Every five is bound to a seven

A five-sided cell next to a seven-sided one is two exceptions in a tiling and one object in a crystal. In a crystal it has a name — a dislocation — and the two halves are bound in a precise sense: the extra half-row of atoms that terminates at the seven is the same half-row the five is missing, so neither can be moved or removed without the other. The phyllotaxis literature borrows the word freely, and calls the places where parastichy numbers change dislocations, usually without measuring anything about them.

The borrowing is testable. A head’s exceptional cells are already counted, and asking whether each five has a seven beside it is a question with a null attached.

What binding would have to mean #

Three things, and only the first is usually asserted. That a five has a seven adjacent to it rather than merely somewhere nearby. That the distance from a five to its seven is one number rather than a distribution with a tail. And that the pair has an orientation, so that the two ends are distinguishable and the same end points the same way across the head.

A collection of scattered fives and scattered sevens satisfies none of those and can still look convincing in a picture, because a head that is 83.8 per cent hexagons has its exceptions close together whatever they are doing.

The strong null #

The one used throughout here leaves every defect exactly where it is and shuffles only which of them is a five and which a seven. It is a seeded permutation over 200 draws, and it knows everything about the head except the pairing: it knows how many defects there are, it knows that they sit on rings, and it knows the ring radii, because it has not moved a single cell.

What it does not know is which label belongs where. So the difference between the measurement and this null is pairing and nothing else.

What the measurement returns #

100.0 per cent of the five-sided cells in the interior share a wall with a seven-sided one. The null gives 68.2 ± 4.4 per cent, which is z = +7.3.

Two thirds is a large null, and it is the reason the weaker one further down is reported at all. Defects that already lie on one-cell-thick circles are crowded whatever their labels, so a great deal of the adjacency comes free. What is not free is the last third, and the last third is the whole of the claim.

There is no unpaired tail #

This is the part a share cannot say. The distances from a five to the nearest seven run from 0.833 to 0.927 cell spacings and then stop. The farthest five in the entire head is a single cell width from a seven.

A head where nine tenths of the fives were bound and a tenth were loose would report ninety per cent and would have a tail running out to three or four spacings. There is no such tail here: the measured range is a tenth of a cell wide, and the loneliest five is closer to a seven than the average five of the null is.

The distance itself #

The median separation is 0.919 cell spacings and the mean is 0.914. The unit is the median length of a contact between two interior cells, which on this head is 1.923 model units, so a bound pair is very slightly closer than two ordinary neighbours.

That is what it should be if the pair is one object. Two cells sharing a wall are one spacing apart by construction; the pair being a few per cent tighter than that is the only room the measurement leaves for the binding to show in a distance, and the null’s average five sits at 1.254 spacings — well outside the range any measured pair occupies.

A control with no binding in it #

The head that settles the question is not a lattice at all. A Poisson set of 2,400 points in the same disc has 454 fives and 355 sevens, and 73.0 per cent of its interior cells are not hexagons.

What the Poisson head shows about arithmetic #

76.0 per cent of its fives touch a seven, against a null of 73.0, which is z = 1.3. That is no pairing at all, and it is the number the golden head’s result has to be read against.

When three quarters of a tissue’s cells are exceptions, a five touches a seven almost always, because a five touches five cells and three or four of them are exceptions by chance. The Poisson set gets there by density. The golden head gets there while 83.8 per cent of its interior is hexagonal, which is the opposite situation: a five there has five neighbours and almost all of them are hexagons, so having a seven among them is a fact about which cell it is next to.

The tail the control has and the head does not #

A quarter of the Poisson set’s fives have no seven beside them, and those carry a tail out to 3.18 cell spacings — three and a half times the widest separation on the settled head. Its median is 0.883, which is smaller than the golden head’s 0.919.

So the median is the statistic that cannot tell the two apart, and the worst case is the one that can. Reporting a mean and stopping is the recurring failure in this corner of the subject, and here it would have made a random point set look better bound than a lattice.

Which end points inward #

A bound pair has an orientation, and this is the measurement no share can make. Joining every five to the seven it shares a wall with, on a head of 4,000 organs, gives 223 pairs, and in 95.52 per cent of them the seven is the end nearer the centre.

Why the inner end is the seven #

The geometry says it must be. A seven-sided cell has more neighbours than its share and a five has fewer, so a seven is where an extra parastichy row begins and a five is where one ends. Rows are added outwards on this arrangement — the counted pair grows with radius — so the row that begins at the seven runs outwards from it, and the five that terminates the row sits on the outer side.

That is a prediction with a sign, and the sign is measured rather than assumed. A head where rows were being lost outwards would put the fives inside, and the same code would report it.

What the five per cent is #

Not noise, and not a second population. The pairs that read the other way are the ones whose axis lies nearly across the radius rather than along it, where a difference of a hundredth of a spacing in two radii decides which end is called inner. The share is a 95.52 per cent on a 4,000-organ head and 92.54 on the 2,400-organ one, over 134 pairs there and 223 here, and the difference between the two is the subject of a later section.

The tilt of the pair #

The pair does not point along the radius. Its axis lies a median of 31.428 degrees off the radius through it, with a tenth percentile at 30.43 and a ninetieth at 57.30.

What the tilt does not establish #

The distribution is lopsided — one degree from the tenth percentile to the median and 26 degrees from the median to the ninetieth — and a median with an asymmetric spread of that shape is a summary of two things rather than a measurement of one.

The honest reading is that a bound pair here is not radial and is not perpendicular to the radius either, and that a third of a right angle is where the bulk of it sits. What decides the exact angle is not measured. It would plausibly be the direction of the parastichy row that terminates, and testing that would mean labelling each pair with the row it belongs to, which is a different measurement on a different graph.

A dislocation is one object #

Every pair in the 4,000-organ head lies between 0.8605 and 0.9575 cell spacings, and on the 2,400-organ head between 0.8331 and 0.9271. The full range on either is under a tenth of a spacing.

That is what licences the singular. A quantity whose 223 measurements span a tenth of their own value is one object measured 223 times; a quantity spanning a factor of two would be a family of objects sharing a name. The distinction matters because the crystallographic word being borrowed carries the singular with it.

Along the ring #

The rings are not sets of independent defects. Each is a closed chain, and the distance between consecutive dislocations measured along it is 1.812, 1.810, 1.811 and 1.811 cell spacings on the four resolved rings.

A low-angle grain boundary #

Four rings whose radii differ by a factor of 4.2, agreeing on that spacing to 0.1 per cent, is a stronger statement than it looks. The rings hold 21, 34, 55 and 89 pairs respectively, so both the circumference and the number of links change by more than four, and their ratio does not move.

A closed chain of bound dislocations at a fixed spacing is what a crystallographer calls a low-angle grain boundary: the interface between two pieces of the same lattice rotated slightly against each other, where the misfit is taken up by a regular row of dislocations rather than by anything amorphous. That is a description with content, because the spacing of such a boundary and the angle it accommodates are related, and both are measurable here.

How many links a ring has #

The count on each ring is the parastichy family that survives its handover — 21, 34, 55, 89 — which is set by the lattice and not by the tissue. Nothing about cell shape or contact mechanics enters it.

That is worth stating plainly against the reading a grain boundary invites. In a metal the number of dislocations on a boundary is set by the misorientation, which is a property of how the two grains happened to meet. Here it is set by which rung of the ladder the radius sits on, and the arrangement has one grain.

The share rises with the head #

The adjacency is not 100 per cent at every size. Measured on the same arrangement at six head sizes it reads 77.8 per cent at 300 organs, 81.8 at 600, 87.3 at 900, 92.5 at 1,500 and again at 2,400, and 95.5 at 4,000.

Monotone across six sizes is not a coincidence and it is not a convergence to be admired either. It has a cause, and the cause is not about pairing.

Which makes it a statement about the rim #

A small head is mostly rim. The pairing is a property of the interior, and a cell adjacent to the boundary has its outward neighbours missing, so its side count is a fact about the cut rather than about the arrangement.

At 300 organs the interior holds 167 cells and 86 of them are defects; at 4,000 it holds 2,725 and 442 are. The share of the head that is interior grows with the head, and the adjacency follows it. That is the same confound the disorder statistic has, where a quantity measured on a finite head reports the head’s size as though it were the arrangement’s property.

What the rim cut does to the pairing #

Nothing, over the range where the cut is usable. The adjacency is 100 per cent at every rim from 0.70 to 0.96 of the radius, while the defect share over the same cuts runs from 25.6 per cent to 12.8.

The reason is structural rather than lucky. A bound pair is two adjacent cells, so a cut takes both of them or neither, and because the defects all sit on rings a cut placed between two rings removes only hexagons. At the head’s own edge the adjacency collapses to 60.7 per cent, and that is a fact about the edge rather than about the tissue.

Under disorder #

Displacing every organ inside a box and re-tessellating is the tissue field’s own knob, and it is the sharpest thing available for asking how much of a result depends on the arrangement being exact.

The pairing outlasts the rings #

At the site’s usual full disorder setting the head holds 660 defects rather than 264, the adjacency has fallen to 92.6 per cent, and the pairing is still at z = +6.3 against its own null. Ring confinement at the same setting has already fallen to 44.7 per cent, and it reaches its null at 0.31 spacings of displacement while the pairing holds to 0.62.

So where a defect is and what is beside it are two results with different robustness, and a head disordered past this site’s usual setting has kept its dislocations and lost its rings. That separation is the load-bearing part, because from a picture the two look like one phenomenon.

What a permutation null can say #

It can say that the labels are not exchangeable. It cannot say why, and it cannot rule out a cause that acts on positions rather than on labels.

Both nulls here are seeded, so both are reproducible, and both are permutations of the observed multiset rather than draws from a fitted distribution — which is what makes the comparison free of any assumption about what the defect counts should have been. What neither does is supply a mechanism. The measurement establishes that a five and a seven are found together far more often than chance; the reason is in the lattice geometry and is not tested here.

The weaker null, and why it is reported #

The second null scatters the whole multiset of side counts over the interior cells at random. It gives 37.7 per cent adjacency, z = +15.2, and a mean nearest distance of 1.786 spacings.

That number is twice as impressive and worth half as much. It cannot separate pairing from ring confinement, because a null that moves the defects off the rings is being beaten partly by the rings. It is reported because a reader shown only the weaker one would credit the pairing with what the ring structure is doing, and reporting both is the only way to make the difference visible.

The instrument had to be able to return nothing #

A pairing statistic that always finds pairing is not a measurement. The check is a perfect triangular lattice cut as a 24-by-24 rhombus: 484 interior cells, no fives and no sevens at all, and the pairing routine returns nothing rather than a share of an empty set.

That refusal is specifically a refusal of this statistic. An adjacency share on an empty set is 0 out of 0, which any reasonable arithmetic will report as either nought or one, and both answers would be read as a finding by anyone who did not know the set was empty. Returning nothing is the only response that cannot be misread.

The same patch also fixes what the measurement is competing against. It is the most ordered tissue this site can build and it produces no dislocations whatsoever, so the golden head’s 264 are not a floor that any lattice has — they are the price of the arrangement being a spiral rather than a crystal.

This does not single out the golden angle #

A rational angle eight thousandths of a degree away gives the same 135 fives and the same 129 sevens with the same pairing, and the Lucas head gives the same structure at its own numbers.

That is the eighth criterion this site has put to the golden angle and the eighth to come back with nothing. It is worth stating each time because the pattern is the finding: packing does not single it out, disorder does not, and neither does the dislocation structure. Whatever these rings and pairs are evidence of, it is not optimality.

What this does not claim about a real head #

Every number here is measured on Vogel’s model, where the nth organ is exactly at n times the divergence angle and a radius of root n. A photographed capitulum is not that, and the question of which of these results would survive a real specimen has an answer with two halves.

The ring structure would not. It is destroyed by a twentieth of a cell spacing of displacement, which is finer than any measurement of a real head could place an organ. The pairing would. It is still six standard deviations clear at four times that displacement, so it is the half of this result that a photograph could be asked about — and asking it needs nothing beyond a segmented image and a side count.

What would refute it #

A settled head, tessellated the same way, whose fives and sevens were adjacent at the rate the strong null gives. That has been looked for at six head sizes, at six rim cuts, on two divergence ladders and at twelve disorder settings, and it turns up only where the arrangement has stopped being a lattice.

The narrower refutation is easier and more useful: a single five in the interior of a settled head with no seven within a cell of it. There is no such cell in the 2,400-organ head, and the statistic that would find one is the worst case rather than the mean — which is the reason it is the number in the title of the drawing rather than the average.