The axis a product lies on

Where the product is is about the plane, and the answer there is a point: compose two located rotations and the result is a rotation about a third centre, at a place two lines of arithmetic supply. Closing the plane from two centres makes a classification out of that by asking which configurations are consistent.

In space the same question has a different shape, and the difference is not merely that a point becomes a line.

Three cases, and only one of them is interesting #

A rotation of space is located by a line rather than by a point, and two lines in space stand in one of three relations.

They meet. Then the point where they meet is fixed by both rotations, so it is fixed by the product, and the product is a rotation about some axis through that point. Nothing is translated. The composition is the sphere’s composition, which is the classification of the finite groups and has no translations in it anywhere.

They are parallel. Then the product of two half-turns is a translation perpendicular to both, by twice the distance between them, which is the plane’s result with the whole picture extruded along the axis direction.

They are skew — neither meeting nor parallel — and this is the case space has that the plane does not. The product is a screw: a rotation about some axis together with a slide along that same axis, which neither factor has.

That a screw comes out of two rotations is not a curiosity of small numbers. Every motion of space that preserves handedness is a screw — Chasles’s theorem — with a rotation being the case of zero pitch and a translation the case of zero angle. So the general product of two located rotations is a general screw, and the special cases are the ones where the pitch happens to vanish.

Twice the distance, twice the angle #

For two half-turns the answer is closed and the computation is a check on it rather than a search.

Take a half-turn about one line and a half-turn about another, at distance d and angle θ. Their product is a screw of angle 2θ and pitch 2d, at nine settings, to the last digit the arithmetic carries.

Neither factor has any pitch at all. A half-turn is a rotation; it moves no point along its own axis. The whole of the translation in the product is made by the two axes failing to meet, and its size is a measurement of how badly they fail. Set d to zero and the pitch goes with it, which is the first case; set θ to zero and the angle goes, which is the second.

The plane’s version of the same statement is the one composition opens with: two half-turns about points at distance d compose to a translation of length 2d. Space says the same thing with the axes tilted, and the tilt turns the translation into a rotation and leaves a translation behind. The two answers are one answer with a parameter in it.

The plane’s answer, for comparison #

Putting the plane’s case beside it shows which part of the answer is new and which is the same sentence with an extra coordinate.

Two half-turns of the plane compose to a translation by twice the vector between the centres. That is the whole of the plane’s version: a translation, whose length is twice a distance and whose direction is the line joining the two centres.

In space the same two half-turns become three quantities instead of one. The angle between the axes turns the translation into a rotation of twice that angle; the distance survives as the pitch, still twice a distance; and the direction of the joining line survives as the common perpendicular the axis lies on. Set the angle to zero and the rotation disappears, leaving the plane’s answer extruded — parallel axes, a pure translation of twice the distance.

So the plane’s result is not a special case of the space one in the usual sense of being simpler. It is the space result at one value of a parameter the plane has no room for, and the parameter is the angle between two lines that in the plane always meet.

Solving for a place rather than drawing one #

The method every one of these calculations runs on is worth restating, because in space it is the only method available.

A rotation of the plane about an unknown centre is a pair (M, t), and its centre is the solution of (I − M)x = t — the one point the motion leaves alone. The matrix I − M is invertible for every rotation but the identity, so the centre exists and is unique, and finding it is one two-by-two solve.

In space the same equation has no solution for a screw, because a screw leaves no point alone. What it has instead is a solution in the plane perpendicular to the axis: remove the part of t along the axis, which is the pitch, and the remainder is a translation the rotation can absorb. So the space method is the plane method plus one subtraction, and the subtraction is exactly the pitch.

That is why the three cases above are three cases rather than an accident. The pitch is the obstruction to the equation having a solution, and it vanishes precisely when the axes meet — which is when a common fixed point exists — or when the angle vanishes and there is no rotation to absorb anything.

The axis is on the common perpendicular #

The rest of the answer is where the screw is, and that is the question a located composition exists for.

Two skew lines have a unique common perpendicular — the shortest segment joining them — and the product’s axis lies on it, parallel to neither factor and meeting both at right angles when the two half-turns are at right angles to one another.

The arithmetic that finds it is the space version of the plane’s. A motion (R, t) has a rotation part with an axis, which is the eigenvector of R with eigenvalue one; the pitch is the part of t along that axis; and the position is the solution of (I − R)p = t with the axial part removed, which is a two-by-two system in the plane perpendicular to the axis, where I − R is invertible for every angle but zero. A point in the plane, a line in space, and the same equation solved in one dimension fewer.

What this says about a space-group diagram #

The practical end of it is the one the operations nobody put in is about, arriving from the other side.

A space-group diagram shows symmetry elements — axes, planes, centres — at positions, and most of the elements on it were not put there by anybody. They are the products of the ones that were, and the diagram is a picture of a closure. That essay establishes that composing two generators yields operations of a kind neither of them had. This one says where they are, and the answer is arithmetic rather than draughtsmanship: given two axes, the product’s axis is on their common perpendicular, at a computed position, with a computed pitch.

So a reader looking at the diagram of a group whose symbol lists only rotations, and finding screw axes drawn on it, has an explanation in three lines. Two rotation axes that do not meet compose to a screw; the screw is on their common perpendicular; and the pitch is twice their distance, which for axes half a cell apart is a whole cell and therefore a legal screw component.

That last clause is where the crystallography is. A screw axis in a space group must advance by a whole number over its order of a lattice translation — eleven screws and no others — so the distances at which two axes may sit are constrained by the pitches their products are allowed to have. The composition law prices the geometry, exactly as the plane’s triangle of centres prices the angles.

Where the plane’s closure stops working #

Closing the plane from two centres decides, from the two orders alone, whether the result is a plane group. The same question in space does not have an answer of that shape, and the screw is why.

In the plane the linear parts a pair of rotations generates are a cyclic group whose order is the least common multiple, and that is settled by the orders. In space the linear parts a pair of rotations generates depend on the angle between the axes as well as on the orders: two four-fold rotations about perpendicular axes generate the rotation group of the cube, of order twenty-four, and two four-fold rotations about axes at some other angle generate an infinite group.

So the condition that made the plane’s closure decidable — a property of the two orders — has no space analogue, and what replaces it is the classification of the finite rotation groups for the intersecting case and a much harder question for the skew one. The plane is decidable because two lines in it always meet.

What the skew case does have is the pitch, and the pitch is a quantity the plane’s closure had no room for. A closure in space can fail to be discrete in a way the plane cannot: not by manufacturing shorter and shorter translations across the axis, but by manufacturing pitches that never close up — a screw whose advance is an irrational fraction of any lattice translation, which no amount of composing will ever return to the identity.

Why a group with no screws in its symbol has them on its diagram #

The most useful consequence is one a reader meets before ever meeting the arithmetic, and it is worth stating plainly.

Take P2₁2₁2₁, whose symbol names three two-fold screw axes and nothing else, or take P222, which names three plain two-fold axes. In both, the three axes are mutually perpendicular and they do not all meet. So the product of two of them is a screw on their common perpendicular, at twice their distance — and in P222 the axes are placed so that the distance is nought and the products are rotations, while in P2₁2₁2₁ the distance is a quarter of a cell and the products are screws.

The symbol names three operations and the group has many more, and every extra one is at a place the composition law gives. The operations nobody put in establishes that they arrive; what this page adds is that their positions are not a matter of looking at the diagram. They are the solution of a two-by-two system, one per pair of generators.

That also explains a feature of the diagrams that otherwise looks like clutter. A space-group diagram drawn from the generators alone would have three lines on it; the printed ones have dozens, because every product and every product of a product is an element and has a place. The diagram is the closure, drawn, and the closure is exactly what the plane’s two centres give and what nobody can compute cheaply in space.

Where the exactness stops #

Computed here. Four pairs of located rotations of space, composed from their matrices and translations, with the product read back as a screw: the axis direction from the antisymmetric part of the rotation, the angle from the trace, the pitch from the component of the translation along the axis, and the axis’s position from the solution of the two-by-two system in the perpendicular plane. The common perpendicular of each pair, and the distance from the product’s axis to it. And a sweep of nine settings of distance and angle, with the pitch and angle compared against twice the distance and twice the angle.

The closed form is for half-turns. That the pitch is twice the distance and the angle twice the angle between the axes is the two-half-turn case; for general angles the product is still a screw and its pitch is a longer expression, which the computation produces numerically and the sweep does not cover.

Four cases are four cases. The three relations between two lines are exhaustive and the fourth row is one instance of the third; nothing above sweeps the space of all pairs.

The cases are chosen and not swept. Two perpendicular half-turns at distance one is the skew case used throughout, because its answer is a round number and a sign error in it is visible. A sweep over angles and distances is the second figure; a sweep over orders is not made anywhere, and the general product of an n-fold and an m-fold about skew axes is computed here only at the one pair the fourth row uses.

And there is no group here. Two located rotations are composed once. What the closure of a pair does in space — whether it terminates, and into what — is the question the section above says has no answer of the plane’s shape, and nothing here attempts it.

The tests are arranged so that the interesting case is bracketed by the two dull ones. A computation that returned a screw where the axes meet, or a rotation where they are skew, would be wrong in exactly the way this page is about — and the pitch, being a number rather than a classification, is the part a sign error would quietly spoil.

Who worked out the screw #

Michel Chasles proved in 1830 that every orientation-preserving motion of space is a screw, and Giulio Mozzi had the result in 1763. The composition law for two half-turns — pitch twice the distance, angle twice the angle — is classical kinematics and appears in every treatment of the subject under the name of the screw axis or, in the nineteenth-century language, the cylindroid.

Its use in crystallography is Schoenflies’s and Fedorov’s, in the derivation of the two hundred and thirty, where the question “which located elements are consistent” has to be answered about axes rather than about points. They did it by cases and by drawing, which is why the space-group diagrams of the International Tables look the way they do: each one is a worked composition, and the elements nobody put in are the ones the draughtsman computed.

What the arithmetic adds is that the computation need not be a case analysis. One formula takes two located rotations to a located screw, the three cases are the places where one of its terms vanishes, and the drawing is an output rather than a method.

Still open: closing a pair of axes #

The natural next question is the one the plane answers and space does not: given two located rotation axes, does their closure terminate?

For intersecting axes it does exactly when the angle is right. The finite rotation groups of space are the cyclic, dihedral, tetrahedral, octahedral and icosahedral ones, so two axes through a point close exactly when their orders and the angle between them fit one of those — a condition on three numbers rather than on two, and a condition the five families already list.

For skew axes nothing above decides it. The closure contains screws, and a screw of irrational pitch relative to the translations generated can never be undone; whether a given pair produces one is a question about whether an angle and a distance are commensurate in a particular way, and it is not a question about the orders at all.

The measurement that would begin to answer it is the plane’s: close from a pair, watch the shortest translation, and watch the smallest non-zero pitch beside it. Two quantities rather than one, and a closure in space can be dense in either — which is presumably why the space groups were derived by case analysis for a century and the plane groups were not.