A head is a lattice two dozen organs at a time

A head is a set of points put two lines of arithmetic behind a sunflower and said, in a section with that heading, that the pattern is a lattice. Almost everything written here since leans on that sentence. A spiral count is a claim about the shortest vectors of a lattice. The largest holes of a head are computed from a lattice’s cell. The defects on a head are predicted from one. And every one of those essays qualifies it in the same word, locally, and moves on without saying how local.

That is a measurable quantity and it has a surprisingly small answer. Take a patch of organs, find the straight lattice that comes closest to them, and ask how far the organs sit from its sites. A head grows outward, so a patch cannot be a lattice exactly. The question is how large a patch can be before the word stops being true, and whether that depends on the angle, on the radius, or on anything else.

A fit that is told nothing #

A fit like this can cheat without meaning to. Every organ on a head has an index, and the index tells which lattice site it belongs to: the difference between two organs’ indices, written in terms of the two spiral families, gives their separation in lattice steps directly. Using that would assume the thing being tested, and it would give an answer that could not be applied to a photograph, where no organ carries a number.

So the labels here come from geometry and nothing else. The organ nearest the patch’s centre and its two nearest independent neighbours give a first lattice. Each organ in the inner quarter of the patch takes the nearest site of that lattice as its label, the lattice is refitted to those labels by least squares, and the labels are read again. As the labels settle, the patch widens to its full size. What comes out is a lattice origin, two lattice vectors, a residual for every organ, and a count of how many organs had to share a site.

The first thing the picture shows is that the fit is good and also visibly not perfect. These are 32 organs at radius 30, about organ 900 on a head, and each sits close to its site but not on it. The misses are not random: they run in a pattern across the patch, pointing one way at one side and another way at the other, which is what a smooth deformation looks like when a straight grid is laid over it. The root-mean-square miss is 0.13 of a spacing. The dial makes the patch larger, and the misses grow quickly.

How far a straight lattice misses #

Reading twelve patches at each of nine sizes and four radii gives four curves that say the obvious thing and one thing that is not obvious. A larger patch misses by more, and a patch further out misses by less. At fifteen organs the miss is 0.21 of a spacing at radius 8, 0.12 at radius 15, 0.06 at radius 30 and 0.03 at radius 60: each doubling of the radius halves it. At fifty-eight organs the same radii give 0.29, 0.30, 0.24 and 0.13.

What is not obvious is that every curve stops rising. They flatten near 0.3 of a spacing and stay there however large the patch gets. That cannot be a head becoming more regular. It is what the fit returns when it has stopped finding a lattice, because 0.3 of a spacing is roughly how far a point dropped at random lies from the nearest site of a triangular grid. Past a certain size, the residual stops measuring the head and starts measuring the arithmetic of rounding.

One law for all of it #

The halving with radius and the steep rise with patch size suggest that one combination of the two might carry everything, and it does. Put $L$ for the radius of the patch and $r$ for its distance from the centre, both in the head’s own units. Then the root-mean-square miss of a straight lattice is

$$\text{miss} = \frac{L^2}{4r}$$

to within a few per cent, over every patch that still holds its labels. At radius 60 with a patch four spacings in radius the fit misses by 0.126 of a spacing and the law says 0.127. At radius 30, three spacings: 0.140 against 0.143. At radius 60, five spacings: 0.197 against 0.198. At radius 30, two and a half: 0.095 against 0.099.

The form of that law has a simple source. A straight lattice can absorb any part of a head’s geometry that is linear across the patch: the local spacing, the local shear and the local rotation are exactly what its two vectors are for. What it cannot absorb is the second-order term. Across a patch of radius $L$ at a distance $r$, circles bend away from a straight line by about $L^2/2r$, and the spacing along the radius changes by a comparable amount, and the residual of a least-squares fit to a quadratic is a fixed fraction of its range. There is no fitted constant in the sense of a tuned parameter: the quarter is what the data show, and the arithmetic of a quadratic over a disc gives a value of that size.

Three quarters of the radius #

The useful form of the law is a count of organs, since a patch is something a person picks out of a photograph by eye, and cells are what the eye counts. On a Vogel head the area per organ is $\pi$, so a patch of radius $L$ holds about $L^2$ organs. Requiring the miss to stay below a tenth of a spacing, about 0.19 in these units, gives $L^2 \le 0.76\,r$.

So a straight lattice good to a tenth of a spacing holds about three quarters of $r$ organs. Since $r = \sqrt{i}$ on this head, that is three quarters of the root of the organ’s own index. Around organ 900, the patch holds twenty-three organs, which is what the bank reads: 0.095 of a spacing at twenty-three organs at radius 30. Around organ 3,600 it holds forty-five, and the bank reads 0.096 at forty-five organs at radius 60.

A head of 2,400 organs, which is a fair size for a sunflower, has a radius near 49, so even at its rim the patch holds about thirty-seven organs out of 2,400. Half way out it holds about eighteen. That is what the sentence the pattern is a lattice comes to when it is measured: a lattice to a tenth of a spacing, about a cell and a half of neighbours in every direction, and then something else.

A tenth is a chosen tolerance, and a tolerance of a twentieth halves the organ count. Any tolerance a person might choose gives a number of organs proportional to the radius, and the word lattice is a claim about the first few cells around any given organ.

Where the labels give out #

The tolerance is a choice. The next threshold is not. As a patch grows, its outer organs drift further from their sites until some of them are nearer to the next site along, and then a label changes hands. Two organs land on one site, a site elsewhere goes empty, and the fit has reached a state no adjustment of two vectors can repair. The transition is sharp. At radius 30 not one of twelve patches has a clash at forty-four organs, one does at fifty-eight, seven do at ninety-one, and all twelve at one hundred and thirty. At radius 60 there are none at ninety-one organs and eleven at one hundred and thirty.

The law says where this should be. A label is at risk once an organ is more than about a quarter of a spacing from its site, since half a spacing takes it to the next, and the largest misses are about twice the root-mean-square. Setting $L^2/4r$ equal to a quarter of a spacing gives $L^2 \approx 1.9\,r$, so about fifty-seven organs at radius 30 and one hundred and fourteen at radius 60. Those are where the bank’s clashes begin. A straight lattice is not a loose description that gets gradually worse; there is a size beyond which it is not a description at all.

The picture of that state is instructive to compare with the first. Through the middle the two look alike. Near the edge the fitted sites have drifted so far from their organs that some of the connecting lines cross, and the eye that was happy to call this a lattice at thirty-two organs would be misled at ninety-one by exactly the part of the image it was looking at. The spirals are still perfectly visible, because the eye follows them as curves rather than holding them against a straight grid.

Is the angle involved #

The essays on holes and defects found that a head’s fine structure belongs almost entirely to its angle, so it would be natural to expect this to as well. It does not. On a head at 137.0° the same patches read 0.060 at fifteen organs where the golden head reads 0.060; 0.140 at thirty-three where the golden head reads 0.140; 0.193 at forty-four where it reads 0.194. Which vectors are shortest differs entirely between the two, and how far a straight lattice misses does not, because the miss is a statement about how the map from index to position bends, and that map has the same square root in it whichever angle is multiplied in. Past fifty-eight organs the two curves do part, 137.0° losing its labels a little sooner, presumably since a long thin cell has less room between sites.

At a rational angle the question does not arise. A head at five thirteenths of a turn gave no usable fit on its patches beyond radius 15, because its organs lie along thirteen straight rays and the two nearest neighbours of any organ in a patch are nearly collinear. A fit cannot start from two vectors that are the same vector.

Coordinates that should have helped #

The natural response to a bending lattice is to change coordinates. A disc is a cylinder in a precise sense: the map taking a plane to the logarithm of radius against azimuth is conformal, it turns circles into straight lines and it preserves shape locally. On a cylinder a pattern of constant rise really is one lattice, exactly and everywhere. So reading a head in those coordinates ought to straighten it.

It does nothing. The same organs with the same labels miss by 0.141 of a spacing, against 0.130 to 0.140 flat. The reason is that a head mapped to a cylinder is not a cylinder of constant rise. Going out by $j$ organs from organ $i$ moves the logarithm of radius by $\log(1 + j/i)/2$, which is $j/2i$ only to first order, and the second-order term is the same size as the bending the map was supposed to remove. The conformal map fixes the circles and leaves the falling rise. That falling rise is what the counts changing with radius consists of, so it is hardly a detail.

The coordinates that do work are not conformal, and no camera offers them. In azimuth against the square of radius, organ $i$ sits at $(i\delta, i)$, a straight lattice by definition. The same thirty-two organs with the same labels fit there to $1.8 \times 10^{-13}$, the rounding of the arithmetic. So a Vogel head is exactly one lattice, all of it, in a coordinate system that stretches a head enormously at its centre and compresses it at the rim, and no patch of a head photographed flat is that lattice for more than a couple of dozen organs.

What this does to the things that lean on it #

The readings that use a lattice were mostly doing the safe thing without having to say so. The closed forms for holes and flips are about a lattice at one rise, and they are applied to a ring, a very thin annulus, so the patch involved is long round the head and a spacing or two wide. A ring’s largest triangle is three neighbouring organs. A patch law at this scale leaves those alone, and the holes of a head agree with the closed form to a few thousandths.

What it does affect is anyone fitting a lattice to a photograph over a larger area. A botanist who lays a straight grid over a patch of a capitulum to read its two lattice vectors will find it fits, with a residual that looks like measurement error and grows with the area. Averaging over more organs does not then give a better reading of the local lattice, because the additional organs add curvature rather than information. The law gives the size to stay inside.

It also puts a scale on what a pair of spiral counts is reading. A counter that takes an annulus of a head is not fitting a lattice, and does not need one to hold over the band: it asks for the median length of a hop of $j$ organs, and a hop’s length is a two-organ measurement. That is a reason the count survives over a band far wider than any straight lattice would. It does not depend on the thing this essay shows is missing.

What this does not establish #

It is a statement about Vogel’s rule. A real capitulum has no reason to follow a square root exactly, and a head with a different radial law has a different second-order term. The law as stated applies only to patches well inside the head, since all twelve patches at each size lay away from the centre and the rim, and at radius 8 the smallest patch is already in the region where the first organs are out of step. The labels are found by the nearest site and a different labelling, by a best assignment over the whole patch, might keep a patch for somewhat longer. And the quarter in the law is measured, with the quadratic argument giving its size and not its value; nothing here derives it.

Nor does a patch’s poor fit say anything against calling the pattern a lattice in a looser sense. The eye’s sense of order is not a least-squares fit. What the measurement rules out is reading the sentence as a statement about straight lines over any area larger than a couple of dozen cells.

How the law could fail #

A patch on a large head that misses by much less than $L^2/4r$ at a radius where the labels hold would mean something other than curvature had been setting the residual. A divergence for which the law failed at small sizes, not only near the size at which labels are lost, would mean the bending was not independent of the angle after all. A fit in the logarithm of radius that did markedly better than a flat one would mean the falling rise was not the reason, and a fit in radius squared with any residual above rounding would mean the organs were not where the rule puts them.

Still open: a head whose radius is not a square root #

Every head here puts organ $i$ at $\sqrt{i}$. Under $r = i^p$ the straightening coordinate is no longer radius squared but $r^{1/p}$, azimuth against that power of the radius, so the exponent that makes a patch’s residual collapse to nothing is itself a reading of the growth law. The next measurement fits a patch across a range of trial exponents and asks whether the residual has a sharp enough minimum to read $p$ from a few dozen organs, how much displacement that minimum survives, and how it compares with the other ways a growth law can be read off a head, from its counts and from its cell areas.