A hundredth of a spacing

Two results about the two meanings of “neighbour” on a seed head were measured on heads whose every organ sits exactly where Vogel’s rule puts it. The empty interval is the rings found that a three-family contact cut names a cell’s Voronoi walls exactly everywhere but a band two thirds of a wall spacing either side of each flip ring, and that off those bands one hop-ratio cut-off, between 1.4302 and 1.4437, separates walls from other contacts on every head from 900 organs to 9,000. The blur was at the centre then found the band the same on every resolved ring of a golden head and a Lucas head.

Both essays said what they did not cover. A real head’s organs are not where the rule puts them; a photograph adds its own error on top. Positional disorder was already known to move the whole-head best cut-off, from 1.47 to 1.81 at a sixth of a spacing. What it does to the band, to the rings and to the cut-off off the rings was not measured. This essay measures it, and the answer is not a widening. It is an order of failure, and a direction.

The displacement, and its unit #

Every organ of the head is moved by an independent gaussian step in each coordinate, seeded so each number here can be reproduced, and both relations are recomputed from the moved points. The size of the step is stated in the ordered head’s own wall spacing — the median length of a wall between two interior cells, about 1.9 in the head’s units — because that is the unit the band is measured in. Displacements run from a four-hundredth of a spacing to eight hundredths, doubling each time, at five seeds each, on golden heads of 900, 2,400 and 9,000 organs and a Lucas head of 2,400.

Those are small numbers. Eight hundredths of a spacing is less than a tenth of a seed’s width. The displacements in the round trip through the counts ran to two and a half spacings, thirty times further, and the counts survived most of it. The question here is about a finer instrument, and it turns out to need a finer scale.

What a small displacement looks like #

At two hundredths of a spacing the head looks unchanged. The two resolved rings are still where the divergence angle puts them, and the disputed cells still sit on them. Of the head’s 150 five- and seven-sided cells, 13 have moved off the band round a ring; of its cells a spacing or more from every ring, none is disputed. Turned up to eight hundredths, 58 of 168 exceptional cells are off the rings and 23 cells between the rings are disputed. Turned down to one hundredth, 5 of 154 have left and nothing between the rings is disputed.

The picture changes slowly and without drama. The number that does not change slowly is the one that cannot be seen on it.

The cut-off is gone by a hundredth #

Off the rings the cut-off exists when the largest hop ratio of any wall is smaller than the smallest hop ratio of any contact that is not a wall. Ordered, that gap is 0.107 on the 900-organ head, 0.064 on 2,400, 0.040 on the Lucas head and 0.014 on 9,000. Displaced, it closes, and it closes on the largest head first.

On the 9,000-organ head a displacement of a four-hundredth of a spacing shuts the gap on four seeds of five, and a two-hundredth shuts it on all five. On the 2,400-organ head and the Lucas head the gap is shut on every seed by a hundredth. The 900-organ head holds out through a hundredth and is shut at a fiftieth. So the single cut-off that separated the two relations everywhere off the rings survives a displacement of somewhere between a four-hundredth and a hundredth of a spacing, depending on the size of the head, and no more.

Why a cut-off is so fragile #

The reason is that a cut-off is decided by extremes. The gap is the smallest non-wall ratio minus the largest wall ratio, taken over every cell off the rings — 5,004 of them on the 9,000-organ head — so a single cell anywhere whose ratio moves by more than the gap closes it for the whole head.

Measured on that head, a displacement of a four-hundredth of a spacing moves a hop ratio near the cut-off by 0.0063 in root mean square and by 0.028 at the worst of 5,637 ratios. The gap it has to survive is 0.014. The typical cell is safe and the worst one is not, and the cut-off answers to the worst one. At a hundredth the same ratios move by 0.026 typically and 0.117 at worst — more than the 900-organ head’s whole gap of 0.107.

The larger head fails first for two reasons that compound: it has more cells, so its worst cell is worse, and its gap was narrower to begin with, because its upper end is set by the outermost ring’s flank and a larger head has a larger outermost ring.

The two ends do not move alike #

The gap closes from both ends, and not symmetrically. On the 9,000-organ head the largest wall ratio off the rings starts at 1.430 and climbs to 1.444, 1.465, 1.505, 1.609 and 1.886 as the displacement doubles from a four-hundredth to four hundredths. Over the same displacements the smallest non-wall ratio falls from 1.444 only to 1.436, 1.429, 1.418, 1.410 and 1.376. The lower end of the window moves seven times as far as the upper.

The asymmetry has a plain cause. The smallest non-wall ratio can only fall by about as much as the displacement shortens a lag, and the fourth family is a long way from being the shortest lag anywhere off a ring. The largest wall ratio can jump: a displacement that flips which diagonal of a cell carries its wall makes a wall of a lag that was not one, and that lag can be as long as the fourth family, near 1.9. A handful of flipped cells anywhere on the head decide the lower end, which is why it climbs so fast and why excluding more cells round the rings does not bring it back.

Setting more aside does not save it #

The obvious repair is to exclude more cells round each ring. Ordered, the heads need 0.68, 0.74, 0.79 and 0.90 of a spacing set aside before a cut-off exists. Displaced by a two-hundredth, the 9,000-organ head needs 1.4; by a hundredth, 2.3. At a fiftieth no seed of the 9,000-organ head opens within three spacings, and at four hundredths no seed of the 2,400-organ head does either — by which exclusion there is almost nothing left between the rings to separate.

So the exclusion does not rescue the cut-off; it grows until it has excluded the head. The cells a wider exclusion removes are not the problem cells. The problem cells are everywhere, because the displacement is everywhere, and the cut-off only needs one of them.

The count holds where the cut-off does not #

A three-family count asks a different question of each cell. It does not compare the cell with every other cell. It asks only whether this cell’s third-shortest lag is a wall and its fourth is not, so a cell fails only when the displacement pushes its own third and fourth ratios past each other.

On the 2,400-organ head the count stays exact off the rings through a hundredth of a spacing — not one of the cells a spacing or more from a ring is disputed on any seed — and at two hundredths it disputes 0.24 per cent of them. The dispute profile against distance keeps its shape and softens its edge. Ordered, the share of cells disputed falls from all to none within five hundredths of a spacing, between 0.60 and 0.65. Displaced by a two-hundredth it falls over about two tenths, from 99 per cent at 0.5 to 5 per cent at 0.7; by a fiftieth, over about half a spacing, from 85 per cent at 0.45 to 3 per cent at 1.0. At all three the halfway point stays near 0.62: the edge widens about where it was rather than moving outward. Only at eight hundredths does the ground between the rings fill in, to about a quarter of the cells 1.5 spacings out.

This is the second time the count has beaten the cut-off. On an ordered head a count was exact from 0.66 of a spacing where a cut-off on distance needed 0.9. Displaced, the advantage is a factor of two to four in the size of error each instrument survives: on the four heads the cut-off is shut on every seed at 0.02, 0.01, 0.005 and 0.01 of a spacing, and the count first disputes one per cent of the cells off the rings at 0.04, 0.04, 0.02 and 0.04.

The margin shrinks as one over the ring #

How much a cell can take is its own margin: how far its third and fourth hop ratios are from the tie that defines a flip ring. At the ring they are equal, and moving away they part. Measured on six-sided cells between 0.9 and 1.1 of a spacing from each resolved ring, the median gap is 0.421 at the golden ring keeping 21, then 0.275, 0.175, 0.110 and 0.069 at the rings of 34, 55, 89 and 144. On the Lucas head it is 0.322, 0.206 and 0.130 at the rings of 29, 47 and 76.

Multiplied by the ring’s family number, the gap is 9.35, 9.63, 9.79 and 9.94 on the golden rings from 34 and 9.34, 9.68 and 9.88 on the Lucas rings, a mean of 9.67. The margin one spacing from a ring is about 9.7 divided by the ring’s number. It has a reason: rings sit a factor of $\varphi$ apart in radius, so a spacing is a smaller share of an outer ring’s radius, and the lattice a spacing from the ring of 144 is nearer its tie than the lattice a spacing from the ring of 21 by the ratio of the two radii.

So the outermost ring fails first #

The margin predicts where a displaced head fails, and it fails there. On the 9,000-organ head at a hundredth of a spacing, the cells one to three spacings from the ring of 144 are 2.0 per cent disputed, those near the ring of 89 are 0.2 per cent, and those near 55 and 34 are untouched. At two hundredths the four rings stand at 12.9, 2.3, 0.8 and 0 per cent; at four hundredths, 41.1, 13.9, 4.3 and 1.4.

At four hundredths each step outward in ring roughly triples the damage — 1.4, 4.3, 13.9, 41.1 — which is what a margin shrinking by a factor of $\varphi$ per ring does to the share of cells a gaussian step can push past it. A displaced head does not widen its band uniformly. It fails from the rim inward, and a larger head fails sooner because it has a larger outermost ring. That is the same reason the cut-off’s gap narrowed with head size, arriving by a second route.

Three things, in a fixed order #

There are three things a displacement can break here, and on all four heads it breaks them in the same order. The cut-off goes first. The rings’ hold on their five- and seven-sided cells goes next: every exceptional cell on an ordered head lies within 0.66 of a spacing of a ring, and a tenth of them have left by 0.04 of a spacing on the 900-organ head, 0.02 on 2,400 and 0.01 on 9,000. The count’s exactness off the rings goes last: one per cent of off-ring cells disputed at 0.04, 0.04 and 0.02.

On the Lucas head the three come at 0.01, 0.02 and 0.04, the same as the golden head of the same size. Nothing about the order depends on the angle, which is what the band’s being the same on both angles would lead one to expect.

What the rings losing their cells means #

The middle failure is the one that looks like a widening band, and it is not quite that either. The five- and seven-sided cells that leave the rings do not drift outward from them in a widening shell. New exceptional cells appear between the rings — on the drawn 900-organ head their number rises from 154 to 168 by eight hundredths — while the rings’ own share of them falls, to half on the 2,400-organ head by the same displacement. The second moment of a cell’s side count, which counts exceptional cells whoever they belong to, stops being a count of ring cells at about the displacement where the rings stop holding them.

So “does the band widen at the rate the rings dissolve” turns out to be two questions with two different answers. The band’s edge softens at about the rate the rings lose their cells. The ground between the bands, where the count and the tessellation agree, holds on for another doubling of the displacement or two.

The head at eight hundredths #

At the largest displacement read the head is visibly different, though not in the way a first guess would draw it. The two rings are still there: most of the disputed cells still lie on them, and the band round each is still where the angle puts it. What has changed is the ground between. Fifty-eight of the head’s 168 exceptional cells are now away from any ring, scattered through the regions that were exact, and 23 cells a spacing or more from every ring have a disputed wall.

This is where the pairing of every five with a seven matters for reading such a head. The cells that leave the rings do not leave singly. Over five seeds at eight hundredths, 319 five- and seven-sided cells lie more than a band’s width from every ring, and 292 of them — 92 per cent — share a wall with an exceptional cell of the other kind. So the off-ring exceptions appear as pairs, as they do on the rings. They are not balanced, though: 177 of the 319 are fives, 141 are sevens and one has some other count of sides. A photograph of a real head showing scattered five–seven pairs between its rings is showing this regime, and the rings that remain beneath them are still the angle’s.

What a hundredth of a spacing is #

These are small displacements. A hundredth of a spacing is a hundredth of the distance between two seed centres. No photograph of a capitulum locates seed centres that finely, and a real head’s own departures from Vogel’s rule are far larger than that — which is the point of the displaced round trip, which found spiral counts surviving displacements of whole spacings.

So the practical content of the result is a ranking of instruments, not a tolerance to aim for. A single distance cut-off separating walls from contacts is a property of an exact lattice and of nothing that could be measured. A three-family count is several times sturdier and still fine by a photograph’s standards. And the rings, which the count’s failures are organised round, are sturdier than the cut-off but not than the count.

What this does not establish #

It does not model a real head’s error. Every displacement here is independent from organ to organ and gaussian; a real head’s are correlated, since growth, drying and pressing move neighbouring organs together, and a correlated displacement should change hop ratios less than an independent one of the same size. The tolerances here are therefore likely to be the floor of what a correlated error would allow, not a measurement of it.

Nor does it follow a head further than eight hundredths of a spacing. Past that the count’s own window starts to need a longer scan of index lags — a displaced 2,400-organ head grows walls reaching a lag of 377 at about a quarter of a spacing — and the three-family reading becomes a different measurement.

What would withdraw it #

A displaced head on which the cut-off off the rings outlasts the rings’ hold on their exceptional cells, or on which that hold outlasts the count’s exactness off them.

A displaced head whose inner rings are disputed before its outer ones.

A resolved ring whose margin one spacing out, times its family number, falls outside 8.5 to 11.

Still open: whether a correlated displacement ranks the instruments the same way #

A displacement that moves neighbouring organs together changes the lengths of short lags less than the positions of the organs, because both ends of a short lag move almost alike. That should favour the cut-off, which reads lengths, over nothing in particular, and it could change the order. The measurement is the same reading with the displacement drawn as a smooth field — a radial stretch, a squeeze along one axis, a twist — at the same amplitudes: whether the cut-off still fails first, and whether a field aligned with the rings moves the band itself rather than blurring it.