The flag reads the angle, not the twist

The round trip from spiral counts to a divergence angle takes a seed head’s counted parastichy pair and returns the interval of angles that would produce it. Recovering the angle built it, a head displaced before it is counted found it robust to organs moved at random, and a twist is a divergence found the one smooth deformation that fools it: a twist of the head about its centre, which changes the angle between one organ and the next and is recovered as a changed divergence.

The count sees the twist first then looked for a check. Two annuli, inner and outer, should recover different angles under a twist of $a \cdot r/R$, whose extra angle falls with radius; on heads of 900, 2,400 and 9,000 organs they never did in time. What caught the twist first, at half a radian to three quarters, was the count itself: some band stopped returning two consecutive Fibonacci numbers — 34 and 89, 89 and 233 — which no untwisted golden head ever showed. It ended by asking about twists of other shapes, $a (r/R)^p$, concentrated towards the rim or the centre.

This essay reads those, and finds that the answer is less about the shapes than about what the flag was flagging all along.

A twist’s change to the divergence #

Vogel’s rule puts organ $i$ at a radius proportional to $\sqrt{i}$, so $(r/R)^2 = i/(N - 1)$ on a head of $N$ organs, and a twist of $a (r/R)^p$ adds $a \, (i/(N - 1))^{p/2}$ to organ $i$’s angle. The step from one organ to the next — the change the twist makes to the divergence where it acts — is about

$$\delta(\rho) = a \cdot \frac{p}{2} \cdot \frac{\rho^{\,p-2}}{N - 1} \quad\text{radians, at } \rho = r/R.$$

Below $p = 2$ the change is largest near the centre, and the linear twist of the earlier essays, with $\rho^{-1}$, is of that kind. Above two it is largest at the rim. And at exactly two the exponent of $\rho$ vanishes: a quadratic twist changes the divergence by $a/(N - 1)$ at every radius, the same between every pair of neighbours from the first organ to the last.

A quadratic twist is a divergence angle #

A constant change to the divergence is not like a different divergence angle; it is one. Twisting a 900-organ golden head by a quarter of a radian at the rim in proportion to $r^2$ puts every organ within $10^{-11}$ of a spacing of where the golden angle plus $0.25/899$ radians — 137.524° — puts it, and the same holds at one radian and four, and on 2,400 organs. There is no difference between the two heads for any instrument to find.

So they count the same pairs. At a quarter of a radian both read 34 and 55 in the inner annulus and 34 and 89 in the outer annulus and the single band: the skipped Fibonacci number the earlier essay called the twist’s flag, on a head that has not been twisted at all. At one radian — 137.571° — both read 21/34 inside and 34/123 outside; at four — 137.763° — both read 34/47 and 34/81.

That settles what the flag is. A count that skips a Fibonacci number is what a head not at the golden angle shows in a band wide enough to hold one of its transitions, whatever made it not golden. The earlier essay’s statement that no untwisted head ever skips was true of the untwisted heads it read, which were all golden.

The flag has a resolution #

If the flag reads the angle, the right question is how far from the golden angle a head must be for it to fire, and that can be asked of untwisted heads directly.

A 900-organ head grown 0.0125° above the golden angle counts consecutive Fibonacci numbers in every band; grown 0.015° above, its outer annulus and single band count 34 and 89. Below the golden angle it tolerates more, to 0.02°, and flags at 0.03°. On 2,400 organs the flag fires at 0.01° above and 0.005° below; on 9,000, at 0.003° above and 0.005° below.

The resolution sharpens with head size, as it should: a larger head’s outer bands sit at higher counts, and a count is worth more the higher it is — the band of angles a pair allows narrows as 221°/mn. The two sides are not symmetric, and which is the tighter changes with the head: above the golden angle on 900 and 9,000 organs, below it on 2,400. The flag fires when a transition — a radius where one family hands over to the next — moves into a band, and how near the nearest transition sits to a band’s edge on either side is a fact about each head’s size rather than about the angle. And the inner annulus of a 900-organ head never flags within a twentieth of a degree above: its counts are too low to see so small a departure.

An early warning, and a steady one #

The flag is not the only reading a head at the wrong angle gives. The round trip itself fails too, eventually: the single band’s recovered interval stops containing the golden angle. On untwisted heads that happens much later than the flag. A 900-organ head is first misled at 0.3° above the golden angle and 0.1° below, twenty and three times the offsets that flag it; a 2,400-organ head at 0.03° above and 0.1° below; a 9,000-organ head at 0.01° and 0.0125°. In every case the flag comes at between a third and a twentieth of the offset that misleads.

The two readings also behave differently as the offset grows. Once a band’s count is flagged it stays flagged at every larger offset read, on every head and on both sides. The misleading reading comes and goes: on the 9,000-organ head the single band is misled at 0.01° above the golden angle, recovers an interval containing it again from 0.0125° to 0.02°, and is misled again at 0.03°. That is the same mechanism in both — a transition moving through the band — but the recovered interval is re-centred each time the counted pair changes, and sometimes the new pair’s interval happens to contain the golden angle again. A flag has no interval to be re-centred.

It is also finer than the interval the round trip reports on the same head. A golden 900-organ head’s single band counts 55 and 89 and returns 137.478° to 137.565°: a band 0.087° wide, with the golden angle 0.057° from its upper edge. The flag fires at 0.015° above, a quarter of the way to that edge. On 2,400 organs the band counts 89 and 144 and returns an interval 0.033° wide, with 0.011° of room above the golden angle, and the flag fires at 0.01°; on 9,000 organs, 144 and 233, 0.010° wide, 0.005° of room, flag at 0.003°. The round trip says the angle is somewhere in its interval; the flag says, at a fraction of that width, whether it is exactly golden.

So as an instrument for asking whether a head is at the golden angle, the skipped count is better than the round trip it was introduced to protect: it fires earlier, and it does not unfire.

Every shape is flagged as a divergence #

Now the twist shapes. A twist of any exponent changes the divergence in the outer annulus by $\delta(\rho)$ at $\rho \approx 0.825$, and if the flag reads the angle, a twisted head should be flagged exactly when that change passes an untwisted head’s resolution.

It is. On the 900-organ head the first twist flagged is a quarter of a radian for exponents from one and a half to four, half a radian for the linear twist, and three quarters for $p = 0.5$; on 2,400 organs, half a radian for exponents two to four, three quarters for one and one and a half, and one radian for a half. In all twelve cases the outer annulus’s divergence change at the first flagged twist is at least the largest offset an untwisted head leaves unflagged, and at the twist before it less than the smallest offset it flags. The linear twist on 900 organs, flagged at half a radian, changes the outer divergence by 0.0193°; the untwisted head flags at 0.015°.

So the shapes differ in when they are flagged only because they put different amounts of divergence change in the outer annulus. A twist concentrated at the rim is flagged sooner; one concentrated near the centre, later, and on 2,400 organs a twist as $r^{0.5}$ needs a full radian at the rim. None of this depends on anything about twisting beyond the change it makes to the angle between neighbours where the counts are taken.

The earlier thresholds, predicted #

The same arithmetic accounts for the linear twists the earlier essay read, including the largest head, which this essay did not twist. A linear twist of $a$ radians changes the outer annulus’s divergence by $a/(2 \cdot 0.825 \, (N - 1))$ radians. Setting that equal to an untwisted head’s resolution gives the twist at which it should first be flagged: between 0.32 and 0.39 radians on 900 organs, 0.52 and 0.69 on 2,400, and 0.52 and 0.78 on 9,000. On the grid of twists the earlier essay read, those fall at half a radian, three quarters and three quarters — which is exactly where it found the flag on every seed.

That also explains the observation it made without accounting for it: the flag needed a larger twist on the larger heads, although their counts are higher and their resolution finer. A twist of fixed size at the rim is spread over more organs on a larger head, so it changes the angle between any two neighbours by less — in proportion to $1/(N - 1)$ — and the resolution sharpens more slowly than that: 0.015° on 900 organs, 0.01° on 2,400, 0.003° on 9,000. The product of the head size and the resolution, which sets the twist that is flagged, rises from 13.5 to 24 to 27. A large head is harder to twist into a skipped count, not easier.

What would tell a twist from an angle #

Only one reading can distinguish a twist from a head grown at another angle: the angle recovered in two places disagreeing, because a twist’s change to the divergence varies with radius and a changed angle’s does not. The ratio of the change in the inner annulus to the outer is $(0.475/0.825)^{p-2}$ — 1.74 for the linear twist, 2.29 for $p = 0.5$, one at $p = 2$, a third at $p = 4$ — so the check has something to find everywhere except at $p = 2$.

It finds it late. On the 900-organ head the two annuli recover separate intervals at eight radians for the linear twist, at six and eight for $p = 3$ and at six for $p = 4$ — every time with the single band already misled — and never at $p = 2$, as they cannot. On the 2,400-organ head they never separate at any exponent within eight radians. The recovery’s intervals are wide enough, and the transitions dense enough, that by the time the two annuli’s angles are far enough apart to see, the head has stopped being countable in one of them.

The flag still comes first #

The earlier essay asked whether a twist concentrated near the centre could mislead the single band without any band’s count skipping. It cannot, at any exponent read. On both heads, at every exponent from a half to four, some band is flagged at a smaller twist than the one that first misleads the single band — on the 2,400-organ head half a radian against one for $p = 3$ and 4, three quarters against two for the linear twist, one radian against four for $p = 0.5$. The 900-organ head twisted as $r^{0.5}$ is never misled at all within eight radians.

So the flag keeps its practical value: a head whose counts all come out as consecutive Fibonacci numbers has not been twisted enough to mislead its round trip. What it loses is its name. A flagged head is a head whose divergence is not golden to within the flag’s resolution, and whether that is because it was twisted by the press, or grew that way, the counts cannot say.

Why this matters for a survey #

A survey counting real heads will find some whose bands skip a Fibonacci number. Read the earlier way, those are damaged specimens — twisted in drying or pressing — and a careful survey sets them aside. Read this way, they are specimens whose divergence differs from the golden angle by more than a hundredth of a degree, which on a large head is a perfectly ordinary botanical fact: how often it is Fibonacci turns on exactly such heads. Setting them aside as damaged would remove from the survey the very departures from the golden angle it exists to measure.

Read that way, a flag is a measurement rather than a defect, and a small one. Its size depends on the head and on where the bands were drawn: on 900 organs a head that counts consecutive Fibonacci numbers in every band is golden to within 0.015° above and 0.03° below, on 9,000 organs to within 0.003° and 0.005°. A survey that records the head’s organ count, the radii of the bands it counted in and the pair in each band has recorded a bound on the divergence that no single count can give, and one a count taken over too few organs would silently lose. A survey that records only “34 and 89, discarded as damaged” has thrown that bound away.

The distinction has to come from outside the counts: a photograph taken before pressing, a record of how the head was dried, or the positions of the organs themselves rather than the spirals through them, which carry the radial profile of the angle that the counts throw away.

What these heads leave out #

They are exact Vogel heads, with radius growing as the square root of the organ’s index. The equivalence of a quadratic twist and a divergence change rests on that law; on a head whose radii follow another law, the twist that mimics a divergence has another exponent, and some twist always does. If the radius of organ $i$ grows as $i^q$, a twist of $a (r/R)^p$ adds $a (i/N)^{pq}$ to its angle, which is a divergence change exactly when $pq = 1$: the quadratic twist for Vogel’s $q = 1/2$, a linear twist for a head whose radii grew in proportion to the organ’s index. They are undisplaced, so the flag’s resolution here is the best case; displacement coarsens it. And the twist shapes are power laws, which a real deformation need not be.

Findings that would overturn it #

A quadratic twist whose organs differ from the equivalently grown head’s by more than rounding. An exponent at which the twisted head is flagged while its outer annulus’s divergence change is below an untwisted head’s unflagged offset, or not flagged when above its flagged one. A twist that misleads the single band before any band is flagged. Each would mean the flag reads something other than the angle.

Still open: the angle’s profile from the positions #

A twist and a divergence differ in one thing, the radial profile of the angle between neighbours, and the counts are too coarse to see it before they fail. The organs’ positions are not: in a photograph each organ has a place, and the angle between the $k$-th and the $(k+1)$-th in birth order is a direct reading of the local divergence — if the birth order can be recovered. The measurement is whether the order can be read back from positions on a twisted head, and at what twist the profile of the angle it gives separates a twist from a changed divergence, on heads of the sizes read here.