The reduction makes two familiar objects into limits of one.

A rotation is a screw of zero pitch. Its axis is a line of fixed points, and the fixed points are what a zero pitch means: a screw with any pitch at all fixes nothing whatever.

A translation is a screw of zero angle. Here the axis is not determined — every line in the direction of travel does equally well — and that indeterminacy is not a defect of the computation but a fact about translations, which is why the reduction reports the direction and not a located line.

Everything between is a screw proper, and the space groups are full of them: of the operations of the forty-five space groups built here, a hundred and twenty-six are screws with a pitch no origin removes.

The improper half, by the same two questions #

A motion that reverses handedness is not a screw, and the reduction that handles it asks the same two questions — the determinant, and what is fixed — with different answers.

If the linear part has a plane of fixed directions it is a reflection, and then the translation splits again: the part across the plane says where the plane is, and the part in the plane cannot be removed. Zero in-plane part gives a mirror; anything else gives a glide.

If the linear part fixes no direction, the matrix minus the identity is invertible and the motion has a fixed point, always. Those are the rotoinversions, including the inversion itself, and the useful consequence is that a rotoinversion never carries a translation that survives a change of origin — which is why a space-group symbol never has a subscript on a bar.

So the classification of every motion of space is two questions and five answers: identity, translation, rotation, screw on one side; reflection, glide, rotoinversion on the other. Six kinds and the identity, and each of them is decided by arithmetic on a matrix rather than by recognising a picture.

What was checked, and what would have gone wrong #

The reduction is a computation and it was run on everything available: every operation of every space group in the library, classified, reduced to axis-angle-pitch, and then rebuilt from those three numbers and compared with the operation it came from.

One thing about that check is worth recording, because it very nearly did not work. The angle of a rotation is usually read off the trace: cos θ = (tr A − 1)/2. That formula is worthless near a half turn, because the cosine is flat there — a trace correct to the last bit of a floating-point number gives an angle wrong by about one part in a hundred million. It is far too small to see in any picture and far too large for an exact comparison: a rebuilt inversion missed the inversion it came from and the check failed on the first group containing a centre. Reading the sine as well, off the antisymmetric part, and handing both to a two-argument arctangent, is accurate at every angle including that one.

The general lesson is one this collection meets repeatedly: a formula that is standard is not the same as a formula that is right at the case in hand, and the case in hand is very often the symmetric one.

What conjugation keeps, and what it moves #

The reduction returns three things, and they are not three things of the same kind.

Take a motion m and look at it from somewhere else — conjugate it by another motion g, forming g m g⁻¹. The linear part becomes gAg⁻¹, which is the same rotation seen in a turned frame, so the angle survives untouched. The pitch survives too: it is the component of the translation along the axis, and g carries the axis and the translation together, so the projection of the one on the other cannot change. The axis is the only part that moves, and it moves exactly where g sends it.

So the reduction splits cleanly into the part that belongs to the operation’s class and the part that belongs to where the operation sits. That is the arithmetic that decides sameness read on a single operation rather than on a whole pattern: two screws in one group are the same symmetry differently placed precisely when some operation of the group carries the axis of the first onto the axis of the second, and the angle and the pitch make that comparison before any axis is looked at.

The practical consequence is what a space-group diagram is allowed to leave out. A diagram does not mark every screw axis in the crystal, because there are infinitely many — the lattice translates each one across the whole of space. It marks one representative and the positions the lattice generates from it, and the reduction is what makes that legitimate: the symbol carries the class datum, the page carries the representative.

One asymmetry is worth stating, because it is where the invariance stops. Conjugating by an improper motion preserves the size of the angle, reverses its sense, and leaves the pitch alone — which is to say it reverses the screw’s handedness. No turning of the frame can do that and every mirror does, and the next section is that one sign.

The sign in the subscript, and the eleven pairs it makes #

The pitch and the angle are two numbers, and between them sits a relation that neither carries alone: whether the climb runs with the turn or against it.

A four-fold screw of quarter-cell pitch is 4₁. A four-fold screw of three-quarter pitch is 4₃ — and three quarters up is one quarter down, so 4₃ is 4₁ turning the other way. No proper motion of space carries one onto the other. Conjugation by a rotation preserves q and n both, and the only thing that exchanges q with n − q is a reflection, which a group containing none does not have to offer.

That is a fact about one operation. It becomes a fact about the classification because a group can be built entirely out of proper motions: sixty-five of the two hundred and thirty space groups contain no operation that reverses handedness, and those are the ones a single hand may sit in. Among them, eleven pairs differ in nothing whatever but the sense of their screws — P3₁ and P3₂, P4₁ and P4₃, P6₁ and P6₅, P6₂ and P6₄, together with the seven further pairs that carry two-fold axes alongside. Twenty-two of the two hundred and thirty groups exist because a screw has a sign.

The exceptions are the screws that are their own mirror image, and the same arithmetic names them: q/n and (n − q)/n are the same pitch exactly when 2q = n, so 2₁, 4₂ and 6₃ are handed neither way. Those are the screws a centrosymmetric group is free to contain, and it is why the list of pairs begins at three-fold rather than at two.

Quartz is where the pair is visible outside a table. Left- and right-handed quartz are P3₁21 and P3₂21: the same lattice, the same atoms, the same distances, and a helix of linked tetrahedra that turns one way in the first crystal and the other way in the second. Nothing internal to either structure prefers a hand — they are exact mirror images — and which one grows is settled by whatever seeded it. What the difference does reach is light, which the two rotate in opposite senses, and the twin laws relating quartz’s own domains, one of which exchanges precisely these two groups.

The reduction sees all of this without being told any of it. It returns a pitch as a signed multiple of the repeat, measured along the axis direction it has just computed, and reversing that direction reverses the sign of the pitch and the sense of the angle together — so the pair is well defined and either member of it alone is not. A check comparing pitches while letting the axis direction float would call 4₁ and 4₃ the same operation and report no pairs at all, which is the shape this error always takes: a quantity compared without the convention that gives it meaning.

The pitch is not free, and that is a fact about lattices #

The reduction returns a pitch for any motion at all, and most values of it are not available to a crystal.

Apply an n-fold screw n times. The turns come to a full circle, so the linear part is the identity, and what is left is n times the pitch — a pure translation, which must be a lattice vector because a symmetry operation of a crystal composed with itself is still a symmetry. So

$$n \times \text{pitch} = \text{a whole number of cells along the axis},$$

and the pitch is q/n of the repeat with q a whole number less than n. That single line, together with the rotation orders a lattice permits, gives eleven screw axes and no others — and it is the reason the pitch is written as a subscript rather than as a measurement.

Nothing in the reduction knows any of this. It happily reports a pitch of 0.37 for a motion that is a perfectly good screw and no crystal’s symmetry. The constraint comes from the lattice, not from the geometry, and keeping the two apart is what makes the second one a theorem.

Composing screws, and where the group comes from #

Two screws compose to a screw, since the product of two proper motions is proper — but the axis of the product is generally somewhere neither factor’s axis is, and its pitch is not the sum of theirs.

That is the mechanism behind the operations nobody put in: a group written down with two generators contains products, and the products have their own located axes. Computing them means composing matrices and vectors and running the reduction on the result, which is what the library here does when it builds a group — so every axis drawn on a space-group diagram in this collection has been located from the operation it belongs to rather than copied from a table.

The mirrors again, and the fourth one #

The screw is also the answer to a question the plane could not raise: how many mirrors does a motion of space need?

In the plane the answer is three, and the argument that gives it also gives space’s: two mirrors compose to a rotation when their planes meet and to a translation when they are parallel, and to nothing else. A screw is neither, so a screw is not two mirrors; it preserves handedness, so its count is even; and four suffice — two planes containing the axis for the turn, two across it for the climb.

Why a helix is allowed to be a crystal #

The physical consequence of all this arrives at once. A screw axis is the only symmetry operation that is compatible with a chain: it maps an infinite helix onto itself, it has no fixed point to pin anything down, and the number of turns per repeat is a rational number the lattice fixes.

That is why a protein’s α-helix and a strand of DNA can be described by symmetry at all, and why the rod groups are a classification rather than a curiosity. A rotation would force every point on the axis to stay where it is; a translation would forbid any turn at all. The motion that does both at once is the one a helix has, and space is where it lives.

What the reduction does not decide #

Three limits, and the third is the one most easily overstated.

It says nothing about whether an operation is a symmetry of anything. The reduction takes a matrix and a vector and returns an axis, an angle and a pitch. Whether a crystal has that operation is a different question, answered by the round trip on an actual set of positions.

It says nothing about which screws a lattice permits. That is the crystallographic restriction for the angle and a separate divisibility argument for the pitch, and together they give eleven screw axes and no others.

And a screw is not a helix. The operation is a symmetry; a helix is an object with that symmetry. A collection of essays about symmetry can slide between the two without noticing, and the difference shows up the moment a real chain is not quite periodic — where the object stops being invariant and the operation does not stop existing.

The size of what the extra dimension bought is worth putting beside those limits. Run the same classification over the seventeen plane groups and there are seventy-eight operations in total, of which twenty-four are reflections, thirty are rotations and seven are glides. There is no screw column at all, and nothing needs a fourth mirror. The forty-five space groups this collection builds contain six hundred and seventeen operations, and a hundred and twenty-six of them are screws — so the motion the plane has no room for is not a marginal addition to the catalogue. It is the second commonest orientation-preserving operation in it.

The order of a screw, which is not the order of its turn #

A screw of pitch q/n is written with a subscript n, and it is easy to read that subscript as the operation’s order. It is not, and the difference is the sharpest single consequence of the pitch being unremovable.

A four-fold rotation applied four times is the identity. 4₁ applied four times is a translation by one cell along the axis — which is a symmetry of the crystal, certainly, but it is not the identity, and applying it four more times gives two cells. A screw has infinite order as a motion of space. The cyclic group it generates is infinite, and every one of its elements moves every point.

What has order n is the screw’s image in the quotient by the lattice translations, which is the point group the symbol’s first character names — one of thirty-two. So the subscript counts turns and the character counts cosets, and the group generated by a screw is infinite while the group generated by its rotation part is finite. That is the same distinction the whole classification runs on — a space group is infinite and its quotient is one of the thirty-two classes — arriving here on a single operation.

The practical form of it is that a screw axis has no site symmetry to offer. A rotation axis fixes a line of positions, and an atom placed on that line has the rotation as part of its own environment; a screw fixes nothing, so no position in the cell has a screw in its site symmetry, and every orbit under a screw is infinite in the axis direction. The multiplicity of a general position and of a position on a screw axis are therefore the same number, which is why a space group’s table of positions never lists a special position for a screw.

How a screw announces itself, since nobody sees one #

Everything above is arithmetic on a matrix and a vector, and the operations themselves are not observable. What is observable is a diffraction pattern, and a screw leaves a signature in it that is as close to a direct reading as the subject offers.

Compose the screw with itself and the turns accumulate while the pitch accumulates too, so an n-fold screw of pitch q/n contributes atoms at heights that are multiples of q/n along the axis. Summing the structure factor along the axial row — the reflections 00ℓ, whose phases depend only on those heights — gives a sum of n-th roots of unity, Σ exp(2πi ℓkq/n), which is zero unless ℓq is a multiple of n. So the axial row survives only where ℓ is a multiple of n divided by the highest common factor of n and q, and the spacing of what survives is that number.

The common factor is the part worth keeping, because it is where the rule stops being “every n-th”. For 2₁, 3₁, 4₁, 4₃, 6₁ and 6₅ the two are prime to one another and the surviving reflections are every n-th: 00ℓ with ℓ even for 2₁, with ℓ a multiple of four for 4₁. For 4₂ the factor is two, and the row survives at every second reflection rather than every fourth — the same condition a 2₁ would impose, which is exactly why 4₂ and 2₁ cannot be told apart on that row alone. 6₂, 6₃ and 6₄ behave the same way, and the three screws that are their own mirror image are the three whose absences are weaker than their order suggests.

That is a systematic absence, and it is the reason a screw can be identified without any structure being solved: the missing reflections are exact zeros, they survive the squaring a detector performs, and the rule that produces them is read straight off the symbol. What the absences cannot supply is the sign in the subscript — 4₁ and 4₃ remove exactly the same reflections, since their subscripts are both prime to four — so the axial row settles the order of the screw and leaves its handedness to anomalous scattering or to the crystal’s optical rotation.

Who found it #

Michel Chasles stated the reduction in 1830, in the middle of a memoir on the geometry of displacements, and it is often given as Chasles’ theorem: any displacement of a rigid body is a screw. Giulio Mozzi had the same result in 1763, which is why the axis is sometimes called the Mozzi axis, and Poinsot restated it for mechanics in 1851. The theorem’s home is kinematics rather than crystallography — it is the reason a rigid body’s instantaneous motion is a twist about an axis — and it arrived in crystallography through the space groups, where a screw is a symmetry rather than a displacement.

Where the ladder goes next #

The reduction gives the operations; the classification asks what sets of them close into a group. Turning and climbing at once takes the screw as a symmetry rather than as a motion, and the operations nobody put in is the observation that most of a space group’s screws arrive as products of generators that contain none.

The other direction is smaller and sharper: with every operation reduced to three numbers, two operations can be compared without comparing matrices, and a group can be described by the axes it has and where they sit. That is what a space-group diagram is, and it turns out to contain the group.