A twist is a divergence

Recovering the angle from the counts built heads at stated divergence angles, forgot the angles, and got them back from the spiral counts alone to about a hundredth of a degree. A head displaced before it is counted then moved every organ by an independent normal error of up to two and a half spacings and found the round trip coarser, then silent, and very rarely wrong by much. A displaced head is counted as its own pair or a neighbour on its own sequence; small heads refuse before they mislead; of 898 intervals returned, 19 missed the true angle and 17 of those by about a tenth of a degree.

That essay closed on the displacements it had not tried. Every error it drew was independent from organ to organ and had no preferred direction. The displacements a real head suffers need not be either: growth, drying and pressing move whole regions together, and a field that moves neighbouring organs alike can move whole families of chords at once. The question was whether such a field can make the recovery confidently wrong where independent error never does.

What independent error did #

The reading being extended has a characteristic shape. At a quarter of a spacing nearly every head is counted as its own pair, 21 and 34. By a spacing, most are counted as the neighbouring pair on the same Fibonacci sequence; by two, nine in ten refuse. The counts move along their own sequence and then stop. They do not jump to a pair belonging to some other angle.

That is the behaviour to test smooth fields against. A field is worse than independent error if it moves a count off its sequence, or returns an interval that excludes the truth, at amplitudes where independent error does neither.

Four smooth fields #

Four fields were applied to golden and Lucas heads of 300 and 900 organs, each with one amplitude.

A squash presses the head along one axis and stretches it along the other with the area kept, the way a head dried against a flat surface is flattened; at the largest amplitude read the aspect ratio is 2.25. A radial spread moves every organ outward by a fraction that grows with the square of its radius, a swelling or drying that is greater at the rim; at the largest amplitude the rim moves out by eighty per cent. A simple shear slides every row along one axis in proportion to its height, up to a slope of 1.6. And a twist turns every organ about the centre by an angle proportional to its radius, so the centre does not move and the rim turns furthest, by up to eight radians.

Each field is deterministic, so each head at each amplitude is one reading rather than thirty. The round trip is unchanged: spirals counted in the band from 0.55 to 0.95 of the displaced head’s radius, the radius and the organ spacing read off the displaced points, and the angle recovered from the counts.

A head pressed flat keeps its angle #

Pressed to an aspect ratio of 2.25, a 900-organ golden head looks like a different object: an ellipse, with spirals that visibly bunch along one axis and spread along the other. The round trip counts it as 34 and 55 rather than 55 and 89 — one step down its own sequence — and recovers 137.52°, with an interval of 0.17° that holds the true angle.

The same is true of every squash read on every head. A 300-organ golden head is still counted as its own 21 and 34 at the largest squash; the Lucas heads drop at most to the neighbouring Lucas pair. The squash is large, far larger than any photograph of a head would show without comment, and the recovery does not notice it except by moving one step down.

The outcomes, field by field #

Laid out together, the four fields divide into three that behave like independent error at its gentlest and one that does not. Across four heads, the squash, the radial spread and the shear — eighty readings between them — never count a pair off the head’s own sequence and never return an interval that misses. The only changes are single steps to the neighbouring pair: a 900-organ golden head spread at the rim by a tenth or more counts 34 and 55, and a Lucas head sheared with a slope of 0.8 counts 29 and 47 or 18 and 29.

Every one of those steps is downward — twenty-five of the eighty readings moved, and all twenty-five to the smaller pair, none to the larger. That has a plain cause in the instrument rather than the plant. The counting band is a circle drawn at 0.55 to 0.95 of the displaced head’s largest radius, and every one of the three fields makes that largest radius larger than the head’s typical one: along the squashed axis, at the spread rim, at the sheared corners. So the band reaches inward, among organs whose counts are lower because counts change with radius, and the pair it reads is the one a smaller radius would show. It is the same move a person makes by counting nearer the centre, and it costs nothing, because the lower pair implies the same angle.

The twist is different from its second amplitude onward. It counts pairs that span the sequence — 34 and 89, skipping 55 — then pairs off it altogether: 34 and 47, 13 and 47, 18 and 65. It refuses where a count shares a factor, 34 and 68 or 18 and 36. And thirteen of its forty-four readings return an interval that excludes the true angle.

What a twisted head is counted as #

The picture shows why the counter is not at fault. Twisted by two radians at the rim, the 300-organ golden head’s spirals are plainly bent: one family is wound tighter and the other opened out. The counter reads 34 and 47 in the band it counts, and the recovery returns 137.80° with an interval of 0.23° — 0.30° from the truth and well outside its own interval.

At half a radian the same head still counts 21 and 34 and recovers the true angle. At one radian it counts 34 and 68, which share a factor, and refuses. At four it counts 13 and 47 and misses by 0.36°; at eight it counts 13 and 26 and refuses again. The failures are not noise. Each is a clean reading of a head that has changed.

A twist changes the angle between organs #

The reason is a line of arithmetic. Organ $k$ of a Vogel head sits at radius $\sqrt{k}$ in the head’s own units, so the step from one organ to the next moves outward by about $1/(2r)$. A twist turns each organ by $a\,r/R$ radians, so it turns organ $k + 1$ further than organ $k$ by about $a/(2Rr)$. That is added to the divergence angle between them.

A twisted head is therefore, locally, a head at a different divergence: the true angle plus $a/(2Rr)$, a little more at the inner edge of the counting band than at the outer. A head that grew at 137.508° and was then twisted by two radians has, in the middle of its band, the spiral lattice of a head at about 137.76°. Counting its spirals and recovering an angle is doing exactly what it should. It is the premise that the lattice records the angle the head grew at that has failed.

Each miss is where the twist says #

That account makes a prediction for every miss, and every miss meets it. The recovered angle is always above the true one, as a twist in this sense adds to the divergence. On the 900-organ golden head twisted by six radians the recovery misses by 0.257° and the twist predicts 0.255°; on the 300-organ Lucas head twisted by eight, 0.948° against 1.022°. Across the twelve misses under five degrees the observed shift is between half and twice the predicted one, the spread being the width of the band the counted pair allows, which quantises the answer.

The thirteenth miss is the one to remember.

Twisted by two radians, the 300-organ Lucas head counts 18 and 65, and the recovery returns 160.56° — sixty-one degrees from the truth, with an interval of 0.08°, as narrow as any correct one. The twist predicts a local divergence of 99.70° to 99.85° across the band, so this is not the twisted divergence recovered; it is something the twist made possible and does not explain on its own.

What happened is visible in the pair, and in where the twist has taken the angle. Across the band the twisted head’s local divergence runs from 99.70° to 99.85°, towards 100°, which is five eighteenths of a turn exactly. Near a rational angle a lattice turns into straight rows, and eighteen of them here: the 18-family is the shortest everywhere in the band. Counted over the inner half of the band alone, the second family is 36, a pair that shares a factor and on its own would be refused; over the outer half it is 65; over the whole band the counter reads 18 and 65. The recovery, asked for the angle at which that pair is the two shortest, found one — 160.56° — and reported it with the confidence the arithmetic allows. A little more twist, and the whole band reads 18 and 36 and the recovery refuses.

So the worst failure is not a twist fooling the round trip into a neighbouring angle. It is a twist carrying a Lucas head — the placement rule’s other branch, whose angle sits about half a degree from five eighteenths of a turn — close enough to that fraction for its spirals to straighten, and the counts of a nearly whorled head no longer describe any spiral lattice.

That is exactly the failure the essay on miscounts found for a counter wrong by one: not a blurred answer but a sharp one somewhere else. There the error was in the person counting; here it is in a head whose angle is not one number. Both give a report that looks like any other and is not.

Why the other three cannot do it #

A change of divergence has to turn every organ, relative to the one before it, by the same amount at every azimuth, because the divergence is the same at every azimuth. The twist does that: it is rotationally symmetric and turns organs along the circles they lie on.

The other three do not. A squash and a shear move an organ at the top of the head differently from one at the side, so the extra angle one organ gains over its predecessor changes sign around the circle, and a count taken over a whole annulus averages it away. A radial spread moves organs along their radii and does not turn them at all; it changes how quickly the rise falls with radius, which moves the counted pair along its own sequence — the same move a disc read as a family of cylinders makes when the counting radius changes — but cannot change which angle a pair implies.

The magnitudes make the point. The largest shear read has a slope of 1.6. A twist of a radian and a half, the smallest that made a 300-organ head miss, has a local shear in the middle of the counting band of about 1.1. It is not the size of the distortion that matters but whether it turns every organ the same way.

Measured against independent error, the twist that does harm is not large either. In the middle of the counting band of a 300-organ head, a twist of a radian and a half moves an organ past its neighbour one spacing further out by about 1.1 spacings; a twist of a quarter radian, which does nothing, by 0.19. Independent errors of a spacing or two move neighbours past each other by as much and more, and the round trip survived them by moving along its sequence. The difference is that independent errors point every way and cancel over an annulus, while a twist’s all point the same way round the circle. A systematic error of a spacing does what a random one of two cannot.

What the round trip can and cannot be asked #

The practical reading is reassuring and has one exception. A head measured from a photograph after pressing, drying or swelling can be expected to give its angle back, a pair a step down its own sequence at worst, with an interval that still holds the truth — unless it has been twisted. A twist is not detectable from the counts, because a twisted head is locally an ordinary lattice at another angle.

The contrast with a finer instrument is worth drawing. The single hop-ratio cut-off that turns counted contacts into cell walls was shut by independent displacements of a hundredth of a spacing; the round trip survived independent displacements of whole spacings and smooth fields that change the head’s shape by half again. The instruments seed heads are measured with here differ in tolerance by a factor of a hundred or more, and the robust ones are the ones that count families rather than measure distances.

In principle it is detectable another way. A twist makes the local divergence depend on radius, falling outward as $1/r$, where a head that grew at one angle has the same divergence at every radius. Recovering the angle in two annuli rather than one, which the essay on miscounts already recommended for another reason, would show a twisted head’s two answers disagreeing, larger nearer the centre.

In practice the arithmetic is against it at these sizes. On a 300-organ head twisted by a radian and a half — the smallest twist that made it miss — the local divergence at 0.6 of the radius exceeds that at 0.9 by only 0.08°, while each annulus’s recovered interval is 0.23° to 0.31° wide. At two radians the difference is 0.11°. On a 900-organ head twisted by six radians it is 0.11° against intervals near 0.08°, which is the first case where the two answers could be told apart at all. So the twist that breaks the round trip on a small head is one the same round trip, run twice, cannot yet see.

What this does not establish #

Whether any capitulum is twisted by as much as a radian. The fields here are chosen for their shape, not measured on plants, and the amplitudes at which a twist does harm are large: the rim turned by 86° relative to the centre at the smallest harmful twist. What the reading establishes is which shape of deformation to look for, not that it happens.

Nor does it combine fields with independent error. A real head is distorted smoothly and displaced organ by organ at once, and the two together may fail sooner than either alone; that was not measured.

What would withdraw it #

A squash, radial spread or shear at any amplitude read that counts a pair off the head’s own sequence, or returns an interval excluding the true angle.

A twisted head’s miss that is below the true angle, or more than twice the shift the twist predicts at the middle of the band.

A twist of a quarter radian or less that moves any recovery outside its interval.

Still open: the head size at which a twist shows itself #

The two-annulus check has an answer on paper and not yet in measurement. The local divergence of a twisted head falls as $1/r$, so the difference between an inner and an outer annulus is fixed by the twist, while each annulus’s recovered interval narrows as the head grows and its counts climb. On 300 organs the difference is smaller than the intervals at every twist that does harm; on 900 it first exceeds them at about six radians. The measurement is the round trip in two annuli on heads of 900, 2,400 and 9,000 organs, twisted and independently displaced together, asking at each size for the smallest twist whose two recovered angles separate by more than their own intervals — the size of head at which the counts alone can see a twist before it misleads them.