The rings are not the transitions

A seed head carries two ladders of radii. One is where the counted parastichy pair changes, which is the site’s oldest measurement and the thing a reader counting a photograph runs into first. The other is where the exceptional cells sit, which is a set of one-cell-thick circles.

Both ladders are geometric, both step by a factor of the golden ratio in radius, and both are computed from the divergence angle with nothing fitted. The obvious conclusion is that they are one ladder seen twice. They are not, and the miss is exact.

What the expectation is worth stating #

It is not a naive guess. A parastichy transition is where the pattern visibly reorganises, and a line of five- and seven-sided cells is what reorganisation looks like in a tiling. The crystallographic reading makes the connection tighter still: a row of bound dislocations is what accommodates a change in lattice register, so a defect line at a transition radius would be the same object described twice.

Two structures that both exist, both step by the same factor, and both have a mechanical story linking them, are the strongest case a coincidence can make. It is worth measuring rather than asserting for exactly that reason.

Where the counted pair changes #

At the radii the cylinder library computes from the rise law, which on a disc is the rise falling as one over radius squared. On the reference head those radii are 3.385, 5.478, 8.861, 14.339, 23.204 and 37.537 model units, and each of them is where the two shortest lattice vectors change places.

That is what a count is. A parastichy family is a set of chains joining organs a fixed number of places apart, and the pair anyone counts is the two families whose steps are the two shortest — so a count changes exactly where those two swap.

Where the neighbours change #

At different radii, and for a different reason. A Voronoi cell’s six neighbours are plus and minus the three shortest lattice vectors, not the two, so the handover that matters to a cell is where the third changes identity.

The radii are 4.304, 6.969, 11.274, 18.238, 29.513 and 47.758. Each of them lies outside the transition below it, and each of them is where a cell gains or loses a side.

What coincidence would have required #

For the two ladders to be one, the second and third shortest lattice vectors would have to hand over at the same rise as the first and second — which means two of the three lengths staying equal across an interval rather than crossing at a point. On a lattice whose vector lengths are smooth functions of the rise, that does not happen except by construction.

So the two events are separated by something, and the only question the measurement settles is by how much. A reader who has met the six sides of a cell as its spiral families has already met the reason there are two events at all: the count reads two families and the cell reads three.

The ratio #

Dividing each ring radius by the transition below it gives 1.2715, 1.2723, 1.2723, 1.2719, 1.2719 and 1.2723. The square root of the golden ratio is 1.27202.

Nothing is fitted #

Both radii come from the divergence angle and the rise law and nothing else. There is no free parameter to have been tuned into agreement, which is what makes a spread of 0.0008 across six rungs a result rather than a good fit.

The rungs span a factor of eleven in radius, from 4.3 to 47.8, and the ratio does not drift across them. A quantity that agreed at one rung and drifted at the others would be an artefact of the innermost geometry; one that holds over a factor of eleven is a property of the ladder.

Half a rung #

Consecutive rungs sit a factor of the golden ratio apart in radius, and the square root of that factor is their geometric midpoint. So the statement is not merely that the ring is outside the transition: the ring sits exactly halfway between two consecutive parastichy transitions, on the logarithmic axis the ladder is uniform on.

The two ladders therefore interleave, alternating all the way out, and the reader looking for the defect line at the radius where the counts change will find a defect-free annulus and will find the line a quarter of the way further out.

The offset is a phase angle, not a scale #

Both ladders step by the same factor, so their relationship has exactly one degree of freedom: where one sits between two rungs of the other. That single number is what the twelve measurements are twelve readings of, and it is the reason a spread of 0.0008 means something — twelve independent chances for that offset to come out anywhere between 1 and 1.618, and it lands on the midpoint every time.

It is the same shape as carrying the cylinder’s rungs onto a cone, where one ladder is mapped onto another by a fixed factor and the spacing survives the mapping intact.

The same on the other branch #

The Lucas head has its own divergence angle, its own families — which are not Fibonacci numbers — and its own rungs. Its rings sit at 9.632, 15.576, 25.203 and 40.780 model units, carrying 18, 29, 47 and 76 pairs.

What the second branch rules out #

That any of this is about Fibonacci numbers. The Lucas ladder’s rungs are the same factor apart, its rings sit the same factor outside its own transitions, and its ring populations are Lucas numbers arrived at by the same code.

So the result is about the relationship between the second- and third-shortest lattice vectors of a spiral lattice, and holds for whatever sequence that lattice’s angle generates. That is the same shape of finding as the branch structure of the ladder itself, where the arithmetic that looks specific to one sequence turns out to be a property of the construction.

The reason, stated #

A count changes where vectors one and two change places. A cell’s neighbourhood changes where vectors two and three do. On a ladder whose rungs are a fixed factor apart, those two events are separated by half a step, and half a step in radius is the square root of the step.

That is the whole of the explanation and it predicts the measured number without any reference to tissue, mechanics or growth. It also predicts that the miss cannot be closed by measuring more carefully, because it is not an error.

What the interleaving does to a reading of a photograph #

A published photograph annotated with a spiral count is a statement about an annulus. A photograph annotated with a defect line is a statement about a different annulus, and the two annuli never coincide.

Anyone reading a real head for both would therefore find, at each rung, a radius where the counts change with no defects on it and a radius a quarter further out where the defects are and the counts are stable. Reporting either as though it located the other is the confusion this measurement exists to remove.

What this does not establish #

That either handover causes the other. Both follow from the same ordering of lattice vector lengths, so they are two consequences of one arithmetic rather than a mechanism and its effect.

Nor does it establish that a real tissue would place its defects at the second handover. The prediction is about a Voronoi tessellation of Vogel’s model, and a tissue whose cells push on each other could plausibly relax a defect off the radius where the tessellation puts it. Nothing here tests that, and the honest statement is that the geometry and the mechanics have not been separated.

What the rim cut is worth #

Every statistic in this field depends on where the rim is cut, because a cell adjacent to the head’s boundary has its outward neighbours missing and its side count is a fact about the cut. The cut is therefore varied rather than defended.

The counts do not move #

From 0.70 of the radius to 0.96, the head reports the same 135 five-sided cells, the same 129 seven-sided ones and the same interior charge of +6. That the counts hold and the share does not has already been read off this table; what has not is why, and the ring result supplies it. Not one cell changes across a cut that adds 1,035 cells to the interior.

Which is a consequence of the rings #

The reason nothing moves is the ring result. The defects all sit on circles, no circle lies between 0.70 and 0.96 of this head’s radius, and so a cut placed anywhere in that range removes only hexagons and only from the outside.

That is a stronger statement than the counts happening to be stable. It says why they are stable, it says where they would move — a cut that ran through a ring would take that ring’s defects with it — and it makes the stability a prediction rather than a reassurance.

Where a cut would move the counts #

The safe band has a computable upper edge rather than a conventional one. The outermost ring on this head sits at 47.758 model units and the head’s own radius is 48.98, so the ring lies between the cut at 0.96 and the cut at 1.00 — which is precisely why the last row picks up a row of extra fives and the five rows above it do not.

That turns the usual cut into a statement with a reason attached. Every ring quoted anywhere in this thread has at least two cell spacings of clear head outside it, and a cut chosen to keep that clearance is choosing where the tessellation is complete rather than choosing a convention. The forced mean of six is no help in making that choice, because it is six on either side of the ring.

The share doubles over the same cuts #

While the counts hold still, the fraction of the interior that is not hexagonal runs from 25.6 per cent at the tightest cut to 12.8 per cent at the loosest. A factor of two, from moving one setting nobody would think to report.

So a defect fraction is a statement about where the rim was cut and a defect count is a statement about the head. Any result quoted as a percentage of cells, on any head in this field, is carrying a hidden parameter that halves it — which is the same failure as a disorder statistic that reports the head’s size rather than the arrangement.

What a result that moves with the rim cut is #

A result about the rim cut. That is the sentence worth carrying out of this section, and it generalises past this field: an instrument setting nobody varies is indistinguishable from a constant of nature until somebody varies it.

The test is cheap. Sweep the setting, and separate the quantities that move from the ones that do not. Here the sweep divides the field’s statistics into two clean halves, and the half that survives is the half worth quoting.

The row that cannot be read #

At the full radius the head reports 224 fives against 130 sevens, an interior charge of +96 and a defect share of 15.0 per cent. Two edge effects were named for it — outward neighbours missing on the boundary cells, and a triangulator that was itself incomplete against the convex hull — and the pairing collapses to 60.7 per cent with them. The second has since been repaired, which recovers twenty-one interior cells and moves the share by a tenth of a point; not one of the twenty-one is a five or a seven, so the counts, the charge and the pairing are the numbers they were and the row is still unreadable on the cut alone.

What the ring result adds is the size of the damage before it is measured. The corrupted row gains its extra fives from one ring and one only, because only one ring lies between the last usable cut and the edge, so the discrepancy is that ring’s own population rather than an unknown amount of noise.

Why that row is drawn at all #

Because leaving it out would make the other five rows look like a survey rather than a choice. The cut at 0.86 is not obviously safe from the inside; what makes it safe is that the numbers hold across every cut short of the edge and fail at the edge for reasons that can be named.

A drawing that showed only the well-behaved range would be asserting that range rather than demonstrating it. The last row is the demonstration, and anything load-bearing is read off the five rows above it.

What disorder does #

Displacing every organ inside a box and re-tessellating separates two results that look like one. Both are about defects, both are measured against the same kind of seeded permutation null, and until the head is disordered there is nothing to tell them apart.

The rings go first #

Ring confinement starts moving at 0.04 cell spacings of displacement — a twentieth of a cell — and is half gone by 0.21, which is the setting this site’s own disorder figures call full. It is statistically indistinguishable from its null at 0.31 cell spacings.

The pairing outlasts them #

The pairing is still at z = +4.1 at 0.40 cell spacings, where ring confinement has been at its null for some time, and it reaches its own null at 0.62 — which is about where a lattice stops being one.

So a head disordered past this site’s usual setting has kept its dislocations and lost its rings. Where a defect is and what is beside it are separate results with separate robustness, and disorder is what makes the separation visible.

What the separation is worth #

It decides which half of the finding could ever be looked for in a photograph. A real capitulum’s organs are not placed to a twentieth of a cell spacing, so the ring structure is below the resolution of any specimen and will stay there.

The pairing is not. It survives four times that displacement, needs only a segmented image and a side count, and has a sign attached — the seven on the inside — that a specimen could contradict. That is the difference between a result about a model and a result a botanist could be asked to test, and the two had to be measured apart before either could be stated.

What disorder does not separate #

The defect count rises with disorder rather than falling: 264 at rest, 660 at the usual full setting and 1,103 at twice that displacement, against 1,215 for a random point set. So disorder is not destroying the defects, it is unbinding them from the ladder that arranged them.

That reading is available only because the count and the confinement are reported apart. A single number combining them would fall smoothly and would look like defects being erased, which is the opposite of what is happening.

What the three tests leave standing #

The interleaving is a property of the arrangement and not of a measurement choice. It holds on two divergence ladders, it holds at every rung of both, it does not move when the rim cut moves, and it is derivable in one line from the ordering of lattice vector lengths.

The rim cut decides a share and no count. Disorder decides which results survive to be looked for in tissue. And the connection a reader arrives expecting — that the rings are the transitions — is refused with a number rather than with a caveat.

What would refute it #

A spiral lattice whose defect rings sat at the radii where its counts change. The ratio would have to come out at 1 rather than at 1.272, and it comes out at 1.272 on twelve rungs across two ladders whose families share no numbers.

The narrower refutation is a rung whose ratio drifts. A ladder whose innermost rungs read 1.27 and whose outermost read 1.30 would say the relationship is asymptotic rather than exact, and the six measured here run 1.2715 to 1.2723 with the smallest value at the innermost rung and the largest at the outermost — a spread eight ten-thousandths wide, which is the kind of agreement that has to be checked for a fitted parameter and here has none to check.

The six run 1.2715 to 1.2723 with the smallest at the innermost rung and no drift across the rest, so what is on offer is not a trend approaching a limit. It is one number, measured twelve times, on two ladders that share no families.