A dislocation that closes

The defects lie on rings measured something that looks like a familiar object, and said so. Every cell of a golden head that is not a hexagon sits on one of a sequence of rings a factor of $\varphi$ apart, each ring is a closed chain of five-sided and seven-sided cells, and every five is bound to a seven with a margin no shuffled null comes near. In a crystal that object has a name, a low-angle grain boundary, made of dislocations. The essay that found the rings wrote that name down, and added that the useful part of it was that it came with an expectation which could be checked. The pairing was checked. The rest of the expectation was not.

A dislocation is a stronger claim than a five next to a seven. A five–seven pair is how a dislocation appears in a triangulation, but what makes the thing a dislocation is what it does to the lattice around it: there is an extra half-row of atoms ending at it, and a lattice is forced to strain and rotate to accommodate that half-row at any distance. A grain boundary is a line of them, and it comes with a misorientation, a real rotation of the crystal on one side against the crystal on the other. Those are all things that can be read from the positions of the organs. If a defect ring is a grain boundary, they should be there.

The test that defines the object #

The textbook test for a dislocation is a Burgers circuit. Walk a closed loop through the crystal, one lattice step at a time, and write down each step as a whole-number combination of the two lattice vectors. In a perfect crystal the steps sum to nothing, because a closed loop ends where it began. Around a dislocation they sum to a lattice vector, the same one however large the loop is, because the extra half-row has to be crossed. That vector is the Burgers vector, and it is the defect’s identity. Nothing about it depends on the number of sides of any cell, which is what makes it a separate test.

Run on a head, the circuit needs a reference lattice, and that has to come from the organs themselves. The patch fit that measured how much of a head is one lattice provides it, and it was built to ignore each organ’s index, so no consistency is being assumed by labelling organs according to how they were generated. The control is a real dislocation: a triangular lattice with each point moved by the textbook elastic field of one edge dislocation, Poisson’s ratio 0.3, and then read by exactly the same steps as a head, its Delaunay triangulation and all.

The two panels are hard to tell apart by eye. Both have a five and a seven side by side; both are surrounded by eight organs; both sit in something that looks like a slightly disturbed triangular net. The first misses by exactly one lattice vector. The second closes.

A loop that cannot hide anything #

The choice of loop matters here, and the first version of this measurement chose badly. It walked round a pair through the organs between 1.2 and 2.3 spacings away, which is a natural radius and on the crystal returned the right answer. On a ring it returned closure on nearly every pair, and that proved less than it appeared to. The pairs on a ring are 1.81 cells apart, so a loop of that size encloses two or three of them, and two dislocations of opposite sign enclosed together close a circuit perfectly well. A wide loop on a ring cannot say whether it found nothing or found things that cancel.

The tightest loop can. It uses the organs that touch either the five or the seven, and only those, in order round the pair. There is then nothing inside but the pair, and no neighbour’s vector to cancel against. It is eight organs long on the crystal and eight on the head.

On the crystal’s pair the tightest loop misses by one lattice vector. On the head it closes every time, on every ring: 29 of 29 five–seven walls at radius 11.27, 46 of 46 at 18.24, 75 of 75 at 29.51, 122 of 122 at 47.76. That is 272 walls and no failure, on a loop that cannot be fooled by cancellation.

The rounding is safe on those loops. On the outer two rings no step is further than a tenth of a lattice unit from a whole number, and even at 11.27 no step reaches the half at which rounding could go the wrong way, 0.44 at worst. Those remainders fall with radius as the head’s curvature does.

The stronger statement #

A loop is one path, and a stronger question can be put. Is there a single consistent labelling of a whole patch: a lattice site for every organ, such that every Delaunay edge steps between its two organs’ sites by exactly the lattice vector its own length rounds to? If there is, then every loop drawn anywhere inside that patch closes, whatever it encloses. A dislocation forbids this. Somewhere there has to be a cut across which the labels jump. In the crystal’s patch 26 of 76 edges refuse the labelling.

On the head’s ring at radius 29.51 not one does: 0 of 5,571 edges across seventy-five patches, each patch centred on a pair and holding several. At 47.76, 0 of 8,811. So through a band that the triangulation marks as a chain of defects, the organs sit on one lattice, with no cut, and the Burgers vector of anything in that band is zero. On the two inner rings the labelling does break, 176 of 2,074 edges at 11.27 and 191 of 3,348 at 18.24, and the cause is measurable: a patch three spacings in radius at those radii is past the size at which any straight lattice holds. Patches halfway between two rings, with no pair in them at all, break more often and for the same reason: 298 of 3,263 edges at radius 14.3, and 164 of 5,561 at 23.2, where the patch over the ring further out broke none.

Read out to a wider circuit, the crystal’s vector stays the same at 3, 4 and 6 spacings, as a Burgers vector has to. On the head all 75 close out to 4 spacings. At 6 only 11 do, since by then the reference lattice has been left behind by the head’s own bending, which has nothing to do with the pair at the middle. The crystal has no such limit because its far field is a lattice that goes on being the same one.

The residual agrees #

A real dislocation also leaves a mark in how well a lattice fits round it. Centred on the crystal’s pair, a patch three lattice spacings in radius misses by 0.214 of a spacing, however far it is from anything else, because the half-row has to be accommodated. So does a patch on a head contain a similar excess when a pair is inside it? It does not. At radius 29.51, a patch over a pair misses by 0.145 of a spacing, and the law for a head’s curvature, with no pair anywhere in it, predicts 0.142. At 47.76, 0.089 against 0.088. A patch halfway between two rings, with nothing in it, at radius 23.2 reads 0.191 against 0.181 and at 37.5 reads 0.110 against 0.112. Every point sits on one curve. Put simply, the fit cannot tell whether it is sitting on a defect ring.

Note how close this came to a wrong conclusion the other way. The head’s residual and the crystal’s are the same order of size, and at radius 18 the head’s is the larger, 0.232 against 0.214. A reading of residuals alone might have called the two equal, or called the head worse. It takes the circuit, or the curve with its law, to separate a residual that belongs to the object from one that belongs to the coordinates.

No rotation at the ring #

A low-angle grain boundary comes with a formula, due to Frank and to Read and Shockley: dislocations of Burgers vector $b$ a distance $D$ apart produce a misorientation of about $b/D$ across the line. The ring at radius 29.51 holds 55 fives on a circumference of 185, so $D$ is 3.37 in the head’s units, and a Burgers vector of one spacing is 1.86. The boundary should rotate the lattice by 0.55 of a radian, thirty-two degrees, in one step. That is not a low angle at all, which on its own should have raised a question.

The head’s lattice does turn against the radius, and quite fast. But it turns continuously, at a rate that has no feature at any ring. Family 55’s vector makes 55.5° with the radius at radius 28 and 60.7° at radius 31, about five degrees in three units of radius, and the same smooth curve runs through 29.51 as through 23 and 35. That curve is computed from the angle alone. With no step in it, there is no misorientation for a formula to predict. Every reading of a lattice change across a ring, the transition in the counts included, is a smooth change that the ring happens to sit on, as the rings’ offset from the transitions already showed.

What a five next to a seven is, then #

If the organs do not know the ring is there, the ring has to belong to something other than the organs’ positions, and it does. Side counts are a property of the triangulation, and a triangulation is a choice among possible triangulations made by a rule. A lattice cell is a parallelogram, and to triangulate it, one diagonal has to be chosen. Delaunay’s rule takes the shorter. A long way inside a band of constant pair, one diagonal is shorter for every cell in the band and every cell has six neighbours. Then, moving outward, the lattice’s shape slowly changes until the two diagonals are equal in length, at which point the cell is a rectangle and its corners share one circle. Beyond that radius the other diagonal is shorter.

Since organs arrive in order and the shape changes with radius, cells do not change over all at once. For one stretch of consecutive organs the two kinds of diagonal coexist, and where a cell with the old diagonal adjoins a cell with the new one, the organ at their shared corner has five or seven triangles round it instead of six. That boundary is the chain of fives and sevens.

So the pair is not a dislocation in the lattice. It is the meeting point between two choices of diagonal on a lattice that has no defect in it. The wall that is about to change hands is identifiable by its label: on the ring at 29.51, the walls nearest to changing are all between organs 34 apart or 144 apart, which are the two diagonals of the cell spanned by 55 and 89. At 47.76 they are 55 and 233, at 18.24, 21 and 89. Each ring is the place where one diagonal gives way to the other.

How near the walls are #

If a ring is a place where a diagonal is changing hands, then its walls should be measurably close to changing hands, and the measure is simple: the difference in length between a wall and the other diagonal of its cell, over the spacing. Sorted, the nearest walls near a ring make a straight line from nothing. The last of the nearest 55 at radius 29.51 is 0.0375 of a spacing from flipping, the last of the nearest 89 at 47.76 is 0.0231, the last of the nearest 34 at 18.24 is 0.0597, and the last of the nearest 21 at 11.27 is 0.0942.

Multiply each by the ring’s own count and divide the position by the same count, and the four curves lie on one line ending at 2. The $k$-th nearest wall to flipping on a ring with $F$ fives is $2k/F^2$ of a spacing away, on four rings whose radii differ by a factor of four. That is the ring’s whole geometry, and it is easier to think with than a chain of dislocations: consecutive organs of a family, each a tiny step closer to the flip than the last.

What this says about the count law #

That line may also explain a law that had no explanation. A ring keeps its count after it loses its order found a ring’s count of cells surviving displacement up to about three over the family entering it, for reasons that essay did not go into, and a Lucas ring gives way where a golden ring of its size does found the same rule at another angle. Now there is a natural scale of the right form. The ring where 144 enter has its 55 walls within 0.0375 of a spacing of flipping; its count survives to about three over 144, 0.0208. The two differ by a factor of 1.73 to 1.80 at all four rings, so they are one law with a constant between them.

And the constant is not mysterious. Comparing a wall and the other diagonal is a comparison of two lengths, and if each of four organs moves independently by $\sigma$ in each coordinate, the difference of the two lengths moves by about $2\sigma$. Set $\sigma$ to three over the family. On the ring where 144 enter, $2\sigma$ is 0.0417 against a margin of 0.0375, and the ratio is 1.11 to 1.16 across the four rings. So a displacement of three over the family is about the displacement at which the noise in comparing a cell’s diagonals becomes as large as the reach of the ring: the whole of a ring’s walls come into play.

That is a suggestion and not a derivation, and it gives no account of why the count, in particular, should last until then and no further. But it does replace an unexplained three with a geometric reason why the number is of that size, and it predicts something that the dislocation picture could not: the tolerance scales with how near the cells are to square, and it does not depend on how large the ring is except through the family.

What the name was worth #

The grain-boundary reading was reasonable. The object looks right, it has the right pairing, and two of the measurements that could have told it apart had been done in the wrong order. It is worth saying what survives. The rings are still real, the charge count is still exact, and every five still touches a seven; those are statements about the triangulation and the triangulation is what they measured. A statement that a five–seven pair is a dislocation, in the sense of a strain field or a Burgers vector or a slip plane, is the part that falls. It should not be used to reason about mechanics. There is no half-row to move and no slip to happen. A real dislocation moves by carrying a strain field through a crystal; these walls change hands with no movement to speak of, and at each flip an organ’s neighbour count changes while the organ itself has not moved at all.

The distinction has a practical edge. In any real tissue the neighbour relation is physical, with walls and contacts, and a transition of that kind costs something. A tissue would not settle on a triangulation by Delaunay’s rule, and which of two near-equal contacts an actual meristem makes is a question the arithmetic of a point set does not answer. The chain of fives and sevens on a real head, if it exists, would be a statement about that decision, and about the head’s physics rather than its lattice.

What this does not establish #

It is measured on Vogel’s rule with exact organs and at the golden angle. On a head grown organ by organ, or displaced, a five–seven pair could carry a real Burgers vector, and it would not be surprising if some did: a displaced head has real strain in it. Nothing here says that no real tissue contains real dislocations. It says the rings of an exact head are not made of them. A head at a noble angle other than the golden, or at 137.3° where the rings are arranged differently, has not been put through this test, though the argument is not angle-specific. The elastic field used for the control is a continuum solution applied to a discrete lattice, and it is accurate enough here only because the question it serves is whether a loop closes.

Measurements that would overturn it #

A tightest loop on a head’s ring failing to close by a whole lattice vector, at a radius where rounding is safe. A labelling that broke on a pair and not at the same radius on a patch without one. A step in any family’s angle against the radius at a ring. Walls near a ring that did not lie on one curve when scaled by the ring’s own count. Any of those would put a real dislocation back into the picture, and the first would be enough.

Still open: the count, computed from the margins #

The line of margins is a precise statement. Organs on a ring sit at known, evenly stepped distances from flipping, and a displacement of a stated size perturbs each comparison by a known amount. In principle that is enough to compute how a ring’s count of cells falls with displacement, rather than reading it off a bank of displaced heads. The next measurement builds that prediction from nothing but the margins and a stated noise: how many walls change hands at each displacement, how many cells of each kind that makes, and whether it recovers the three over the family, the order of the losses and the trade of one pair for two hexagons that the ring’s count law measured by displacing heads directly. If it does, that law has a derivation. If it comes out at another number, whatever the margins omit is what the law is about.