A head displaced before it is counted

Recovering the angle from the counts built heads at four stated angles, set the angles aside, counted the spirals from the points alone and recovered each angle to about a hundredth of a degree. It was a clean test of the round trip, and it was clean in a way no plant is: every organ sat exactly where Vogel’s rule put it.

A real head is not like that. Primordia are pushed by their neighbours as they grow, a head dries and warps, and a photograph adds its own error to every position. So the question that essay left is what displacement does to the round trip — whether it makes the recovered angle wrong, and if it does, by how much and how often.

Displacing a head #

The heads here are the same Vogel heads, of 300 and 900 organs, at the golden angle and at the Lucas angle. Every organ is moved independently by a normal displacement in each direction, of a stated share of the mean spacing between organs, and the displaced head goes through the round trip exactly as before: spiral counts in the band from 0.55 to 0.95 of the radius, the radius and scale read off the displaced points, and the angle recovered from the counts. Thirty heads are displaced at each of nine sizes, from none to two and a half spacings.

At one and a half spacings a head has no visible spirals at all. The undisplaced head on the left counts 21 and 34 and recovers 137.515°; the displaced head on the right counts 55 and 89, and no divergence angle makes that pair the shortest at that radius, so the recovery refuses.

What the counter reads #

The counter does not look for curves. For each offset in the order of arrival, it takes the median distance from every organ in the band to the organ that many arrivals later, and the counted pair is the two offsets whose median distances are deepest minima. That is a median over hundreds of organs, which is why a displacement smaller than a spacing does very little to it.

On the head of 900 organs the three deepest minima are at 55, 89 and 34 undisplaced, and still at 55, 89 and 34 displaced by a spacing. Displaced by two spacings they are at 34, 89 and 55. Displacement adds the same random part to every distance, which lifts the whole curve and shrinks the gaps between the minima, until the order of the minima is decided by the random part rather than by the lattice.

A large head keeps its sequence #

Up to half a spacing every one of the thirty heads of 900 organs at the golden angle counts 55 and 89. At three quarters of a spacing and at one, two heads in thirty count 34 and 55 instead. At two spacings 73 per cent still count 55 and 89, 10 per cent count 34 and 55, and 17 per cent count 34 and 89 — two Fibonacci numbers that are not neighbours, which is what a count straddling a transition gives. None refuses until two and a half spacings, and then one head does.

The heads of 900 organs at the Lucas angle are steadier still: all thirty count 47 and 76 up to a spacing, and at two spacings 70 per cent do, the rest counting 29 and 47 or 29 and 76.

A small head moves sooner #

A head of 300 organs has fewer organs in its band and its median distances are less settled. At a quarter of a spacing one of the thirty golden heads already counts 34 and 55 rather than 21 and 34. At half a spacing half of them do, and at a spacing eighty per cent do, with two more counting 21 and 55. From a spacing and a quarter onward the heads start to refuse.

The small heads at the Lucas angle hold their pair, 29 and 47, to three quarters of a spacing, and then begin to refuse without much moving first.

Which family comes in #

The pairs a displaced head moves to are not arbitrary, and they obey one rule on all four heads. Every moved pair, below two spacings, includes the family whose chord was third shortest on the undisplaced head. On the head of 900 golden organs that family is 34, and the moves are to 34 and 55 and to 34 and 89. On the head of 300 it is 55, and the moves are to 34 and 55 and to 21 and 55. On the Lucas heads it is 29 and 18.

That is what shrinking gaps between minima would do. The family nearest to taking a place in the counted pair is the one whose chord was nearest the pair’s already, and displacement is enough to let it in. So a displaced head’s count moves along its own sequence, to a pair the counts change with radius would have shown one ring inward or outward.

Why some heads move sooner #

How soon a head moves is set by two numbers the undisplaced head already carries. The first is how close the third family’s chord is to the counted pair’s. On the head of 900 golden organs, in the counted band, the median chords are 1.005 spacings for 55, 1.177 for 89 and 1.230 for 34, so the third family is only 0.053 of a spacing behind the second. On the head of 300 golden organs the gap is 0.063. On the Lucas heads it is 0.322 and 0.216 — five and three times wider — and that is why the Lucas angle’s heads hold their pair to a larger displacement.

The second is how many organs the median is taken over: 540 in the band of a 900-organ head and 179 in the band of a 300-organ head. A median over a third as many organs moves further for the same displacement, so on a gap of nearly the same width the small golden head’s second and third minima should trade places sooner than the large one’s — and they do, at a quarter of a spacing against three quarters. The two numbers are separate levers. The gap is set by the angle and where the band sits on it; the organ count is set by the size of the head. A Lucas head of 300 organs has the small head’s count and still holds its pair to three quarters of a spacing, because its gap is the wide one.

The interval narrows as the count moves up #

A moved count is usually a higher pair, and a higher pair pins the angle more tightly: a pair is worth a band $221°/mn$ wide, so 21 and 34 allow about 0.31° and 34 and 55 about 0.12°. On the small golden head the mean width of the recovered interval falls from 0.310° undisplaced to 0.221° at a spacing and 0.127° at two, as more heads count 34 and 55.

The misses sit in the same stretch. On the small golden head the first intervals to exclude the true angle appear at a spacing and a quarter — two of them — and six at a spacing and a half, which are the displacements where the moved pair has already become the usual reading and the intervals are at their narrowest. By two spacings only three heads are still recovered at all, so the 0.127° there is an average over three and says little by itself. The misses themselves say which reading produced them: all ten on the small golden head are 34 and 55, the moved pair, read at the head’s inner radius. None is the undisplaced 21 and 34. So the miss is not the counter losing its pair; it is the counter finding the next pair early, at a radius where the ideal head would not yet show it, and the band for that pair at that radius sitting a tenth of a degree below the truth. What has not been checked is why the offset is always downward — all ten recover between 137.405° and 137.435° — and a head displaced in a second random draw at the same size would be the first test of whether that direction belongs to the angle or to the thirty seeds used here.

And then it refuses #

The recovery refuses when no pair can be counted or when no divergence angle makes the counted pair the shortest at the counted radius. The heads of 900 organs almost never refuse: none at the Lucas angle and one at the golden angle, at two and a half spacings. The heads of 300 organs refuse a third of the time at a spacing and a quarter at the golden angle and at a spacing at the Lucas angle, and 90 and 87 per cent of the time at two spacings.

A refusal is a reading, and recovering two numbers from the points is the essay that treats it as one. It says that the counted pair is inconsistent with the counted radius, which is exactly what a pair moved by displacement often is: the refusal that caught the counter returning the wrong pair is the same check doing the same work on a different cause.

How far the angle moves #

On the heads of 900 organs the median error stays at the undisplaced head’s to a spacing and a half — 0.012° at the golden angle and a few thousandths at the Lucas angle — and rises to 0.017° at two spacings. On the small golden head it grows from 0.007° as soon as the count moves to 34 and 55, to 0.038° at half a spacing and 0.078° at two. It grows because the moved pair is being read at a radius where the head showed a different pair, and the band that pair allows at that radius sits a little off the true angle.

How often it misleads, and by how much #

Across all four heads and all nine displacements, 898 heads were recovered and 19 of their intervals exclude the true angle: 7 of 269 on the large golden head, 10 of 188 on the small golden head, none of 270 on the large Lucas head and 2 of 171 on the small Lucas head.

Seventeen of the nineteen miss by less than 0.11°. On the small golden head all ten are 34 and 55 read at the inner radius, recovering 137.405° to 137.435° against the true 137.508°. On the large golden head six of seven miss by about three hundredths, at two and two and a half spacings. Those are near misses: intervals a tenth of a degree wide that sit a tenth of a degree off.

The two far misses #

Two misses are not near. A head of 900 golden organs displaced by two and a half spacings counted 34 and 89 at a radius where that pair is consistent with 84.9°, and a head of 300 Lucas organs displaced by two spacings counted 47 and 58 and recovered 130.3°. Both are narrow intervals far from the truth, which is the dangerous kind, and both need a displacement of two spacings or more — a head with no visible spiral in it.

Two in 898, at displacements where most small heads refuse and every head looks like noise, is a low rate. Both are the kind of miss a second reading at another radius exists to catch — a pair far from its neighbours on the sequence — though whether a second band would have caught these two has not been measured here.

Against a counter wrong by one #

A count that can be wrong by one found that a person’s miscount of a high pair does something specific: it moves the report to a pair whose band is as narrow as the true one and tens of degrees away, so a wrong 34 and 55 becomes a confident 109° or 113°. Displacement does something different. It moves the count along the head’s own sequence, to a pair whose band is also near the true angle, and when it cannot do that it refuses.

The difference is in what each error acts on. A miscount changes a number after it has been read, and the neighbouring number belongs to a different lattice. Displacement changes the positions before they are read, and the counter’s median over hundreds of organs still reads the lattice those organs were placed on — until it cannot, and then the recovery says so.

What a person photographing a head would need #

Very little beyond what the round trip already asks for, which is the useful conclusion. A head of 900 organs photographed with every organ misplaced by a spacing — far worse than any careful photograph — returns the same angle as a perfect one. A head of 300 organs photographed that badly returns a neighbouring pair and an angle about a twentieth of a degree off, or refuses.

The one thing worth adding is a second band. The near misses are all a moved pair read at the wrong radius, and the two far misses are pairs that do not continue the sequence either side of them, so counting a second annulus and requiring the two recovered intervals to overlap is the natural check. It is the check a person should add, and it is not measured here: how many of the nineteen misses it would remove is the next measurement rather than this one.

What this does not say #

It does not say that real displacement is independent from organ to organ or normal. A head crushed along one axis, or a young head whose inner organs are displaced more than its outer ones, is a different experiment with a different answer. The band counted is the round trip’s own and is not varied. Nothing here measures a plant; the heads are ideal Vogel heads with a stated error added.

Nor does it say anything about error made while the head grows. The displacement here is added once, after every organ is in place, and the organs it moves were placed at the exact angle. A growing apex whose angle wanders from one organ to the next is a different error, and noise at the apex is not a slow rate: across a hundred and sixty noisy stems one changed branch, at an amplitude where the pattern was already coming apart. Noise in the growth mostly leaves the pattern’s sequence alone and degrades the pattern itself, while displacement after the fact leaves the sequence alone too and moves only which pair a counter reads. A counter would meet both in a real photograph, and nothing here separates them.

The claim, reduced #

Displaced before it is counted, a head’s round trip gets coarser, then silent, and very rarely wrong by much. Heads of 900 organs keep a pair from their own sequence to two spacings and more; heads of 300 organs move sooner and then refuse. Every moved count brings in the family whose chord was third shortest. Of 898 heads recovered, 19 intervals miss the true angle, 17 of them by under 0.11°, and the two far misses need two spacings or more.

What would withdraw it #

More than one head in thirty at a quarter of a spacing counted differently from the undisplaced one. A large head missing the true angle at a spacing and a half or less. A moved count that does not include the third-shortest family. A miss of a degree or more below two spacings. Each is checked whenever the heads are displaced.

Still open: displacement with a direction #

Every displacement here is independent and has no preferred direction. The displacements a real head suffers need not be either: growth, drying and pressing can each move organs in a preferred direction, and a head pressed flat is squeezed along one axis. Each of those is a correlated field, and a correlated field can move whole families of chords together rather than adding the same random part to every one.

The measurement is the same round trip with displacements drawn as a smooth field — radial, along one axis, and along the spiral families themselves — at the same sizes: whether the counted pair still moves only to a neighbour on its own sequence, and whether a displacement aligned with a family can make the recovery confidently wrong at sizes where independent displacement never does.