The band moves, it does not blur

A seed head’s organs can be called neighbours in two ways — by the shortest index lags a count keeps, or by the walls their Voronoi cells share — and the two relations agree everywhere except in a band two thirds of a wall spacing either side of each flip ring. A hundredth of a spacing displaced the organs and found three things fail in a fixed order: the single hop-ratio cut-off off the rings goes first, then the rings’ hold on their five- and seven-sided cells, then the three-family count’s exactness away from them. Every displacement there was independent from organ to organ.

It ended on the obvious objection. A real head is not displaced organ by organ. Growth, drying and pressing move neighbouring organs together, and a displacement that moves both ends of a short lag alike changes that lag’s length much less than it moves the organs. This essay repeats the reading with displacements that are correlated — a smooth random field, and three fields a whole head suffers — and finds that the order does not so much change as dissolve.

A field that moves neighbours together #

The smooth field is a gaussian random field in each coordinate, built from 128 plane waves, with a correlation length of $\ell$ wall spacings and a root-mean-square step of $\sigma$ spacings — the same size, stated the same way, as the independent step it is compared with. Two organs a distance $d$ apart move with correlation $e^{-d^2/2\ell^2}$, so at a correlation length of one spacing a cell’s neighbours move about half alike and at eight spacings almost exactly alike. At $\ell = 0$ the reading uses the earlier essay’s own independent step, and on a 2,400-organ head with seed one it returns that essay’s banked readings to the last digit.

Correlation lengths of one, two, four and eight spacings were read, at steps from a two-hundredth of a spacing to a third, three seeds each, on golden heads of 900 and 2,400 organs.

What the walls feel is strain #

The instruments here all read distances between neighbours, so the quantity that matters is how much a step changes a wall’s length: the strain. An independent step of $\sigma$ changes it by $1.42\sigma$, since both ends move independently and a wall is about one spacing long. A smooth field changes it by $0.87\sigma$ at a correlation length of one spacing, $0.49\sigma$ at two, $0.26\sigma$ at four and $0.135\sigma$ at eight — about the step divided by the correlation length, from two spacings up. The 900-organ head gives the same numbers to within a hundredth.

So a smooth field of eight spacings at a step of a third of a spacing strains the walls by 4.3 per cent, about what an independent step of three hundredths does. That is the first thing correlation does, and it is the expected one: every tolerance stated in steps grows roughly in proportion to the correlation length.

The census stays where it was #

The second thing is not expected, and it changes what the tolerances mean.

An independent step adds disputed cells. On the 2,400-organ head, which carries 353 cells with a disputed wall and 264 five- and seven-sided cells unmoved — all of them, as the defects’ own reading found, within the bands round the rings — an independent step of four hundredths — a strain of 5.7 per cent — raises the disputed cells to about 394, and a step of eight hundredths to 560. The 900-organ head goes from 209 to about 257. The band round each ring broadens and new cells in dispute appear between the rings.

A smooth field of eight spacings adds none. At every step read, up to a third of a spacing, both heads keep their count of disputed cells within one and their count of fives and sevens within two: 352 or 353, and 264 to 266. A field of four spacings keeps them to a step of 0.16. A twist, turning the rim by as much as 0.4 radians and straining the walls by 9.6 per cent, keeps them exactly. The head carries the same census whatever the field does to it.

Where the census goes #

If the census is kept, the cells in it must be somewhere, and the obvious guess is that the band has moved.

On the ring of 55 the unmoved head has 34 disputed hexagons, all within 0.16 of a spacing of the ring — the census every resolved ring carries: the family number before 55. Twisted by a tenth of a radian at the rim, the head still has 34, packed just as tightly, centred a quarter of a spacing further out. At a twist of 0.2 they sit half a spacing out; at 0.4, a spacing and a quarter, with none left where the ring was.

A smooth field of eight spacings keeps the 34 and spreads them, to half a spacing either side at a step of 0.16 and more at 0.32, because the field strains different parts of the ring differently and each part of the band moves its own way. Independent steps are the only moves that change the number: 44 hexagons near the ring at a step of 0.02, and 60 at 0.04. The smooth field bends the band. The independent step blurs it.

Where the twist puts the ring #

A twist has a closed form, and it is the one a twist is a divergence found for the counts. Turning an organ at radius $r$ by $a\,r/R$ adds $a/(2Rr)$ radians to the divergence between neighbours there, so a twisted head is locally a head at a slightly different angle, and its flip rings should sit where the ties of that angle fall. Solving for each ring’s radius at its own local divergence gives a prediction, and the band sits on it. The ring of 55 moves 0.12 of a spacing at a twist of 0.05, 0.25 at 0.1 and 0.53 at 0.2 — the closed form says 0.120, 0.249 and 0.535. At 0.4 the closed form says 1.23 and the 34 hexagons are centred at 1.24.

The band does not blur on the way. It keeps its width, about three tenths of a spacing, and its count, and slides as a block onto the ring its own geometry now asks for. What the earlier essay called the rings losing their fives and sevens is, under a twist, the rings moving away from where they were drawn.

Three rings, two directions #

Not every ring moves the same way. Under a twist the ring of 21 moves out, the ring of 34 in, and the ring of 55 out again: at a twist of 0.2, 0.07, −0.17 and 0.53 of a spacing, against 0.07, −0.16 and 0.54 in closed form.

The alternation is arithmetic. A flip ring is where two of the head’s lattice families tie, and the families a ring trades are consecutive Fibonacci numbers, whose ratios approach the golden ratio from alternate sides. A small increase in the divergence moves the lattice towards one tie and away from the next, so the ring of one parity moves outward and the ring of the other inward. The outer rings move furthest even though the twist’s extra divergence is smaller there, because the ties of the outer rings lie closer together in divergence — a reported pair pins the angle to about $221°/mn$ — so a smaller change of angle is a larger step between ties.

A radial spread, which pushes every organ outward by a factor that grows with its radius, moves every ring outward — 0.10 and 0.53 of a spacing for the rings of 34 and 55 at a spread of a tenth — and also keeps their hexagons.

Why a radial spread moves the rings outward #

The radial spread’s direction can be read off its shape. It moves an organ at radius $r$ to $r(1 + a r^2/R^2)$, so round a circle it stretches the head by $1 + a r^2/R^2$ while across the circle, between one radius and the next, it stretches it by $1 + 3a r^2/R^2$. The lattice is pulled apart three times as much radially as round the head. On a Vogel head the radial spacing between successive organs, set against the circumference, is what falls as the head widens — it is the local rise — and a flip ring sits where the rise has fallen to its tie. Stretching the head radially raises the local rise everywhere, so every tie is reached further out, and every ring moves outward. The outer rings move furthest because the stretch grows as the square of the radius.

The twist has no such single direction because it changes the divergence rather than the rise, and a change in divergence helps one family of each tie and hurts the other. The ring moves towards the family the extra angle favours, and the favoured family alternates from ring to ring.

What a moved band does to the fives and sevens #

Every five-sided cell on these heads shares a wall with a seven, and the pairs lie on the rings. A band that blurs breaks the pattern: under independent steps new fives and sevens appear between the rings, and the earlier essay counted 58 of a 900-organ head’s 168 away from any ring at eight hundredths of a spacing. A band that moves carries its pairs with it. Under a twist the 2,400-organ head keeps exactly its 264, and they go where the hexagons go: the 110 fives and sevens round the ring of 55, centred 0.17 of a spacing out on the unmoved head, are centred 1.37 out at a twist of 0.4, beside the 34 hexagons, and within three tenths of a spacing of where the ring used to be there is now no disputed cell of any kind.

That makes the fives and sevens the most direct witness to which kind of error a head carries. Their number says whether the band has blurred; their position, set against where the head’s divergence puts its rings, says whether it has moved.

Read against strain, the order dissolves #

The earlier essay’s three instruments can be put to the correlated heads unchanged, and their failures read as strains rather than steps. The cut-off off the rings is helped by correlation. On the 2,400-organ head it is shut at a strain of 1.4 per cent under independent steps and at 1.8, 2.0, 2.1 and 2.2 per cent as the correlation length rises from one to eight spacings; on the 900-organ head, 2.9 per cent rising to 4.7. The earlier essay guessed this — a cut-off reads lengths, and a smooth field changes lengths less than positions — and it holds. It is still the fragile instrument it was on the whole head: it answers to the worst cell anywhere, and a strain of two per cent somewhere is enough.

The count is not helped. Its failure strain on the 2,400-organ head is 5.7 per cent under independent steps and 3.9 to 4.3 from two spacings of correlation up; on the 900-organ head it falls from 11.6 to 4.0–4.7. So the count’s margin over the cut-off, four times the strain under independent steps, falls to about two on the larger head and to nothing on the smaller: from two spacings of correlation up, the 900-organ head’s cut-off and count fail at the same step on the grid.

The reason is the section before. The count is judged on cells a spacing or more from where the unmoved rings are, and under a smooth field nothing is added to the census — the band simply arrives there. When the ring of 55 has moved half a spacing, the cells a spacing outside its old position are half a spacing from its new one and some of them are in dispute. The count has not failed; the ring has come to meet it. The same is true of the rings’ hold on their fives and sevens, which under a smooth field of eight spacings fails at a step half the cut-off’s on the 900-organ head, because a moved five or seven is scored as one that left.

So the earlier essay’s order — cut-off, rings, count — was the order in which a blurred band spills across fixed boundaries. A band that moves crosses all three boundaries together.

A squash is the exception #

The fields that move rings cleanly are the circular ones: a twist and a radial spread act the same way all round the head. A squash, which compresses the head along one axis and stretches it along the other, does not, and it is the one field here that breaks the census.

Up to an aspect ratio of about 1.04, a strain of 1.4 per cent, the squash keeps 353 disputed cells and 264 fives and sevens. At 1.10 it has 364 and 275; at 1.21, 440 and 355. Before it breaks the census it distorts where the band lies: read against the radius each organ had before the squash, the rings keep their 13, 21 and 34 hexagons at an aspect ratio of 1.10, but read against circles drawn on the squashed head the ring of 55 keeps only 16. The band has become an oval, shifted differently at each angle round the head, and a circle through it catches only part.

What a correlated error means for a photograph #

The practical content of the earlier essay was a ranking: a distance cut-off is a property of an exact lattice, a three-family count is several times sturdier. That ranking was right for the error it measured, and a photograph does add independent error to every organ’s position. But a head’s own departure from Vogel’s rule is mostly correlated — a flattening, a shrinkage, a twist from uneven growth — and for that kind of error the ranking says less than it seemed to. Both instruments survive strains of a few per cent, and what they then report is not disorder but a band that has moved.

So a reading of a real head’s disputed cells has a question to ask before it counts them: are they more than the rings should carry, or the same number in the wrong place? The second moment of a cell’s side count, which counts fives and sevens wherever they are, is blind to a moved band and sees only a blurred one, which makes it the census-keeper here.

A head’s error, read from its band #

The result turns a nuisance into an instrument. If a head’s error is mostly correlated, its disputed cells are not noise to be filtered out before the rings are read; they are the rings, displaced by an amount the local geometry sets. A twist of 0.2 radians at the rim is a rotation nobody could see on a photograph of a 2,400-organ head — the rim turns by eleven degrees and the centre not at all — and it moves the ring of 55 by half a spacing, which the 34 hexagons round it report to within a hundredth.

Read the other way, the offset of each ring’s band from where the head’s own divergence puts it is a measurement of the head’s local divergence at that radius, one per ring. That is the same quantity the round trip through the counts recovers from an annulus, measured instead from where the cells in dispute sit, and it comes with a check the counts do not have: the census. A band that keeps its number of disputed hexagons and its fives and sevens has moved; one that has gained cells has blurred, and its position means less.

What these fields leave out #

It does not model a real head’s error, which is presumably correlated and independent at once. Strains add, so the tolerances of the two combine, but the census result does not: any independent part adds disputed cells, and a real head’s count of fives and sevens will be above its rings’ census by however much independent error it carries.

Nor does it read the Lucas head, the 9,000-organ head under a smooth field, or correlation lengths beyond eight spacings. The last is a head moving as a whole, which by construction changes nothing.

Findings that would overturn it #

A correlation length at which the cut-off off the rings fails after the count. A smooth field of eight spacings that changes a head’s disputed cells by more than a handful. A twisted ring more than a tenth of a spacing from where its local divergence puts it in closed form. Any of the three would mean the band’s motion is not what the arithmetic says.

Still open: the band on a head that grew #

Every head here was placed by Vogel’s rule and then moved. A head that grew — each organ placed where the ones already present leave room — carries its own departures from the rule, and they are neither independent nor a field applied afterwards. The measurement is the same census on grown heads: whether each ring carries its closed-form count of disputed hexagons and fives and sevens, and whether its band sits where the local divergence and rise of the grown lattice put the ring. If it does, a grown head’s defects are a moved band and can be read as a map of its local divergence; if it carries more, growth has disorder of its own.