The positions read the twist's shape

A seed head’s counts pin its divergence angle to an interval. Recovering the angle built that round trip, a head displaced before it is counted found it robust to organs moved at random, and a twist is a divergence found the one deformation that fools it: a head turned about its centre, which changes the angle between one organ and the next and is read as a changed divergence.

The flag reads the angle, not the twist then settled what the counts can do about that. A band that skips a Fibonacci number flags a head whose divergence is not golden — 0.015° off on 900 organs, 0.01° on 2,400, 0.003° on 9,000 — whatever made it so. A twist of $a\,(r/R)^p$ changes the divergence between neighbours by

$$\delta(\rho) = a \cdot \frac{p}{2} \cdot \frac{\rho^{\,p-2}}{N-1}$$

at a distance $\rho$ from the centre, so a twist and a head grown at another angle differ only in the radial profile of that change, and the counts could not see the profile before they failed: their two annuli separated at six to eight radians on 900 organs and never on 2,400. The essay ended by pointing past the counts to the organs’ positions, which carry the profile if the birth order can be recovered. This essay reads them.

The birth order is not in the photograph #

On a Vogel head organ $i$ sits at radius $c\sqrt{i}$, so sorting the organs by radius puts them in birth order, and the angle between each organ and the next is the local divergence, read directly. That is the reading the last essay hoped for, and it fails at once.

Consecutive organs at the rim of a 900-organ head are $c(\sqrt{i+1} - \sqrt{i})$ apart in radius: a hundred and sixth of a spacing. On 2,400 organs a hundred and seventy-fourth, on 9,000 a three hundred and thirty-sixth. Displace every organ by a thousandth of a spacing and the 900-organ head keeps its rim in order; by a hundredth, 75 per cent of consecutive pairs keep their order in radius, 66 on 2,400 organs and 59 on 9,000; by a tenth, 53, 52 and 51, which is a coin.

A tenth of a spacing is gentle. It is less than the displacements the counts shrug off, which run to whole spacings, and far less than the error anyone locating the centres of florets on a photograph should claim. So the order in which the organs were laid down is not recoverable from where they sit, and an angle between organs $k$ and $k + 1$ is not a thing a photograph offers.

The family is in the photograph #

What survives displacement is the lattice, and it survives because it is a statement about many organs at once. The counts already use it: the counter finds the parastichy numbers $m$ and $n$ of a band. On a Vogel head every organ satisfies a relation that uses those numbers and nothing else. Write $\beta$ for the angle by which the $m$-th organ along a family has turned, wrapped to a half turn: $\beta = m\delta - 2\pi k$ for whole $k$. Then organ $i$, at angle $i\delta$ and with $(r/c)^2 = i$, has

$$m\theta - \beta\,(r/c)^2 = m i \delta - (m\delta - 2\pi k)\,i = 2\pi k i \equiv 0 \pmod{2\pi}.$$

So the average of $e^{\,i(m\theta - \beta r^2/c^2)}$ over the organs has modulus one at the right $\beta$ and falls away from it. The sum needs each organ’s angle and radius, the family’s number and the head’s scale $c$, fitted from the sorted radii; it never needs to know which organ is which. Given the peak, the divergence is $(\beta + 2\pi k)/m$.

The picture is the outer annulus of a 900-organ golden head, displaced by a tenth of a spacing: 382 organs, counted as 55 and 89. The counts’ recovered interval runs from 137.4801° to 137.5492°, seven hundredths of a degree wide. Inside it the two families’ phase sums each rise to one sharp peak, and their sum peaks at 137.50773°, against the golden angle’s 137.50776°. The counts supply the integers and the window; the positions supply the value inside it, three orders of magnitude finer.

Why the two families are read together #

Each family’s phase sum has side lobes as well as a peak, spaced by about $360°/(m \cdot \Delta u)$, where $\Delta u$ is the number of organs the annulus spans in birth order: a few hundredths of a degree for the inner annulus of a 900-organ head. On an undisplaced head the side lobes are low and never compete. Displace the organs and every family’s peak falls — to about 0.8 at a tenth of a spacing in the outer annulus, and to between 0.15 and 0.4 at a quarter — and the family with the larger number, whose phase angle multiplies every organ’s error in angle by that number, falls furthest.

Read alone, it is then caught. On the inner annulus of a 900-organ head displaced by a quarter of a spacing and grown 0.01° above golden, the 55-family’s own peak fell on a side lobe on two seeds in ten, reading 0.0956° and 0.1747° above golden instead of about 0.01; the 34-family read all ten within a hundredth of a degree. Averaging the two readings would have carried those errors into the result. Summing the two families’ squared moduli before finding the peak does not, because the two families’ side lobes are spaced differently and never coincide: the joint reading put all ten seeds between 0.0077° and 0.0126° above golden. Every reading below is the joint one.

What the reading is worth #

On an undisplaced head the reading is exact. Heads of 900 to 9,000 organs grown at the golden angle, or a hundredth or a twentieth of a degree off it, are read to a ten-millionth of a degree in both annuli.

Displaced, the error grows with the displacement and shrinks steeply with the head. On 900 organs the outer annulus reads to $7 \times 10^{-5}$ degrees at a twentieth of a spacing, $1.2 \times 10^{-4}$ at a tenth and $6.8 \times 10^{-4}$ at a quarter; on 2,400 organs to $1.7$, $3.6$ and $14 \times 10^{-5}$; on 9,000 to $2.4 \times 10^{-6}$ at a tenth. The skipped count’s resolution on undisplaced heads of the same sizes is 0.015°, 0.01° and 0.003°. At a tenth of a spacing the positions read the divergence 125 times finer than the flag resolves it on 900 organs, 278 times on 2,400 and more than a thousand on 9,000.

Two things make the reading sharpen so fast with head size. The families counted are higher — 55 and 89 at the rim of 900 organs, 144 and 233 at the rim of 9,000 — and a family’s reading of the divergence is $m$ times as fine as its reading of $\beta$. And the annulus holds more organs, each adding to the same sum. This is what a count is worth again, one level down: the count’s worth grows as the product of the two counts, and a position reading built on the count inherits the growth and adds the organs.

The profile, ring by ring #

With a reading that fine, the profile a twist leaves is not a subtle thing. Read in six rings a tenth of the radius wide, a 900-organ head twisted by half a radian at the rim draws $\delta(\rho)$ ring by ring.

Undisplaced, the linear twist reads 0.0397° above the golden angle in the innermost ring and 0.0177° in the outermost, falling as $1/\rho$; the twist as $\rho^4$ reads 0.0104° inside and 0.0518° outside, rising as $\rho^2$. Every ring’s reading matches $\delta(\rho)$ averaged over the ring’s organs to within three per cent at every exponent. The square law reads 0.0319° in all six rings, which is $a/(N - 1)$ in degrees, and is the same reading a head grown 0.0319° above the golden angle gives.

Displaced by a tenth of a spacing the curves keep their order and their slopes; the innermost ring, which holds the fewest organs and the lowest counts, scatters most, by about half a hundredth of a degree. At a quarter of a spacing the inner rings scatter by up to three hundredths, and the curves cross there; the outer rings still read their shape.

Two annuli, as the counts used them #

To compare with the counts on their own ground, the positions are read in the same two annuli the counts used — 0.35 to 0.6 of the radius and 0.7 to 0.95 — and a twist is said to be seen when the inner reading minus the outer is larger than it ever is on an untwisted head of the same size and displacement. The untwisted heads are grown at the golden angle and at a hundredth and a twentieth of a degree either side of it, on every seed read, so that the threshold includes whatever a changed angle does to the difference. On 900 organs at a tenth of a spacing the largest such difference is a thousandth of a degree; on 2,400, a third of that; on 9,000, a hundredth of a thousandth.

At a tenth of a spacing a 900-organ head’s positions tell the twist from a changed angle at a tenth of a radian for the linear twist and for exponents of a half and three, at a quarter for one and a half, and at a twentieth for four. A 2,400-organ head does it at a twentieth of a radian for every exponent but one and a half, which needs a tenth. A 9,000-organ head does it at a hundredth, the smallest twist read. The skipped count, which says only that the head is not golden, first fires on an undisplaced head at a quarter of a radian to three quarters on 900 organs and at half a radian to one on 2,400.

So at every exponent read the positions separate the shape at or before the twist at which the counts first notice anything, and on the larger heads by a factor of ten. The counts’ own two annuli, set the same question, answered at six to eight radians on 900 organs and not at all on 2,400. The count sees the twist first could only say what the counts could see; what the counts cannot see, the organs they are counted from can.

A quarter of a spacing #

Displacement costs the reading precision but does not blind it. At a quarter of a spacing a 900-organ head separates every shape but the square by half a radian — a quarter for exponents of a half, three and four — and a 2,400-organ head by a quarter, a tenth for exponent four. A 9,000-organ head needs a twentieth. The counts’ interval widens at that displacement too — displaced heads count pairs further from their own — but the reading’s only use of it is as a window, and a wider window costs the positions nothing but search.

One shape at a quarter of a spacing on 2,400 organs is never separated, the exponent of one and a half, and the reason is worth knowing: at half a radian and one the counts on some seeds return a pair with no recoverable interval, and the positions, which search inside the counts’ window, return no reading. The positions refine what the counts supply. Where the counts supply nothing — the outer annulus of a 900-organ head grown a twentieth of a degree below golden, which counts a pair with no recoverable interval — the positions say nothing either, and the reading is honest to refuse rather than search the whole circle for a peak.

The one twist no reading separates #

The square-law twist is the exception at every size and every displacement, and it is not a failure of the reading.

Twisted as the radius or its cube, the two annuli’s difference climbs out of the untwisted heads’ band by a tenth of a radian on every seed and keeps climbing, to a hundredth of a degree by half a radian. Twisted as the square, every seed’s difference stays inside the band at every twist up to a radian, and undisplaced the two annuli agree to a millionth of a degree. The flag essay showed why: a square-law twist puts every organ within $10^{-11}$ of a spacing of where a head grown at the golden angle plus $a/(N-1)$ puts it. There is no second head for any reading of any precision to find.

That is a property of the radius law and not of the square. If organ $i$’s radius grew as $i^q$, a twist of $a\,(r/R)^p$ would add $a\,(i/N)^{pq}$ to its angle, and the invisible twist would be the one with $pq = 1$. On any real head some shape of twist is, organ for organ, a changed angle, and the positions separate everything else.

What a survey could record #

The flag essay ended by arguing that a flagged head is a measurement, not a defect: a head counted as 34 and 89 is not golden to within a stated resolution, and a survey that sets it aside as damaged throws that away. The positions sharpen the measurement and add a second one. From a photograph good to a tenth of a spacing, a 900-organ head’s outer annulus states its divergence to a ten-thousandth of a degree, and the two annuli state whether the divergence changes across the head.

A head whose two annuli agree grew at one angle, or was twisted as the square of its radius, and nothing distinguishes those. A head whose annuli disagree was twisted — by pressing, drying or uneven growth — and the sign and size of the disagreement say which way the twist is concentrated. How often it is Fibonacci turns on heads a little off the golden angle; with positions, a survey can keep the ones that grew off it and set aside only the ones that were bent there.

What these heads leave out #

They are Vogel heads, with radius exactly $c\sqrt{i}$, and the relation between angle and radius rests on that: on a head whose radius law differs, the sum must use that head’s own law, which has to be fitted first, and a law misfitted by a fraction shifts the reading by a fraction of $\beta/m$. The displacement is independent from organ to organ; a smooth field of displacement moves neighbours together and would move a family’s phase coherently, which is a different test. And the head’s centre and scale are taken as given, where on a photograph both must be found — the centre to much better than a spacing, since an off-centre reading turns a twist-free head’s angles by an amount that varies round it.

The twists are power laws, and a real deformation need not be one. What the reading establishes is narrower and firmer: that the radial profile of the divergence is readable from positions far below the size at which counts can say anything, and that the square law is the one shape it cannot see.

Findings that would overturn it #

An undisplaced Vogel head whose ring readings depart from $\delta(\rho)$ averaged over the ring by more than a few per cent. A square-law twist, at any size or displacement read here, whose two annuli differ by more than an untwisted head’s. A displacement of a tenth of a spacing at which consecutive organs at the rim of a 900-organ head keep their radial order more than two times in three. Each would mean the reading or the arithmetic here is wrong.

Still open: the centre the photograph does not give #

Everything here takes the head’s centre as known. A photograph gives a scatter of florets, and the centre must be estimated from them — by the symmetry of the rings, by fitting the radius law, or by the point about which the families’ phase sums are most coherent. An error in the centre adds to every organ’s angle a term that varies round the head, and a centre off by a fraction of a spacing may look like a twist concentrated near the centre. The next measurement is the same reading with the centre estimated from the displaced points, asking how well the centre can be found, how large an error in it reads as a twist of each shape, and whether the positions still separate a small twist from a changed angle once the centre is not given.