An edge that knows which side it is on

The count says how many and not where cut rhombuses out of a kagome lattice. The kagome has three joints and six bars in every cell, so it counts to nothing. A patch cut from it has exactly as many mechanisms as its edge has lost bars, and the rank agrees at every size. Where the mechanisms sit depended on the lattice’s geometry. A straight kagome, whose bars line up into lines across the patch, kept nearly half its mechanism weight in the middle. A twisted one, with every up-triangle turned by the same angle, kept a twentieth. In both, the weight was spread round the edge roughly as the missing bars were.

It ended by naming a distortion for which that last part should fail: move each up-triangle’s joints unevenly, not all by one turn. Then one edge of a strip should carry more mechanisms than its missing bars account for and the opposite edge fewer. If that imbalance is a whole number that survives smooth changes and jumps only at straight configurations, it is a property of the cells, not of the edge.

That is what happens, and the number can be read off one cell.

A strip, and what the cut says #

The strip is ten cells wide and periodic along its length, so it has two long edges and no ends. Each long edge has lost the same bars: the down-triangles that would have joined the strip to the lattice above and below. By the count, the strip has as many mechanisms as those lost bars — 24 on a strip twelve cells long, two of them translations — and nothing in the count distinguishes the two edges.

The distortion is the simplest one with the unevenness in it. Each up-triangle’s joint number k is moved a distance $x_k$ along the triangle’s side that leaves it anticlockwise. With all three moves equal, every up-triangle turns and shrinks alike, and this is the twisted kagome of the patch essay again. With the moves 0.1, 0.1 and −0.1, the third joint of every triangle goes the other way.

The drawing is at that second setting, and the joints are drawn by weight: the diagonal of the projector onto the mechanisms, which does not depend on how the mechanisms are written down. Nearly all the weight is on the top edge. The top half of the strip carries 19.99 of the 24 mechanisms and the bottom half 4.01. Move the slider to a third move of +0.1 and the two halves carry 12 each, exactly.

One wavevector at a time #

A strip that repeats along its length can be taken apart by wavevector, the way a vibrating string is taken apart into its harmonics. At each wavevector q along the strip — here the twelve values 2πm/12 — the strip’s compatibility matrix becomes a small complex one, one row per bar in a column of cells and one column per coordinate in it. Its null space is that wavevector’s mechanisms. There are exactly two at every wavevector: the count, one per edge. At q = 0 those two are the strip’s two translations, which move every joint alike and belong to neither edge.

Wavevector by wavevector the picture is sharp. With the three moves of one sign, the two mechanisms’ weight falls away exponentially from both edges, one mechanism on each. With the third move reversed, it falls away from the top edge only: 1.983 of the two are in the top half of a strip twenty-four cells wide. With the first two reversed instead, it falls away from the bottom only. A mechanism decays into a lattice exponentially and at a rate set by the geometry, so its share of either half settles on a whole number as the strip widens.

At widths from four to forty cells, the top half’s share is 1.000 at every width for moves of one sign. For the reversed third move it runs 1.36, 1.66, 1.83, 1.92, 1.98 and 2.00 to 1.9993 at forty cells, and for the reversed first two it runs down to 0.0007. The two uneven strips are not sharing their mechanisms in some new proportion. Each gives one edge both and the other none.

The drawing’s 19.99 of 24 is the same statement summed over wavevectors, with two things added. The two translations, at q = 0, are spread evenly over the strip, one in each half. And close to q = 0 the decay length grows, so the wavevectors nearest it leak across the middle of a finite strip. With eleven wavevectors carrying two each on top and the translations split, an infinitely wide strip would put 23 of the 24 in its top half; at ten cells wide the leak near q = 0 accounts for the difference between that and 19.99.

The number in the cell #

The strip was one route: build an edge and look. The other route never builds an edge. One cell of the infinite lattice has three joints and six bars, so at each pair of Bloch phases ($\theta_{1}$, $\theta_{2}$) its compatibility matrix is six by six. Where the determinant of that matrix vanishes, the infinite lattice has a mechanism of its own with that wavevector. Everywhere else it has none, and the determinant is a complex number that is not nought.

Take $\theta_{1}$ once round from 0 to 2π with $\theta_{2}$ held fixed. The determinant goes round a closed loop in the complex plane and winds a whole number of times about the origin. Do the same with $\theta_{2}$. Those two windings cannot change as the distortion is varied unless the loop is forced through the origin, which is to say unless the lattice acquires a mechanism of its own somewhere in its zone. They also depend on how the cell was drawn, so they are only meaningful against a reference. Here the reference is the twisted kagome, whose windings are 2 and −1, the same for either sign of twist.

The eight sign patterns give three kinds of answer. All three moves of one sign give the twisted kagome’s windings, so no polarization. The six mixed patterns each give a polarization that is not nought: (1, −1), (−1, 1), (−1, 0), (1, 0), (0, 1) or (0, −1). Its second part is the one that counts on this strip’s long edges, which run along the lattice’s first direction. The prediction is that the top half holds one less than that second part: two where it is −1, none where it is +1, one where it is nought. The strip, computed without the determinant, matches it at every one of the eight patterns to 7 × 10⁻⁴, and that residual is the forty-cell strip’s finite width.

Two of the six mixed patterns have a second part of nought. For those the strip’s long edges share equally, as the uniform ones do, but the lattice is still polarized, along the strip rather than across it. A strip cut the other way — its edges along the second lattice direction — would see their first part instead. No single strip sees everything. The pair of windings does.

Where the number can change #

A whole number can only change by jumping, and it can only jump where the determinant passes through nought. That is a mechanism of the infinite lattice itself, a bulk mode that belongs to no edge. The open question guessed that this happens at the straight configurations, where the patch essay found mechanisms reaching all the way in.

Along this path the determinant stays clear of nought except at one point. Its smallest magnitude off the zone’s centre is about 5 × 10⁻⁴ at either end and falls smoothly to 6 × 10⁻⁵ a hundredth from the middle. At a third move of exactly nought it is 3.5 × 10⁻¹⁹. The polarization is (1, −1) at every step below nought and (0, 0) at every step above.

What is special about nought is visible in the lattice. Each of the kagome’s straight lines of bars passes alternately through two of a triangle’s three joints. The lines of the third family run through joints one and two, and joint one is moved along that very line. With joint two not moved at all, every line of that family is straight again. So the step happens exactly where one family of bars lines up, and nowhere else on the path. The same holds for the other two moves. The planes where any one move is nought are the straight configurations, and they divide the space of distortions into the eight octants of the table.

That is the answer the question anticipated. The imbalance is a whole number — one mechanism per cell of edge, in either direction. It does not change as the distortion is varied smoothly within an octant. It changes only as the lattice passes through a configuration with a family of straight bars. And it is carried in the cells: the determinant that gives it has no edge in it at all.

Why the count could not see it #

Every count in this field so far has been a difference: freedoms less constraints, joints less bars. A count is the same for every geometry of the same graph, and the strip’s two edges have the same graph. So no count can tell the edges apart, and the patch essay’s count said nothing about where the mechanisms were. Reading it as if it did is the first of six things a network is not: a count that is right about a difference, read as an answer about places.

The winding is a different kind of number. It is still an integer and it is still robust, but it is computed from the geometry — the directions of the six bars in one cell — and it can change when the geometry passes through a special configuration. The count answers how many. The winding answers which side, and it answers it from the same one cell that the repeating-cell essay used to count the lattice’s freedoms. That essay counted what the cell allows the lattice as a whole; this counts where the cell sends what is left at an edge.

There is a way of putting the two together that does no violence to either. The count assigns each edge its lost bars. The polarization moves a whole number of them per cell from one edge to the other, and the total is conserved because the strip’s mechanisms are still exactly what Maxwell’s count says they are. The top edge’s 23 and the bottom edge’s 1, for a strip twelve cells long, are the cut’s 12 and 12 with 11 moved across. The twelfth wavevector, the one that carries the translations, stays shared.

The other null space stays empty #

A count of mechanisms is only exact when no bar is redundant. A constraint that has been said already put it as two null spaces. The mechanisms live in one. The other holds the combinations of bar tensions that balance with no load, the states of self-stress, and the count is really mechanisms less self-stresses. A straight kagome has self-stresses in abundance: every straight line of bars can be put in tension end to end, and that is exactly why the patch essay’s straight lattice had mechanisms reaching into its middle.

The polarized strips have none. At every wavevector the compatibility matrix has full rank in its rows, so every bar is independent, and the count of mechanisms is exactly the count of lost bars — two per wavevector, twenty-four on the strip. That is what makes the imbalance a statement about where, not about how many. Nothing has been added and nothing taken away. The polarization moves mechanisms from one edge to the other with the total held by a count that is right about a difference, and the count never had anything to say about the difference between the edges.

It is also why the straight configurations are where the number can change. A family of straight bars carries a self-stress along every line, and a self-stress in the bulk is the other face of a mechanism in the bulk: the same zero of the determinant, read along rows rather than columns. Crossing a straight configuration opens that pair for an instant, and a whole mechanism per cell can pass from one edge to the other through it. Away from it the bulk is tight in both directions, and the edges have nothing to trade through.

A number from geometry, not from a graph #

Which diagonal rigidifies a grid found that the answer to a question about bars was a fact about a graph: whether the grid’s rows and columns form a connected graph. The polarization is the opposite kind of fact. The strips at +0.1 and −0.1 have the same graph — every bar joins the same two joints in both — and differ only in the directions of the bars. The count and the name disagreed only where geometry was special; here geometry that is not special at all, anywhere in an open octant, decides something no count can.

What a designer could do with a side #

A structure built from a polarized lattice has a soft edge and a stiff one, and which is which is decided by the sign pattern of a distortion in every cell rather than by anything done at the edge. Cut two strips of the same material and flip one over, and the soft edge follows the material. Join two regions of opposite polarization and the seam between them either collects mechanisms from both sides or is left with none, depending on which way the two polarizations point.

That is the sense in which the edge knows which side it is on. It does not know from its own bars, since the two edges are cut alike. It knows from the cells behind it, and it knows exactly, as a whole number that no smooth change can alter.

What this does not settle #

One family of distortions. Every lattice here moves each up-triangle’s joints along the triangle’s own sides. A general distortion also moves the down-triangles independently. That is a larger space with the same eight octants in it, or not; the straight configurations are still where the windings can change, but whether they still divide it into octants is not measured.

Small distortions. The moves are 0.1 against sides of length 1. Larger moves bring the joints of neighbouring triangles close, and the lattice can reach a configuration where two bars cross or a triangle folds flat. Nothing here follows the windings that far.

Finite strips near q = 0. Towards the zone’s centre the edge mechanisms’ decay length grows, and a strip of any finite width shares those wavevectors between its edges. The summed counts in the drawing are therefore short of whole numbers by an amount that shrinks only as the strip widens. The per-wavevector shares, away from the centre, are the whole numbers.

Still open: a seam between two polarizations #

Join a lattice with polarization (1, −1) to one with (−1, 1) along a line, and the seam has one side that each region sends its mechanisms towards.

The distinct argument there would be the seam’s own count. The polarization says mechanisms per cell are moved towards the seam from both sides, or away from it from both, so a seam should collect two per cell of its length, or lose them and be left with states of self-stress in their place. That is a statement about a line in the middle of a lattice, where no bar has been lost at all. The measurement would be a strip made of two halves with opposite distortions, the weight across it at each wavevector, and the stresses the seam can carry against the ones the count allows.