The centre the spirals give

The positions read the twist’s shape found a way past the counts’ coarseness. The organs’ birth order is not in a photograph, but each counted family is, and the sum of its phase angles over the organs peaks at the local divergence with no organ named. Read in two annuli, that sum tells a twisted head from one grown at another angle at a tenth of a radian of twist on 900 organs displaced by a tenth of a spacing, and at a twentieth on 2,400 — at or before the twist at which a skipped count first says anything at all.

Every one of those readings took the head’s centre as given, as recovering the angle and every reading after it had. A photograph gives a scatter of florets, and the centre is a number that has to be found from them. The essay ended on the danger: 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 might look like a twist concentrated near the middle. So there are three questions — how large a centre error reads as a twist, how well the centre can be found, and whether the positions still separate a small twist from a changed angle once it has been.

What a misplaced centre does to an angle #

Measured about a centre that sits a distance $e$ from the true one in a direction $\psi$, an organ at radius $r$ and angle $\theta$ appears turned by about $(e/r)\sin(\theta - \psi)$ and moved in radius by about $-e\cos(\theta - \psi)$. Round any annulus the turn averages to nothing, since the sine takes every sign; what it leaves is a spread, largest where $r$ is smallest. And each family’s phase sum weights every organ’s angle by the family’s number $m$, so the spread it sees is $m e / r$.

That is where a twist would hide. A twist of $a\,(r/R)^p$ with a large exponent changes the divergence most at the rim, one with a small exponent most near the centre; a centre error disturbs the inner annulus more than the outer, and could be mistaken for the second. The threshold the twist is judged against is the largest difference between the two annuli that any untwisted head shows, read about its true centre: a thousandth of a degree on 900 organs, a third of that on 2,400.

How far off is too far #

On 900 organs a centre a fiftieth, a twentieth, a tenth or a sixth of a spacing off leaves every one of ten heads inside the threshold: the difference between the annuli stays near half a thousandth of a degree, where it is with the centre right. A quarter of a spacing takes five heads of ten past it, and half a spacing seven. On 2,400 organs a quarter of a spacing leaves every head inside, and half a spacing takes one of ten past.

The error lands where the arithmetic says. At a quarter of a spacing on 900 organs the inner annulus reads 1.35 thousandths of a degree off the golden angle and the outer 0.29: the inner organs are nearer the centre, so the same misplacement turns them further. The larger head tolerates more because its inner annulus is further out in spacings — the ratio $e/r$ that matters is smaller for the same $e$.

So the tolerance is about a sixth of a spacing on the smaller head and a quarter on the larger. A centre found to that has cost the reading nothing.

Three ways to find the centre #

Three are natural. The centroid is the mean of the organs’ coordinates. The radius law says the $i$-th organ sits at $c\sqrt{i}$ from the centre, so the sorted squared radii should lie on a straight line against their rank, and the centre can be put where they lie straightest. And each family’s phase sum has a centre: the counted families are coherent about the true centre and less coherent about any other, so the centre can be put where the inner annulus’s two families, searched over the divergences the counts allow, are most coherent. The inner annulus is used because it is where a centre error bites and because it is the part of a head a damaged rim leaves.

The centroid and the families’ phase sums find it; the radius law does not. On 900 organs displaced by a tenth of a spacing the centroid lands 0.012 of a spacing off, the families’ phase sums 0.011, and the radius law 0.38 — thirty times as far, and past the sixth of a spacing that the reading tolerates. On 2,400 organs the centroid lands 0.007 off, the families’ phase sums 0.006 and the radius law 0.42. The two that work differ in how they respond to displacement: at a twentieth of a spacing the families’ phase sums find the centre to 0.0053, half the centroid’s 0.0109, and at a quarter of a spacing to 0.066, four times the centroid’s 0.017. The families’ phase sum is built on the lattice, and the more the organs wander off it, the more its peak blurs; the centroid is built on the outline, which displacement barely touches.

Every one of these errors except the radius law’s is well inside the tolerance, and that is the first answer: on a whole head the centre can be found from the organs to a small fraction of what the reading can afford to lose.

Why the radius law cannot see the centre #

The radius law looks like the obvious fit, since it is the law the head was built by, and the reason it fails is structural. Move the centre by $e$ and every squared radius changes by $-2er\cos(\theta - \psi)$ plus $e^2$. The first term is positive for half the organs and negative for the other half, and sorting the radii — which is all the law can do, since it does not know which organ is which — mixes the two halves back into a distribution whose shape the first term hardly changes. The law sees the centre only at second order, and at second order the organs’ own displacement drowns it.

Along a line through the true centre of a 900-organ head displaced by a tenth of a spacing, the law’s misfit wanders between 0.64 and 1.09 of its value at the true centre and is least 0.63 of a spacing away from it. The coherence of the inner annulus’s two families does the opposite: it peaks at the true centre and has fallen to a sixth of that peak a quarter of a spacing away, with side lobes a tenth as high half a spacing out. It sees the centre at first order, because a misplaced centre turns the angles it is built from, and the angles are what the radius law throws away.

What the radius law’s centre costs #

A centre a third of a spacing off is past the tolerance on 900 organs and near it on 2,400, and the reading shows it. About the radius law’s centre, four of ten untwisted 900-organ heads displaced by a tenth of a spacing read as twisted, their two annuli differing by more than any head read about its true centre; on 2,400 organs, four of ten again. The inner annulus pays for it, as the arithmetic of $e/r$ says it must: it reads the golden angle to 1.84 thousandths of a degree about the radius law’s centre on 900 organs, against 0.50 about the centroid, and to 1.27 thousandths on 2,400 organs against 0.064 — twenty times as coarse. At a quarter of a spacing of displacement the 900-organ inner annulus about the radius law’s centre is 0.093° off, seventy times the centroid’s error.

About the centroid and about the coherence centre, no untwisted head of either size reads as twisted, and both annuli read as finely as about the true centre: 0.50 and 0.12 thousandths of a degree on 900 organs, 0.064 and 0.036 on 2,400. The count’s worth grows with the head, and so does the reading built on it; a centre found to a hundredth of a spacing keeps every bit of that growth.

The centroid is a statement about the rim #

The centroid’s success is less obvious than it looks. A Vogel head’s organs are spread at uniform density — radius $c\sqrt{i}$ is exactly one organ per equal area — so any disc of organs well inside the rim, about any point in it, has that point for its centroid. The organs deep inside carry no information about where the centre is. The centroid of the whole head works because the head has a round outline, and it is the outline that places the mean. Undisplaced, the golden spiral’s own asymmetry puts it a hundredth of a spacing off the true centre on 900 organs, which is most of what it gets wrong displaced.

That makes the centroid exactly as good as the rim, and a photograph’s rim is the part most often damaged, obscured or outside the frame.

Found by the centroid, the twist is still separated #

Read about the centroid instead of the true centre, the untwisted heads set a threshold of 0.91 thousandths of a degree on 900 organs against 1.00 about the true centre, and 0.32 on 2,400 against 0.34. The first twist every seed separates is then the same at every exponent on 900 organs — a tenth of a radian for exponents of a half, one and three, a quarter for one and a half, a twentieth for four — and no later on 2,400, where the fourth power is separated at a fiftieth of a radian instead of a twentieth. The square law is still never separated, for the reason the flag essay gave: a square-law twist is a changed angle organ for organ, and no centre makes it anything else.

So on a whole head the centre costs nothing. The positions tell a twist’s shape from a changed angle with the centre found from the positions as well as with it given.

A head with half its rim missing #

Remove the organs beyond seven tenths of the radius over half the head — a head bent at its edge, broken, or half outside the photograph — and the centroid goes with the rim. On 900 organs it lands 3.1 spacings off, on 2,400 organs 5.1. Read about it, untwisted heads differ between their two annuli by up to 0.45°, four hundred and fifty times the difference that declares a twist, and not one of fifty cropped 900-organ heads reads within a hundredth of a degree of the angle it grew at.

The families’ phase sums do not care. Searched over a grid round the centroid, re-centred on its best point until the best lies inside it, the inner annulus’s coherence finds the centre to 0.011 of a spacing on 900 organs and 0.006 on 2,400 — as well as on a whole head, because the inner annulus is untouched by the crop and the spirals in it are centred where the head grew. The dial shows the same: at a twentieth of a spacing of displacement the coherence centre lands 0.0044 off and at a quarter 0.048, while the centroid stays three spacings away at every displacement.

What the crop costs, and what it does not #

The crop has a cost of its own, and it is not the centre’s. Whole, forty of fifty untwisted 900-organ heads read both annuli within a hundredth of a degree — the ten that do not are the heads grown a twentieth of a degree below golden, whose outer counts admit no interval. Cropped, twenty-eight do, and the loss is the same about the true centre as about the coherence centre: with half the outer annulus gone, the counter behind the reading miscounts that annulus on twelve more heads — all ten grown a twentieth of a degree above golden, and two others — whatever centre it is given. On 2,400 organs the cropped heads read exactly as the whole ones do, thirty of fifty, because half an annulus of a larger head still holds enough organs to count.

Among the heads that do read, the coherence centre is as good as the true one: the untwisted heads set a threshold of 1.13 thousandths of a degree about it on 900 organs against 1.16 about the true centre, and 0.32 against 0.33 on 2,400. So on a cropped head the positions separate a twist wherever the counts can be read, and the centre is not what limits them.

Two finders as a test of the rim #

The two finders that work are built on different things — the centroid on the outline, the coherence centre on the lattice inside it — and that difference is itself a measurement. On whole heads they agree: across ten 900-organ heads displaced by a tenth of a spacing the centroid and the coherence centre land between 0.006 and 0.025 of a spacing apart, and across ten 2,400-organ heads between 0.002 and 0.015. With half the rim gone they land 3.0 to 3.2 spacings apart on 900 organs and 5.1 on 2,400, a separation two hundred times the largest seen on a whole head.

So a worker who runs both needs no separate judgement of whether a rim is intact. A disagreement of more than a few hundredths of a spacing says the outline is not the head’s, and says which finder to believe; an agreement says the outline is round and either will do. It is the move the count sees the twist first made with two annuli: two readings of one head, built on different parts of it and set side by side, so that each reports on the other.

What a survey should do #

Take the centroid on a whole, round head; it is within a hundredth of a spacing, and nothing downstream can tell it from the true centre. On a head with any part of its rim missing or hidden, find the centre from the spirals: search the inner annulus’s coherence over a few spacings round the centroid and take its peak, which is also a check on the centroid, since a peak far from it says the rim is not what it seems. Never fit the radius law for the centre; it will return an answer, and the answer will be a third of a spacing off in a direction set by noise.

And read the coherence centre’s own coherence. A peak as high as the head’s displacement allows says the families were counted right about it; a low one says the counts in the inner annulus are wrong, which no choice of centre repairs.

What these heads leave out #

They are Vogel heads with a round outline, displaced by independent gaussian steps, and the crop is one shape: half the rim beyond seven tenths of the radius. A real head’s outline is not a circle, and photographed off its axis it is an ellipse. An ellipse still has a centroid at its centre, but the radius law and the relation between angle and radius both assume a circular head, and a squashed one turns every organ’s angle by an amount that varies twice round the head rather than once — a different error from a misplaced centre and one this reading does not model. A smooth field of displacement would move the inner annulus coherently and could move its coherence centre with it; the displacement here is independent from organ to organ.

Findings that would overturn it #

A centre error of a tenth of a spacing that takes a 900-organ head’s two annuli past the untwisted threshold. A radius-law centre that lands within a tenth of a spacing on displaced heads. A twist separated later about the centroid than about the true centre on a whole head. A cropped head on which the coherence centre lands more than a tenth of a spacing off while the inner annulus is intact. Each would mean the reading here is wrong.

Still open: a head photographed from the side #

A photograph taken off the head’s axis squashes it by the cosine of the tilt across one bearing, as a section seen from the wrong angle found for a shell. The squash turns each organ’s angle by an amount that goes round the head twice, and it changes the radius law from a circle’s to an ellipse’s. The next measurement is the positions’ reading of a head viewed off its axis by a few degrees: how large a tilt reads as a twist, whether fitting the ellipse with the centre recovers the reading, and whether the square-law blind spot survives a squash — since a tilted head is not a changed angle organ for organ, a twist the round head hides might show on an oblique one.