A seam with no bar missing

An edge that knows which side it is on cut a strip from a kagome lattice whose triangles had been distorted unevenly. Both long edges lose the same bars, so the count says they share the strip’s mechanisms equally, and they did not. At every wavevector along the strip, the two mechanisms went to one edge or the other, by a whole number read off one cell: the winding of its compatibility determinant round the Brillouin zone, the lattice’s polarization. The upper edge took 1 − P and the lower 1 + P, with P the polarization’s component across the strip.

That essay ended with a construction that separates the polarization from edges altogether. Join a lattice with one polarization to a lattice with the opposite one along a line, in the middle of a strip, and ask what the line holds. The line loses no bars. Every joint on it keeps its four, two to each side, so nothing about it is a boundary in the count’s sense. If the line holds anything, the polarization put it there.

It holds exactly the difference of the two polarizations. One way up that is two mechanisms per wavevector, and the other way up it is two states of self-stress.

What the count says about a seam #

Maxwell’s count for a planar framework is twice the joints less the bars. A kagome lattice has three joints and six bars in every cell, so it counts to nought. A strip cut from it counts to two per wavevector, because each long edge has lost one bar per cell. A constraint that has been said already put that count in its exact form: mechanisms less states of self-stress, the two null spaces of the compatibility matrix.

A seam changes neither term. The strip here is sixty cells wide. Its lower thirty rows have every up-triangle’s joints moved along the triangle’s sides by 0.3, 0.3 and −0.3, and its upper thirty by the opposite, −0.3, −0.3 and 0.3. The bars between row 29 and row 30 join a joint placed by one distortion to a joint placed by the other. They are ordinary bars of slightly unusual length, and none is missing. So the strip still has two more mechanisms than states of self-stress at every wavevector, and the count has nothing to say about where they are.

The polarization says where. The bottom edge of the lower lattice takes 1 + p, from the edge essay. The top edge of the upper lattice takes 1 − p′. Their sum is two, less the difference p′ − p, and since the total is fixed at two, the seam takes p′ − p. With p = −1 below and p′ = +1 above, that is two: the seam holds two mechanisms and the edges none. The other way up it is −2, which a region can hold only as a surplus of states of self-stress: two tensions on the seam, and two mechanisms on each outer edge to keep the total at two.

Measured one wavevector at a time #

The strip is periodic along its length, so its compatibility matrix splits into one small complex matrix per wavevector. The mechanisms at a wavevector are that matrix’s null space, and each is weighted by the rows of joints it moves. The states of self-stress are the null space of its conjugate transpose, weighted by the rows of bars that carry them. Each set is summed over three bands: the bottom quarter of the strip, the middle half with the seam in it, and the top quarter.

Every row agrees with the prediction. With the lower lattice at −1 and the upper at +1, in either of the two pairs of distortions that give those polarizations, the seam holds 2.00 per wavevector and each edge 0.00. The other way up, the seam holds −2.00 and each edge 2.00, and the strip has four mechanisms and two states of self-stress at every wavevector. With the two lattices’ polarizations equal, the seam holds 0.00 and each edge 1.00, which is the undistorted strip’s even split. In every row there are exactly two more mechanisms than states of self-stress, as the count requires.

Row by row, the states sit where the bands say and decay away from where they sit. The seam’s two mechanisms peak on the rows either side of the seam and fall into both lattices by a factor of ten every 2.6 rows. The seam’s two self-stresses, the other way up, have the same profile in the bars, while the four mechanisms rise towards the two outer edges. With both halves alike, one mechanism falls away from each edge and nothing is left on the seam but rounding.

Tension with no load, where no bar is spare #

The seam’s self-stresses are the result the count could least have predicted, so they are worth looking at directly.

A state of self-stress is a set of bar tensions in equilibrium at every joint with no external load. At a single wavevector it is a wave of tension along the seam, each cell’s bars carrying the same pattern shifted in phase from the next; adding the wave at q to its partner at −q gives a real pattern of tension and compression that repeats along the seam at the wave’s length. Usually one shows a bar that is not needed, the second diagonal in a braced square or a bar closing a triangle that was already rigid. Here nothing is spare. Every joint on the seam has the four bars every kagome joint has, and the lattices on either side are Maxwell’s balanced count cell by cell. The tensions exist because the two polarizations disagree about which side owes the other a bar. Locally, each lattice behaves as though its edge were cut and the mechanisms should be at the seam, and the two claims cancel into a redundancy.

That has a practical reading. A self-stress is a way to pre-tension a framework, and a framework with a state of self-stress along a line can be made stiffer along that line by tightening its bars, with no reaction at the supports. Twelve bars and a symmetry found a mechanism in a framework whose count said none. This is the complementary case: a redundancy along a line that the count says is exactly balanced.

The same bookkeeping, done locally #

The result is a local version of a rule that framework analysis uses globally. What decides whether it moves introduced the count and its failure. A constraint that has been said already repaired the failure: mechanisms less states of self-stress is exactly the count, for the whole framework, with no exceptions. What the global rule cannot do is apportion. A framework whose count is two can have two mechanisms, or three mechanisms and one self-stress, or a hundred of each, and the rule is satisfied in every case.

The polarization turns the global rule into a local one. Every region of a polarized lattice has its own balance, read off its boundary: an edge owes one plus or minus the polarization per cell, and a seam owes the difference of the polarizations on its two sides. The balances add up to the global count, which is why the strips above always have exactly two more mechanisms than self-stresses in total. They also say where the difference sits, which the global count cannot.

A shared load is a shared motion met the same bookkeeping on a platform held by two legs of three joints each. There the count was nought, so every redundant load path the two legs shared came with a freedom, one for one. The seam is that statement spread along a line. Its count is nought, since no bar is missing or added, and its polarizations decide whether the zero is two mechanisms or two self-stresses, with the edges taking the other side of the balance. In both cases a count of nought is not an absence. It is a balance that geometry can break in either direction, and does.

What a designer could do with a seam #

The edge essay ended its practical section by noting that a polarized lattice lets a designer choose which edge of a sheet is soft. The seam adds a second choice: where in the middle of a sheet the soft line or the pre-tensioned line runs.

A seam with two mechanisms per wavevector is a line along which the sheet can buckle or crease at no cost to first order while both halves stay stiff. A seam with two self-stresses per wavevector is a line that can be pre-tensioned by tightening bars on it, with no reaction needed from the sheet’s supports, and a pre-tensioned line stiffens a framework against the motions its tension resists. Which of the two a seam is depends only on which lattice is on which side. The same pair of distortions gives the soft line one way up and the tense line the other.

The seam can also be moved. A polarization changes only where a family of bars lines up straight, and a distortion field that passes through straight between two regions puts the seam wherever it passes. A sheet whose distortion could be set locally, by an actuator on each triangle or by a swelling material, could move its soft line by changing where the distortion changes sign. The cell that repeats for ever treated a lattice’s cell as the thing a designer specifies. The seam makes the arrangement of cells part of the specification, because what a line holds depends on the cells on both sides of it.

None of that is measured here beyond the counts and the profiles. The practical claims need finite motions, elasticity and a way of changing the distortion, and all three are outside what this essay computes.

Every wavevector but the longest #

The counts in the table are taken over five of the strip’s seven wavevectors, from π/2 to 3π/2, and the two left out are left out for a reason worth measuring.

At π/4 and 7π/4, the wavevectors nearest the centre of the zone, the seam-down strip shows no states on the seam and just one mechanism at each edge. The seam-up strip still shows its two, slightly spread. The reason is the decay length. Every localized state here decays away from its seam or edge over a length that grows without bound towards the zone’s centre, because a very long wave is barely distinguished from a translation of the whole lattice, which is a mechanism everywhere. At π/4 that length is already long enough for the seam’s states and the edges’ states, thirty rows apart, to reach one another. They pair up, and the strip shows only the two states the count forces.

So a seam’s count is a statement about wavevectors whose states fit between the seam and the edges. That is the same caveat the edge essay needed, and it applies more strongly here because a seam has two neighbours, one edge on each side, instead of one.

Exact only in a strip wide enough #

The seam-down strip has a sharper version of the same effect even at the wavevectors that count. Its edges want two mechanisms each and its seam two self-stresses, but a finite strip has only the two mechanisms the count guarantees exactly. The others are states that would be exact in a semi-infinite lattice and are only nearly exact here, because the edge’s extra mechanisms and the seam’s self-stresses reach one another across the thirty rows between them.

The measure is the smallest pivot of a complete-pivoting elimination that is above rounding: the size of the smallest combination of rows that is not quite nought. It falls by a factor of ten for every 11.2 cells of width, from 2 × 10⁻³ at 30 cells to 6 × 10⁻⁷ at 70. Across half the width that is a decay length of 2.44 rows in each state’s amplitude, which matches the profile’s factor of ten every 2.6 rows in weight, since weight is amplitude squared. The seam’s states and the edges’ are the same states seen from two ends of the gap between them.

This is why the table’s counts use a tolerance. A state is counted as a mechanism or a self-stress when its pivot is below a thousandth of the largest. At sixty cells the paired states sit at 5 × 10⁻⁶, well inside that tolerance, and the bulk’s smallest pivots sit near 0.7, well outside it. The count does not depend on where the tolerance is placed between those two values. It would depend on it in a strip half as wide.

Why the distortions are 0.3 #

The edge essay moved each joint by 0.1. The seam essay moves them by 0.3, and the reason is the same decay length. At 0.1 the lattice is closer to the undistorted kagome, whose straight lines of bars carry states along their whole length, and the localized states decay over about four rows in amplitude where at 0.3 they decay over two and a half. That difference is large across thirty rows. In a strip sixty cells wide at 0.1, the seam-down strip’s paired states are exact only to 2.6 × 10⁻³, above the tolerance the table counts at, and the count would come out wrong for a reason that has nothing to do with the seam. At 0.3 they are exact to 5 × 10⁻⁶ and the three regions are well separated.

The polarizations are the same at 0.1 and 0.3; they are whole numbers that change only where a family of bars lines up straight, and nothing between the two distortions does that. What changes is how much room the states need. The count says how many and not where found the straight kagome’s mechanisms spread through the whole of a patch. The polarized lattices put them on lines, and a larger distortion puts them on narrower lines.

What this does not settle #

Seams at an angle. The seam here runs along the strip’s periodic direction, where the polarization’s second component counts. A seam cut along another lattice direction reads a different component, and a seam at a general angle reads a combination. Only the one direction is measured.

Finite motions. Every mechanism here is infinitesimal, a null vector of the compatibility matrix. Whether the seam’s mechanisms extend to finite motions, and whether the seam’s self-stresses survive when the lattice is loaded and deforms, is not addressed. It moves to first order and not at all is the standing warning that the two can differ.

Stiffness. A state of self-stress can pre-tension the seam, and how much stiffness that buys depends on the bars’ elasticity, which nothing here models.

Still open: a seam that turns a corner #

A seam can be closed into a loop: an island of one polarization in a sea of the other. Its boundary then runs along every lattice direction in turn, and each stretch of it reads its own component of the polarization difference. The counts along the different stretches need not agree, and the island’s total must still satisfy Maxwell’s count, which for a closed seam in an infinite lattice is nought.

The distinct argument there would be an island of polarization (1, −1) inside a lattice of (−1, 1), solved as a finite patch rather than a strip. Two things would come out of it: where the island’s boundary holds mechanisms and where it holds self-stresses, stretch by stretch, and whether the corners, where one stretch meets the next, carry something of their own. If the stretches’ counts cancel round the loop, the corners carry nothing. If they do not, the corners must hold the difference, and a corner is a single joint.