The mechanisms a count cannot see

Maxwell’s count is one subtraction. A periodic framework with n joints and e bars in a fixed cell has 2n velocities to choose and e constraints on them, minus the two rigid translations that are not motions of anything — so 2n − e − 2 is how many more freedoms there are than constraints, and if it is positive the framework must move.

The count’s weakness is in the same line as its strength. It is a difference, and a difference is silent about what it is a difference of. A framework with one mechanism and one state of self-stress has a count of nought, and so does a framework with neither, and the count cannot tell them apart. What it reports is the net of two quantities, either of which can be interesting.

The row that shows the price #

The bathroom net — four joints and six bars in a square cell, the net of a herringbone floor — has a scalar count of 2 × 4 − 6 − 2 = 0. Read as a verdict that reads “isostatic”: as many constraints as freedoms, nothing left over, no motion and no redundancy.

Its rigidity matrix says otherwise. Reduced exactly over the rationals, it has a kernel with one vector in it beyond the translations and a left kernel with one vector in it. One mechanism and one self-stress, and 0 = 1 − 1.

Nothing is wrong with Maxwell’s count. It is exactly right and it is answering a different question from the one a reader of it usually has, which is whether the framework is rigid. The repair is not a better count. It is to do the same subtraction in a place where cancellation is harder.

A subtraction between representations #

Every object in the count is acted on by the framework’s own point group. The joints of one cell are permuted; the bars are permuted; the velocities transform as the two translations do. So each of the three is a representation of that group, and the count is the dimension of a virtual representation:

Γ(m) − Γ(s) = Γ(joints) × Γ_T − Γ(bars) − Γ_T

Evaluated at the identity this reads 2n − e − 2 and nothing has been gained. Evaluated at every operation of the group and decomposed into irreducible representations, it is a list of integers rather than one integer — and a mechanism cancels a self-stress only when the two belong to the same irreducible representation.

That is Fowler and Guest’s observation, and the reason it works is that a symmetry which forces a mechanism rarely forces the self-stress into the same place.

A character is a count of things left where they are, so all three are cheap.

Two of the three counts have a trap in them, and both traps are about identification.

A joint carried into the next cell is the same joint. The framework is periodic, and the object being permuted is the set of quotient vertices rather than the set of points of the plane, so the character counts vertices with σ(v) = v and pays no attention to which cell the image landed in. Counting the points instead gives zero for every operation that is not the identity, and a decomposition of nonsense.

A bar traversed backwards is the same bar. The coordinate a bar contributes is its extension, and an operation that swaps a bar’s two ends leaves the extension exactly where it was. Counting only the bars whose ends are preserved individually would miss every two-fold axis through a bar’s midpoint, which on a framework with any symmetry at all is most of them.

The third count needs no care and is the one place the geometry enters. Γ_T’s character at an operation is the trace of its matrix — 2 for the identity, −2 for the half-turn, 0 for a quarter-turn, 1 for a three-fold and so on — and it is the only quantity here that is not combinatorial.

What the decomposition says #

The bathroom net’s point group is 4mm, which has five irreducible representations — four of dimension one and one of dimension two. Its Γ(m) − Γ(s) decomposes with a +1 in one of them and a −1 in another.

A positive multiplicity is a mechanism that must exist, and a negative one is a self-stress that must exist. The two cannot cancel because they are in different representations: a motion transforming one way under the quarter-turn is not the negative of a stress transforming another way, however the totals add up. So the symmetry-extended count predicts one mechanism and one self-stress — and the rigidity matrix, which knows nothing about characters, finds exactly one of each.

The prediction is a lower bound and not an equality, and the distinction matters. Γ(m) − Γ(s) is a difference of representations, so a +2 in some row could be three mechanisms and one self-stress rather than two mechanisms and none. What the decomposition establishes is that at least that many mechanisms exist in that representation, and at least that many self-stresses in the rows that are negative. It is a certificate rather than a census, and the certificate is what the scalar count could not give at all.

Why a representation cannot be argued out of existing #

It is worth being explicit about why the decomposition is a proof of a mechanism rather than a suggestion of one, because the argument is short and it is the whole justification for the method.

The mechanisms of a framework form a vector space: sums and multiples of mechanisms are mechanisms, since the constraint that every bar keeps its length is linear in the velocities. The point group acts on that space, because moving the framework by a symmetry and then flexing it is the same as flexing it and then moving it. So the mechanisms carry a representation of the group, and so do the self-stresses, and both are honest representations with integer multiplicities in every row of the character table.

The identity Γ(m) − Γ(s) = Γ(joints) × Γ_T − Γ(bars) − Γ_T then holds as representations and not merely as dimensions, because it is a statement about the rigidity map: the joint velocities carry Γ(joints) × Γ_T, the bar extensions carry Γ(bars), and the map between them commutes with the group. A linear map that commutes with a group has a kernel and a cokernel that are themselves representations, and the difference of the source and target is the difference of those two — which is the rank–nullity theorem with a group along for the ride.

So a positive entry cannot be argued away. If the multiplicity of some representation in Γ(m) − Γ(s) is +1, then the mechanisms contain that representation at least once more than the self-stresses do, and a representation cannot appear a negative number of times. Something is there.

The converse trap is worth naming too: a multiplicity of nought does not say that nothing is there. It says that whatever mechanisms exist in that row are matched one for one by self-stresses in the same row, which is exactly the situation the scalar count is in, one row at a time. The extension does not remove the cancellation; it makes the cancellation happen in smaller compartments, and the smaller the compartments the less can hide.

The kagome net, where the total is negative and it moves anyway #

The bathroom net is the clean case. The kagome net is the useful one, because its scalar count is not merely uninformative but actively misleading.

Three joints, six bars, a scalar count of 6 − 6 − 2 = −2. Read straight, that says the framework is over-braced by two and has no freedom left — which is what a reader of the count concludes, and it is wrong. The kagome framework has a mechanism, and this collection has drawn it moving: the triangles counter-rotate and the whole net folds without any bar changing length.

The decomposition has a +1 in one row, so a mechanism is forced. The negative entries account for three self-stresses, and 1 − 3 = −2 recovers the scalar count exactly.

This is the case that shows what the extension is for. A count of nought is at least ambiguous; a count of minus two looks like a definite answer, and the definite answer is false. Nothing about the scalar count warns a reader which situation they are in, and the decomposition does — with no more information going in, since the characters are counts of fixed joints and fixed bars and the framework’s own operations.

The star net makes the same point with the sign reversed. Its scalar count is +1, which reads as one mechanism and nothing else, and its decomposition has a negative entry: there is a self-stress as well, and the exact calculation finds two mechanisms against one self-stress rather than one against none.

The rows with nothing forced, which are most of them #

Six of the nine frameworks in the table have a decomposition with no positive entry at all, and it is worth saying what that does and does not establish.

The square net as a framework — one joint, two bars — has a scalar count of −2 and a decomposition that is negative in two rows and nought elsewhere. It has no mechanism, and the exact calculation agrees. But the decomposition did not prove that: a row with multiplicity nought could have held a mechanism and a self-stress together, and a row with −1 could have held two self-stresses and one mechanism. The extended count certifies the self-stresses and is silent about the rest.

That asymmetry is the honest shape of the method and it is the reverse of what a reader expects from a count. The scalar count’s positive values are its certificates — 2n − e − 2 > 0 really does force a motion — and its zero and negative values are the ambiguous ones. The extended count turns each of those into a list, and every entry of the list has the same logic as the whole count did: positive is a certificate, negative is a certificate of the other thing, nought is silence.

What it never gives is a proof of rigidity. No count of this kind can establish that a framework is rigid, because a framework whose every representation cancels may still have a mechanism paired with a self-stress in each row, and because infinitesimal rigidity is not finite rigidity anyway — the exact rank calculation settles the first and says nothing about the second. The nine rows above are settled by the rank, and the characters are what says which of them had to come out that way.

Where the group comes from, which is not from a table #

One thing here is worth stating because it is where this essay’s machinery meets an earlier repair.

The point group used above is the framework’s own, detected on its barycentric placement — and until this phase that detection depended on the basis the net’s voltages had been written in. A framework handed a group that is too small does not fail loudly: the characters are computed over fewer operations, the table has fewer representations, the decomposition comes out with fewer rows, and every number in it is a perfectly consistent statement about the smaller group.

What would be lost is precisely the content. A mechanism forced by a four-fold axis is invisible to a calculation whose group has no four-fold axis in it, and the extended count would then agree with the scalar count and add nothing. So the correction that made a net’s group a property of the net rather than of its description is a precondition for this whole calculation rather than a tidy-up beside it.

The same applies to the placement. The barycentric placement is where the group is largest, so it is where the extended count is strongest — and a framework drawn at any other placement of the same net has a subgroup of that group, a coarser decomposition, and a weaker certificate.

What a bar-and-joint framework is, and what it is not #

One clarification, since the objects here are easy to over-read.

A bar is a distance constraint and nothing else: two joints at a fixed separation, free to rotate about each other. That is a good model of a rigid rod pinned at its ends, and a poor model of a chemical bond, which resists bending as well as stretching. So a framework with a mechanism is a framework that can move if only the distances are held, and a real structure with the same net may be perfectly rigid because its angles resist.

The counting still says something about the real structure, and what it says is where the softness is. A framework’s mechanism is the direction in which only the weakest restoring forces act, which is why the kagome geometry appears in discussions of soft modes and negative thermal expansion. A mechanism is a prediction about what is floppy, not about what falls down.

And a self-stress is the reverse: a set of bar tensions that balance at every joint with no load applied. A framework with one can be pre-stressed, which is how a tensegrity stands up and how a real crystal can carry an internal strain that no external force explains. The two objects the scalar count confuses are therefore not merely different in sign; they are different phenomena, and telling them apart is most of the reason to run the calculation.

What the count refuses #

The last is the one that makes this an argument rather than a formalism. Character theory predicts a self-stress; the rigidity matrix is exact linear algebra over the rationals with no group in it at all; and the two are checked against each other on every framework here. A prediction from symmetry that were never compared against a direct calculation would be a decoration on the count rather than a repair of it.

The third is the one that makes it worth doing. If no framework in the collection had a scalar count of zero and a motion, the whole extension would be an elaborate way of restating the subtraction, and the assertion fails if that ever becomes true.

Where the exactness stops #

Computed here: each net’s own point group from its barycentric placement; the permutation of quotient joints and of quotient bars induced by every operation; the three characters; their combination; the character table of the group, built from its own matrices rather than looked up; the decomposition into irreducible representations; and, independently, the rank and kernels of the exact rigidity matrix over the rationals.

A fixed cell, throughout. Every count here forbids the cell to change shape. Allowing it adds three columns for the metric and changes every number — the mechanism count is a property of the pair (net, cell) and not of the net — and the symmetry-extended version of the flexible-cell count needs the strain’s own representation added to the sum, which is not done here.

One wavevector, and it is the zone centre. The characters count joints and bars of one cell, so the motions this finds are the ones that repeat every cell. A mechanism with a longer wavelength is invisible to it, exactly as it is invisible to the scalar count, and the kagome net’s whole line of zero modes is found by the supercell ladder rather than by this.

A difference of representations is a lower bound. A multiplicity of +k in a row guarantees k mechanisms in that representation and permits more, paired with self-stresses in the same row. Every framework here happens to realise its bound exactly, which is a fact about these nine frameworks and not a theorem.

Where the ladder goes next #

Back, to the count this one repairs: the count that promises a mechanism, where the subtraction is set up and its exceptions first appear, and to what a group does to a function, where representations are introduced as the thing a symmetry acts on.

Sideways, to the mechanism the kagome net actually has: a mechanism that is a wave, where the count is taken cell by cell and the answer turns out to be a line in reciprocal space rather than a number.

Onward, to what a forced mechanism does to a crystal: an order parameter is a representation, where a mode’s representation decides what it can couple to, and which modes a site can carry, which is the same decomposition applied to displacements rather than to mechanisms.