A fold that keeps its symmetry

The rank of a rigidity matrix reports how many infinitesimal mechanisms a framework has: assignments of velocity to the joints under which no bar’s length is changing to first order. That is a real thing to know and it is not the thing anybody wants to know.

What is wanted is whether the framework can actually move — whether the velocities integrate into a path along which the bars stay the length they started. An infinitesimal mechanism need not: a framework can have a first-order motion that stops immediately, the way a straightened elbow can be bent in only one direction while the derivative points both ways.

One framework where the motion goes all the way #

The kagome net is three joints and six bars in a cell, and its bars are exactly the edges of a lattice of corner-sharing triangles: up-pointing and down-pointing, alternating, each sharing its three corners with three of the other kind.

Because the bars are the triangles’ own edges, a triangle is rigid. So any motion of the framework must move each triangle rigidly — turning it and sliding it, but never deforming it — and the only question is whether the turnings can be made consistent.

They can, and the construction is the whole answer. Turn every up-triangle through an angle θ about its own centre and every down-triangle through −θ about its own. The shared corners stay shared: two triangles meeting at a point turn in opposite senses, so the point they share moves to the same new place from both sides. Nothing has been deformed, no bar has changed length, and the framework has moved through a finite distance.

That is a finite mechanism, and the difference from an infinitesimal one is the difference between a structure that has a soft mode and a structure that has a transition available to it.

What happens to the symmetry along the way #

Here is where the framework question meets this collection’s own subject, and it is the reason the essay is in this field rather than in a book about structures.

At θ = 0 the framework is the kagome net at its equilibrium placement, whose group is p6m, of order twelve. At any θ strictly between zero and sixty degrees the group has fallen, and the interesting question is which half of it goes.

There is an argument that answers it in one line and gets it wrong, and it is worth setting out because it is the argument this essay used to make. Turning the up-triangles one way and the down-triangles the other is a chiral operation, so the mirrors must go and the rotations must stay. That reasoning treats a mirror as something that has to preserve the sense of every turn, and a mirror does no such thing when it also exchanges the two families of triangles: a reflection that carries every up-triangle to a down-triangle carries a turn of +θ to a turn of −θ, which is exactly what the down-triangles are doing.

So the test has to be made rather than argued, and it can be made exactly. The framework along the motion is its starting configuration plus the mechanism times a small angle, so an operation of p6m still maps it onto itself, to first order, precisely when it leaves the mechanism vector unchanged. The mechanism space here is one-dimensional, so every operation acts on it by a number, and the number is plus or minus one.

The answer is p31m, and it keeps mirrors. What the twist destroys is the six-fold rotation at the hexagon centres, the two-fold at each shared corner, and the three mirrors that run through the triangle centres. What it keeps is the two three-folds and the three mirrors that run through the shared corners.

The rule the table obeys is legible once it is stated. Sort the operations by two questions — does it exchange the up-triangles with the down-triangles, and does it reverse orientation — and an operation survives exactly when the two answers agree. A three-fold fixes both families and preserves orientation: survives. A mirror through a shared corner exchanges the families and reverses orientation: survives. A six-fold exchanges them and preserves orientation, so it takes a triangle turned by +θ to a place where a triangle turned by −θ is wanted; the two-fold at a corner does the same; and a mirror through a triangle centre maps each triangle to itself while reversing its turn. Those three fail, and there are six of them once the cosets are counted.

That is a first-order statement, and the finite motion agrees with it. Building the twisted framework out of rigid triangles — turning the up-triangles by +θ, the down ones by −θ, and letting the lattice contract by cos θ, which is exactly the condition that keeps the shared corners shared — and then testing the point set operation by operation at θ = 0.37 returns the same six. So the group along the motion is p31m, and not merely p31m to first order.

The wrong version of that check is worth a sentence, because it looked convincing. Turn each triangle about its own centre and hold the lattice fixed, and every shared corner splits in two — by a third of a bar length at this angle — because the two triangles meeting there disagree about where it went. Average the two positions and the bars come back to unit length, which is enough to make the construction look right, and the point set is not the twisted framework and does not have its symmetry. The contraction of the cell is not a consequence of the motion; it is a condition on it.

So the motion is a descent of symmetry parameterised by an angle, and it is exactly the shape of a displacive phase transition: a structure sits in a high-symmetry configuration, a soft mode carries it away, and the group falls to a subgroup while every atom stays bonded to the atoms it was bonded to. p6m to p31m is a descent of index two, which by the counting of domains means such a transition makes two domain states and one wall between them.

And at θ = 60° the framework closes up again into a configuration with the full symmetry restored, with every second triangle turned inside out. The path runs from one symmetric configuration to another through a continuum of less symmetric ones, which is the shape a soft mode has when it is followed far enough.

First order, and the gap it leaves #

It is worth being precise about what a first-order calculation can and cannot promise, because the gap is where most of the interesting cases live.

An infinitesimal mechanism is a solution of a linear system. A finite motion is a solution of the full non-linear system — the bar lengths are quadratic in the coordinates — and the linear system is its derivative at one configuration. Every finite motion gives an infinitesimal one by differentiating; the converse fails, and it fails in a way that has a name.

A framework whose infinitesimal mechanisms all fail to extend is called infinitesimally flexible and rigid, and the standard example is three bars in a straight line with a joint in the middle: the middle joint can move sideways to first order, because moving it sideways changes the bars’ lengths only at second order, and it cannot move at all in fact. The derivative points somewhere the function does not go.

So the kernel of a rigidity matrix is an upper bound on the motions, never a guarantee of them. What closes the gap in a particular case is an argument or a construction, and this essay supplies a construction. There is a general theorem in the background — a framework at a generic configuration is finitely flexible whenever it is infinitesimally flexible — but the configurations that interest crystallographers are precisely the non-generic ones, the ones with symmetry, so the general theorem is exactly no use here.

That is a recurring shape in this collection. The results that hold generically are the ones that fail on the symmetric cases, and the symmetric cases are the subject. It is the same reason a fingerprint that gives the right answer can be wrong about the cases that matter: agreement on the general case is cheap.

The count depends on the cell, and Maxwell’s does not #

The sharpest thing this ladder has to say is a discrepancy, and it takes three lines of a table to show.

Compute the kagome framework’s mechanisms in a cell of one: one mechanism. In a cell of two: two. In a cell of three: three. Maxwell’s count says one in every case, because doubling the cell doubles the joints and the bars together and the arithmetic cannot notice.

What is being sampled is a line of zero modes. The kagome framework has a mechanism at every wavevector along a whole direction of reciprocal space, not merely at the origin, and a calculation in a cell of size k sees k of them because a cell of size k can represent motions of k different periodicities. The number found is therefore a property of the pair (framework, cell), and quoting it without the cell is quoting half a result.

This is a general hazard and not a curiosity of one net. Any framework whose soft modes fill a line or a plane of reciprocal space will report a mechanism count that grows with the supercell, and any framework whose soft modes are isolated will not. Running the ladder is how the two are told apart.

The two tables carry the same numbers, and that is worth stopping on rather than passing over. Both ladders are cells stretched in one direction only — one cell, then two, then three, then four along a — and both report one mechanism, then two, then three, then four, with Maxwell’s count sitting at one throughout and the self-stresses running nought, one, two, three behind it. Two frameworks that look nothing alike, built on different lattices from different numbers of joints and bars, produce identical ladders. What the ladder is measuring is not the framework but how many wavevectors along one line the cell can represent, and both of these have a soft mode at every one of them.

The honeycomb is the control, and it is worth having for the same reason every refusal in this collection is worth having: a growing count means nothing unless a count that grows with Maxwell’s line is available beside it. Two mechanisms in a cell of one, three in a cell of two, four in a cell of three, five in a cell of four — and Maxwell’s number is two, three, four, five as well. The count grows here too. What it does not do is fall behind, because the honeycomb has no self-stress at any of these sizes, and it is the gap between the two columns rather than the growth of either that says a mode is being missed.

Why a triangle is the right unit #

The kagome twist is the cleanest case of a rigid unit mode, and it is clean because the rigid unit is visible: a triangle of bars is rigid, so the framework decomposes into rigid pieces joined at points, and the motion is a rotation of the pieces.

Real framework crystals are built the same way and the pieces are polyhedra. In a silicate the SiO₄ tetrahedron is stiff and the Si–O–Si angle at a shared oxygen is soft, so the structure is rigid tetrahedra sharing corners, and its low-energy motions are rotations of the tetrahedra. The counting done in the previous rung is usually done on that decomposition rather than on individual bonds, and the answer means the same thing.

The practical payoff is that these modes are where the thermodynamics goes. A structure with rigid unit modes has an anomalously large density of low-frequency vibrations, which shows up in its heat capacity; it often has a displacive transition at a temperature set by where those modes go soft; and it can have negative thermal expansion, because turning the units to fill space more efficiently pulls the structure in as the amplitude grows.

What the twist does to the diffraction #

A structure moving along this path would announce itself, and it is worth saying how, because it connects the framework question to the field this collection spends most of its time in.

The twisted framework is periodic at every θ — the motion carries the whole lattice with it — so it diffracts sharply throughout. What changes is the cell, which contracts, and the intensities, which redistribute as the atoms move. A structure caught mid-twist looks like an ordinary crystal with an ordinary space group; nothing about the sharpness of its reflections reports that it is sitting on a soft mode.

What does report it is the group. At θ = 0 the group is p6m and at any intermediate angle it is p31m, so the systematic absences change, and reflections forbidden in the high-symmetry configuration appear weakly as the angle grows. Their intensity is a direct measure of the amplitude, which is how order parameters are measured in practice.

That is a case of the general rule that a symmetry lost is a symmetry that shows up as new reflections. It is worth noticing that the framework calculation predicts which symmetry will be lost — the six-fold rotation and the two-fold, with three of the six mirrors surviving — before any diffraction is computed at all. And it predicts it from a kernel vector, which is a piece of linear algebra with no crystallography in it: the same calculation that says how many motions there are says which operations each motion is compatible with.

What is exact here and what is measured #

This essay makes two kinds of claim and it is worth keeping them apart, because the whole habit of this collection is that a claim carries the method that produced it.

Exact: the rank, the mechanism counts, the self-stress counts, the identity between them, and the velocity vectors drawn on the joints. All of that is rational arithmetic on the equilibrium placement, with no tolerance anywhere.

Measured: the finite twist. A rotation through a general angle is not a rational operation, so the path is computed in floating point and the bar lengths are measured along it. The largest deviation over the whole path is quoted with the figure, and it is at the level of the arithmetic’s own precision rather than at the level of anything about the framework — but it is a measurement and it is reported as one, because a claim that a motion is exact should be backed by an argument and not by a number that happens to be small.

The argument, for the record, is the one given above: the triangles are rigid because their edges are bars, and a shared corner is reached from both sides by the same rotation. That is a proof, and the numbers are a check on the code rather than on the geometry.

Two structures on the same path #

The endpoints of the twist are worth naming, because both are objects this collection has already met and the path between them is a new relation.

At θ = 0 the framework is the kagome net: corner-sharing triangles, group p6m, three joints and six bars per cell. At θ = 60° the triangles have turned until each shares an edge rather than a corner with its neighbours — the holes have closed, every second triangle is inside out, and the framework has collapsed onto the triangular net with the cell at half its original edge and a quarter of its area.

So the twist is a continuous path between two of this collection’s registry entries. That is unusual: nets are combinatorial objects and there is normally no sense in which one can be deformed into another, because deformation is a geometric operation and a graph has no geometry. What makes it possible here is that the framework supplies the missing geometry — one length per edge — and the path is a path in that extra structure, with the graph unchanged throughout.

It is a useful reminder of what a framework is for. A net is a graph; a framework is a net with a metric attached to its edges and nothing else; and the space of frameworks on one net is where all the motion in this pair of essays happens.

Where the twist is met #

The kagome lattice’s mechanism has been rediscovered repeatedly, which is a good sign that it is a real object rather than an artefact of one formalism.

It is in the mechanics literature as a bar-and-joint framework on the isostatic boundary; in the physics of frustrated magnets as the reason kagome antiferromagnets have an extensive ground-state degeneracy; and in the materials literature as the mechanism behind a family of framework structures that contract on heating. The name comes from a Japanese basket weave, and the pattern of that weave is the net.

For this collection the interest is narrower and sharper: it is a case where a group falls to a subgroup along a path that can be written down, with every step of the fall computable and the endpoint symmetric again.

The twist contracts in every direction at once #

The path has a property worth extracting as a number, because it is the one an engineer would measure and it is unusual.

Along the twist the cell shrinks isotropically: the framework contracts by the same factor in both directions, since the six-fold symmetry that survives forces the two cell edges to stay equal. Stretch such a framework in one direction and, following the path, it expands in the other as well.

A material behaving that way has a negative Poisson’s ratio — it gets fatter when pulled rather than thinner — and for a mechanism contracting isotropically the ratio is exactly −1, which is the extreme value the elasticity of an isotropic material permits. That is not a small effect at the edge of a range; it is the boundary of what is possible.

The mechanism responsible has a name outside crystallography — rotating triangles, alongside rotating squares and the other members of the same family — and materials built to have it are made by cutting the pattern into a sheet rather than by finding a crystal with it. The arithmetic is the same arithmetic: a framework with a finite mechanism whose motion is a rotation of rigid units, and a cell that follows.

What the framework calculation supplies is the sign and the extreme value, both from the geometry alone. What it does not supply is how stiff the material is, how far along the path it can be pushed before something yields, or whether a real crystal with this net does any of it — all of which are measurements.

Where the twist is a real transition #

The kagome twist is a two-dimensional model, and its three-dimensional relatives are the reason rigid-unit modes are studied at all.

Quartz’s transition at 573 °C is a rigid-unit rotation. The high-temperature form has the higher symmetry, the low-temperature form is reached by turning the silicon-oxygen tetrahedra slightly about shared corners, and the tetrahedra themselves do not deform — which is exactly the kagome picture with tetrahedra in place of triangles and one more dimension to turn in. The transition is displacive rather than reconstructive for that reason: nothing has to break.

Cristobalite and the other silica forms behave the same way, and so do a large number of framework silicates and their aluminate relatives. The general observation is that a structure built of corner-linked rigid polyhedra has mechanisms wherever the counting permits, and that those mechanisms are the transitions the material actually has.

And the negative expansion above appears here too. A framework whose units rock without stretching pulls itself inward as the rocking amplitude grows with temperature, so several of these materials contract on heating over a range — the same geometric fact as the negative Poisson’s ratio, driven by thermal amplitude rather than by an applied force.

Where this ladder goes next #

That closes the two rungs on frameworks. The connection to run forward is to the descent of symmetry and to how many domains a transition makes: a mechanism supplies the direction a structure’s symmetry can fall in, and the subgroup calculation supplies the count of how many equivalent ways it can fall. Neither implies the other, and a full account of a displacive transition needs both.

The connection to run backwards is to what counts as a bond. Every framework in these two essays was built from a net, and every net was built from a cutoff — so a framework’s mechanism count inherits that arbitrariness, and a structure whose bonding is ambiguous has a rigidity that is ambiguous in exactly the same way.