The defects lie on rings

A golden head of 2,400 organs holds 1,631 interior cells and 264 of them are not hexagons. Counting them and adding up their charges says how many there are and what kind they are. It says nothing about the one property a picture would show immediately.

They are not scattered. Every one of the 264 sits within a cell’s width of one of eight circles, the circles are computed from the divergence angle before any cell is looked at, and between two of them there is a band of 422 consecutive cells in which every single one is a hexagon.

What “on a ring” is being asked #

A claim that exceptions lie on circles is easy to make and easy to make vacuously. Eight circles through a disc will pass near a great many points, and a curve fitted to a cloud goes through the cloud by construction.

So the claim is made in the order that can fail. The radii are computed first, from the divergence angle alone, with no cell counted and nothing adjusted. Then every exception in the head is measured against them, and the measurement is scored against a null that scatters the same exceptions over the same cells.

The prediction comes first #

The rule is the site’s oldest and it is stated in the counting field rather than the tissue one. A spiral head is a lattice whose local geometry changes with radius, and the counted pair changes as it does — 13 and 21 near the middle, then 21 and 34, then 34 and 55.

Those changes happen at radii the model computes. On a cylinder the same ladder is a ladder of rises, each rung a factor of 1/φ² below the last, and a disc’s rise falls as one over radius squared, so the rungs land a factor of φ apart in radius.

Six neighbours are three vectors #

Here is the step that turns a counting result into a tissue prediction. A cell in a lattice has its Voronoi region bounded by the perpendicular bisectors of the vectors to its neighbours, and in a generic two-dimensional lattice the six that survive are plus and minus the three shortest lattice vectors.

Three vectors, six walls, six sides — which is the mean Euler forces anyway, arriving from the geometry instead. The consequence is that a cell’s neighbourhood is decided by a list of three, and it changes when the list does.

The third vector, not the second #

A parastichy count changes when the two shortest vectors change places. A cell’s neighbours change when the third changes identity, because that is the one entering and leaving the list of six walls.

Those are different events at different radii, and the difference is the reason this essay is not a restatement of a counting result. Where they sit relative to one another is a separate measurement; what matters here is only that the tissue follows the third and the counted pair follows the second, so the radii that predict a ring are not the radii anyone has published.

From a rise to a radius #

Each handover of the third vector happens at a rise, and a disc has a rise: at radius r, the lattice’s local rise is c²/4πr², which is the same law the ladder is walked with on a cone and on a disc. Inverting it turns each handover rise into a radius.

The result is a ladder of eight radii inside the reference head’s cut, each about 1.618 times the one below it. Nothing in the derivation mentions a cell, a Voronoi region or an exception.

Nothing is fitted #

That is worth being explicit about because the alternative is so much easier. A ring radius could be read off the exceptions themselves by taking their mean radius in a band, and the result would be a ring at every radius where there happened to be exceptions.

No number below was obtained that way. The radii come from the divergence angle and the rise law; the measured centres are computed afterwards and reported as a discrepancy from the prediction, which is the only arrangement in which a small discrepancy means anything.

The measurement #

Every interior exception’s distance from the nearest predicted radius, in units of the cell spacing — the median length of a contact between two interior cells, which is 1.923 model units on this head.

All 264 sit within one spacing of a ring. The worst offset in the whole head is 0.637 of a cell spacing and the median is 0.386. There is no tail: the farthest exception from its ring is closer to it than two cells are to each other.

A null that already knows the counts #

The comparison is a permutation. Keep the 1,631 interior cells and the 264 exceptions, scatter the exceptions at random over the cells, and ask what share lands within one spacing of a ring. Two hundred draws.

The answer is 32.9 per cent, plus or minus 1.4. A third of a randomly scattered set falls near a ring anyway, which is the number the claim has to beat and the number nobody would have guessed — eight circles at a spacing of a factor of φ cover a good deal of a disc.

Twenty-two standard deviations #

Observed 100 per cent against 32.9 ± 1.4 is z = +22.8. That is not a marginal result and it is not a result that needs a significance test to be believed; the test is there to price the alternative rather than to establish the finding.

The honest way to read a z of 22.8 is that the null is the wrong model rather than an unlikely one. Nothing about a scattered set of exceptions produces 264 out of 264, at any sample size.

What the null does not test #

It tests position and only position. Because it leaves the number of exceptions alone and leaves the interior cells alone, it cannot say anything about why there are 264 rather than 300, and it deliberately says nothing about what sits beside each one — that is a different question with its own null.

It is also a null with a designed weakness: it already knows the head. A null built from an easier comparison would score higher and mean less, which is the lesson a single background sample taught this site on a different measurement.

The ladder, ring by ring #

Taken to 4,000 organs, where the head reaches four rings with clear space around them, each ring can be read on its own. The innermost of the four sits at a predicted radius of 11.274 and holds 42 exceptions: 21 fives and 21 sevens. The next, at 18.238, holds 34 and 34. The next, at 29.513, holds 55 and 55. The outermost readable one, at 47.758, holds 89 and 89.

Equal numbers of both kinds on every one of them, which is why the interior charge of the whole head came out at six rather than at anything larger.

The family a ring keeps #

Those counts are not free numbers. 21, 34, 55, 89 are exactly the parastichy families that survive each handover — the member of the counted triple that is common to the list before and the list after.

So a ring does not merely hold exceptions; it holds one per member of a spiral family, and the lattice decides how many. This is the same closure that makes a third contact family a prediction rather than a measurement, turning up in the tissue as a count of cells.

Rings a factor of φ apart #

The measured centres of consecutive resolved rings stand in the ratios 1.6177, 1.6182, 1.6182. The ladder’s own step is φ, and φ is 1.6180.

Three ratios agreeing with a constant to four figures is a stronger check than any single ring’s position, because a systematic error in the radius law would have to be proportional to the radius to leave those ratios alone.

A ring is one cell thick #

The radial thickness of a ring — the spread of its exceptions about its own centre — is 1.30 cell spacings, and it is 1.29 or 1.30 on every one of the four.

Across those four the radius grows by a factor of 4.2, from 11.3 to 47.8. A feature that keeps a fixed width in cells while its radius quadruples is a feature of the lattice’s local geometry rather than of the head, and it is about as sharp as a structure made of cells can be: one cell.

Where the ladder stops being readable #

The predicted ladder has more rungs than four. Inside 11.274 there are five more, at 6.969, 4.304, 2.664, 1.639 and 1.025, and every one of them has exceptions near it.

But the counts there are wrong: 30 exceptions at 6.969 where the family is 13, split 17 and 13 rather than evenly; 19 at 4.304 where the family is 8. Reported without a caveat those rows would look like a prediction failing at small radius.

Which is a resolution limit and not a departure #

A ladder step is a factor of φ in radius, so at radius r the next ring inwards is 0.618r away. Below about r = 6 that gap is under two cell spacings — and a ring is one and a third cell spacings thick.

Two rings a cell and a half apart, each a cell and a third wide, are not two measurable objects. The inner rows are overlapping rings read as one, which is the same arithmetic that decides how much of a head a single parastichy count can honestly be taken from. Nothing there is evidence against the ladder and nothing there is evidence for it.

When a ring counts as resolved #

The rule is stated rather than chosen after the fact: a ring is resolved when its nearest neighbour is more than two cell spacings away, when it has at least two spacings of clear head outside it, and when it holds at least four exceptions.

Four rings of the nine pass on a 4,000-organ head. Every claim above is made on those four, and the five that fail are drawn rather than dropped, because a reader shown only the passing rows cannot see what the criterion did.

Between the rings there is nothing #

The other half of “on a ring” is the half a scatter plot hides, and it is the stronger half.

On the reference head the widest stretch of radius holding no exception at all runs from 19.44 to 28.30 — 4.61 cell spacings, sitting between the ring at 18.2 and the ring at 29.5. Inside it are 422 cells and every one of them is a hexagon.

Not a low rate. None.

Four hundred and twenty-two hexagons #

Against the null, 422 cells with no exception is the part that is hardest to produce by any mechanism that puts exceptions somewhere with a preference. A rate of 16 per cent applied to 422 cells expects about 68 of them.

At 4,000 organs the same band is wider still: 1,221 cells across 8.50 cell spacings, again with none. The empty band grows with the head because the ladder is geometric and the gaps between rungs grow with radius.

The outer third of the head #

Read as a fraction rather than as a band, the outer 30 per cent of the reference head’s radius — from 34.3 out to the cut at 42.1 — contains no exception at all.

That is a third of the head’s radius and rather more than a third of its area. Anyone measuring this tissue in the outer annulus, which is where a photograph is sharpest and where the cells are most nearly equal in size, would find a perfect hexagonal sheet and conclude the arrangement has no defects in it.

The radial profile #

Cut the head into annuli of equal area and count the exceptions in each. Equal area rather than equal width, so that every bin holds about the same number of cells and an outer ring is not flattened by a wide bin.

The profile is not a decaying curve with structure on it. It is a set of spikes standing on zero, with the spikes at the predicted radii and long runs of empty annuli between them, each holding hundreds of cells.

The Lucas control #

The result so far could be an artefact of Fibonacci numbers being everywhere in this subject. The control that answers that is a head at the Lucas angle, 99.502°, whose contact families are 4, 7, 11, 18, 29, 47, 76 rather than the Fibonacci ones.

The same code, given that angle, predicts a ladder at 9.632, 15.576, 25.203 and 40.780 rather than at the golden radii, and on a head deep enough to resolve those four it finds 18, 29, 47 and 76 pairs on them. All of a Lucas head’s exceptions lie on its own rings, worst offset 0.62 cell spacings; the 2,400-organ one drawn here carries 230 of them across nine predicted circles.

Which is what says the instrument reads the lattice #

A measurement that returned Fibonacci numbers on both heads would be reporting the folklore. This one returns Lucas numbers on the Lucas head, at Lucas radii, from code that was not told which angle it was given.

That is the same control the site uses whenever a count comes out Fibonacci — the ladder is a property of the branch and not of the sequence, and an instrument that cannot tell the branches apart has not measured anything. The rings are a consequence of a lattice having a third-shortest vector, and every lattice has one.

The spacing along a ring #

One more number, because it is the sharpest in the set. The exceptions on a ring are spaced around it, and the gap between consecutive ones is 1.812, 1.810, 1.811 and 1.811 cell spacings on the four resolved rings.

A spread of 0.1 per cent across rings whose radii differ by a factor of 4.2. The ring at 47.8 has four times the circumference of the ring at 11.3 and four times as many exceptions on it, spaced identically. Whatever sets that spacing is local, and it is the same everywhere in the head.

What a ring is, in one sentence #

A closed chain of pairs of exceptions, one pair per member of the surviving spiral family, at a fixed spacing of 1.81 cells, one cell thick, at a radius the divergence angle predicts.

In crystallography that object has a name — a low-angle grain boundary, made of dislocations — and the useful part of the name is that it comes with an expectation this measurement can be checked against. The pairing it implies is measured separately and against its own null, because a structure that looks like a known object is the easiest thing in the world to describe as one.

What this does not support #

It does not single out the golden angle. A rational 137.5° gives an identical set of rings, and the Lucas angle gives the same structure at its own numbers, so whatever the rings are evidence of it is not an optimality anybody has been able to measure.

It does not claim a real capitulum has these rings. Every number here is measured on Vogel’s model, where organ i sits exactly at i times the angle and a radius proportional to the square root of i. Ring confinement is the first thing scatter destroys: four hundredths of a cell spacing of displacement already begins to move exceptions off their rings, and at a fifth of a spacing half of them are off. A photographed head would need its organs located to a few per cent of a cell before this could be looked for at all.

And it does not measure the outermost ring of any head. The ring nearest the edge is clipped by the rim cut, and at the full radius the numbers are corrupted. Two causes were named when this was written and only one of them is left: cells at the hull lose outward neighbours they should have, which is the cut and is unavoidable, and the triangulator’s super-triangle was built too small, which was a bug and has since been repaired. Every ring quoted above has at least two cell spacings of clear head outside it, so neither cause ever reached a number in this essay.

What would refute it #

An exception, anywhere, at a radius the ladder does not name. There are 264 chances on the reference head and 442 on the larger one, and the criterion is a fixed distance rather than a fitted band, so a single one at 24 would end the claim.

A second refutation is available and is more interesting: the empty bands. If a head were found whose exception rate in the band from 19.4 to 28.3 were merely low rather than zero, the picture would change from rings to a radial preference, and every statement above about spacing and thickness would become a statement about a distribution’s shape.

What is left over #

Two things this measurement raises and does not settle. It says nothing about which of a ring’s exceptions is a five and which is a seven, or whether the two kinds are bound to each other; the counts are equal on every ring, which is suggestive and is not evidence.

And it leaves the relation between these radii and the published ones open. The tissue’s handover is the third vector’s and the counted pair’s is the second’s; both are rungs of one ladder, and where the two sets of radii sit relative to each other is an exact number worth measuring rather than a remark. What is settled here is only the thing the title says: the exceptions are on circles, the circles were computed in advance, and between them the tissue is a perfect hexagonal sheet.