A growth law moves the counts

The counts change with radius showed that one head gives 21 and 34 near its centre and 55 and 89 near its rim, and that the changes fall at radii anyone can compute. Transitions a factor of $\varphi^2$ apart pinned them down: a stem’s transitions are a factor of $\varphi^2$ apart in rise, and a disc’s rise falls as one over radius squared, so a disc’s transitions are a factor of $\varphi$ apart in radius.

That last step leans on one part of Vogel’s model, the one part with any biology in it. Organ $i$ sits at radius $\sqrt{i}$ because that keeps the area per organ constant, and it is exactly what a head that adds organs at a steady rate over a growing disc would do if its disc grew in just the right way. Nothing guarantees that. A head is a set of points noted that the square root was a stated assumption, and a real capitulum’s organs can crowd towards the rim or spread out towards it as much as its development allows.

So the factor of $\varphi$ belongs to a square root, not to a head, and the obvious next question is what it becomes under any other radial law. The answer turns out to be a reading: a number that can be taken from a photograph, tells how the head’s radius grew, and is remarkably hard to spoil.

The arithmetic first #

Put organ $i$ at radius $i^p$. A head’s local lattice depends on how far the next organ moves round and how far it moves out, as a share of the circumference it sits on. Around organ $i$, one step outward is $p\,i^{p-1}$, and the circumference is $2\pi i^p$, so the rise is

$$t = \frac{p}{2\pi i}.$$

The exponent and the organ’s index enter as a ratio. Nothing else about the law appears in the lattice at all. The pair a counter reads depends only on the rise and on the divergence, so a head’s pair at organ $i$ under exponent $p$ is the same as a square-root head’s pair at organ $2p\,i$.

Two predictions follow, and neither has a free number in it. First, whichever pair is being watched, it arrives at an organ in proportion to the exponent: twice the exponent, twice the organ. Second, consecutive transitions are a factor of $\varphi^2$ apart in rise and so in organ, whatever the exponent. Since radius is $i^p$, that makes them a factor of

$$\varphi^{2p}$$

apart in radius. At a half that is the familiar $\varphi$. At 0.4 it is 1.47, and at 0.7 it is 1.96.

Three heads with one angle #

The three settings on the dial look like three kinds of head, and that is what makes them worth staring at. At an exponent of a half, the organs are evenly spread and the transition circles sit where the essays on this subject have always put them. Turn it down to 0.4 and the organs crowd towards the rim, with four changes of pair inside the head where there had been three; the pair 89/144 now arrives at organ 1,084 and it lies inside this 1,200-organ head. At 0.6, the organs spread out towards the rim, the same 1,200 organs reach a much greater radius, and only three changes fit.

None of these heads has anything different about its divergence. A person told only that a head reads 89 and 144 at its rim could not conclude from that alone anything about how many organs it has.

Each pair is delayed by one factor #

The count here is the usual one: the median length of a hop of each size, over a window of organs, with the two shortest taken as the pair. It is given no angle and no exponent. Sliding the window out through 8,000 organs gives the organ at which each pair first appears.

At an exponent of a half the pair 34/55 arrives at organ 187, 55/89 at 517, 89/144 at 1,356 and 144/233 at 3,635. Each is 2.6 to 2.8 times the last, as it should be, since $\varphi^2$ is 2.618. At 0.4 they arrive at 145, 404, 1,084 and 2,905, the same steps from a point further in. At 0.7 they arrive at 276, 740, 1,941 and 5,088.

On a logarithmic axis the three staircases have the same steps and the same widths, and one is the other moved sideways. That is the first prediction’s visible form. The only thing that the exponent can do to a sequence of transitions in index is multiply it by a constant.

How exact the proportion is #

To test that constant it helps to take each pair’s arrival against the same pair’s arrival at a half. For 55/89 that gives 0.78, 0.87, 1.00, 1.09, 1.20 and 1.43 at exponents of 0.4, 0.45, 0.5, 0.55, 0.6 and 0.7. Twice the exponent would be 0.80, 0.90, 1.00, 1.10, 1.20 and 1.40. For 34/55: 0.77, 0.89, 1.00, 1.13, 1.23, 1.47. For 89/144: 0.80, 0.89, 1.00, 1.12, 1.22, 1.43.

Those are as close as the instrument could be expected to get. The window slides in steps of a few per cent and a transition is read at the middle of the first window that returns the new pair, and every number is uncertain by a step of that size. The slight excess at 0.7 is common to all three pairs and could be real. A pair’s arrival is not a sharp event: a band around each transition reads ambiguously and at larger exponents the radius changes faster across a window, which may well shift what a window reports. It is a small effect on a large proportion, and nothing here tries to separate it.

The practical reading of this is restrained, because it requires knowing an organ’s index, and on a head a person can only get that by counting organs from the centre. It says something else, though: a pair and an organ’s index together give the exponent. A head counted out to organ 517 that reads 55/89 there has an exponent of about a half; if it reads 34/55 there, it has one nearer 0.7 or more. That is not how anyone would measure a head. It does show how much of a growth law is already packed into a count.

The radii alone #

The second prediction needs nothing of that kind, since it uses only radii. On a logarithmic axis, the radii where the pair changes lie on a straight line, and the slope of the line is $2p \log\varphi$.

Read from the 8,000-organ heads: at 0.4 the transitions after the first lie at radii 7.3, 11.0, 16.4 and 24.3, ratios of 1.51, 1.48 and 1.48, where $\varphi^{0.8}$ is 1.47. At a half they lie at 13.7, 22.7, 36.8 and 60.3, ratios of 1.66, 1.62 and 1.64 against $\varphi$. At 0.55, ratios of 1.72, 1.72, 1.70 against 1.70. At 0.7: 1.99, 1.96, 1.96 against 1.96.

The earliest ratio is the least certain in each row, and the first transition of all, 21/34, was left out entirely. A window holding too few organs can’t return a count at all, so near the centre the window has to widen, and with it the place the transition seems to fall. On these heads leaving it in pushes the read exponent up by about two hundredths. The figure leaves it out and the reason should be kept in mind: this reading works from the second change of pair outwards. It needs a head with three or four changes in it, which any head of more than a few hundred organs has.

The reading itself #

So take the ratio between successive transition radii, average it, take the logarithm and divide by $2\log\varphi$. That returns 0.416 for a head built at 0.4, 0.464 at 0.45, 0.514 at 0.5, 0.559 at 0.55, 0.610 at 0.6 and 0.706 at 0.7.

The reading is about a hundredth high across the whole range, and that small excess is a real and consistent bias. Its sign fits the same cause as the inner transition: windows are wider relative to the ring near the centre, so each pair’s radius is read fractionally later near the centre than near the rim. In a stated instrument with a stated bias of one per cent, that is a good result. A person working with a photograph and a counter has an exponent within two hundredths of the truth, using a quantity no cell area enters.

Compare what the exponent has been made to mean elsewhere here. A spread of cell areas relates to the exponent by a formula, gives two answers for one measurement, and fails as soon as the head is disordered at all. The patch residual sees a head’s curvature and has nothing to say about which law made it unless its coordinates are searched. The ratio of radii has a third property: it does not care what the organs’ cells look like.

How much a window can blur it #

The instrument’s own resolution can be stated, which keeps the agreement above from looking better or worse than it is. The window here steps through organ index by about 2.3 per cent at a time, and a transition is reported at the middle of the first window that returns its pair. So an arrival organ is known to about one step, and a radius, being the organ raised to the exponent, to about 1.2 per cent at an exponent of a half. A ratio of two radii is then uncertain by about 1.6 per cent, which at a ratio of 1.62 is an uncertainty of about 0.016 in the exponent from any one ratio, and around 0.01 over three.

The scatter in the rows above is about that size: 1.66, 1.62 and 1.64 against 1.618, then 1.51, 1.48 and 1.48 against 1.47. So the differences between one ratio and the next are at the level of the step, and nothing here can be read into them. The steady excess of a hundredth over the whole range is larger than the error on its own average, so it is the window’s bias rather than its scatter. A finer step would reduce the second and would not, by itself, remove the first.

A rim’s pair as a limit on the exponent #

The proportion has a blunter use which needs no ratio. Arrival organ goes as twice the exponent, and on a head whose exponent is a half 89/144 arrives at organ 1,356. Set against a head’s total, that makes a rim’s pair a limit: a head of 1,200 organs can only read 89/144 at its rim if $2p \times 1{,}356$ is below 1,200, which requires an exponent under about 0.44. Conversely a head of 1,200 organs that still reads 34/55 at its rim would need an exponent far above any on this list, since 55/89 arrives at organ 517 at an exponent of a half and at 740 at 0.7.

So the two numbers that are most often known for a capitulum, its organ count and the pair at its rim, already bound its growth law on one side, with no radius measured. The bound is loose, since a head that crowds its rim arrives at its pair early and can then hold it for a long way, and being past a transition only says the exponent is small enough. It is a check, and it is cheap, and a published pair inconsistent with a published organ count at any plausible exponent would be worth a second look.

What displacement does to it #

Every other reading of a head’s fine structure has had to be tested against displaced organs, and several have failed quickly: the reading of an angle from a head’s holes is largely gone at a hundredth of a spacing. This one is not of that kind. On a head built with an exponent of 0.45 it returns 0.464 with nothing moved, 0.464 at two hundredths of a spacing, 0.460 at a twentieth, 0.464 at a tenth and 0.473 at a fifth. Built at 0.55: 0.559, 0.567, 0.559, 0.555 and 0.596.

Those are single seeds, and the movement among them is the noise of where a window happens to change, about a hundredth up and down. The first thing to move visibly is the 0.55 head at a fifth of a spacing, up by four hundredths. A count is a median of hop lengths over a whole band of organs, so the displacement that ends most fine-grained readings barely touches it, and a transition’s position is decided by many hundreds of organs.

Two limits have to be stated. The heads here are shaken independently, organ by organ; a correlated error that slides whole bands could do something this does not show. And nothing here measures the error in a human counter, as distinct from a median over a thousand hops. A person counting a real head has their own difficulties near a transition, where the pair is ambiguous, and a transition radius read by eye could easily be out by more than one of these windows.

What it means for a photograph #

Suppose someone has a photograph of a large capitulum, and they count the spirals in several thin annuli. They find the pair changes at three radii. Dividing each radius by the last gives two ratios, and those give an exponent.

The attraction of that reading is what it does not require. It needs no organ’s index, no count from the centre and no estimate of the angle. Its unit does not enter: a ratio of radii is the same in millimetres as in pixels. It needs no cell to be segmented. A head with a growth law near a half reads a ratio near 1.62, a head that crowds its rim reads a smaller one, and a head that spreads reads a larger one. Compared with the usual practice of reporting one pair for a whole capitulum, it asks for perhaps three times the counting and returns an independent fact about development.

A published count is a statement about one annulus. Three of them, with their radii, are a statement about how the head grew. Whether anyone’s data contains enough transitions to try this on is another matter, because a survey reports its radii only rarely, and nothing here makes up for that.

Which also says what φ was #

There is a small reinterpretation owed in all of this. The factor of $\varphi$ between rings and transitions has been treated throughout as a fact about the golden head. It is two facts multiplied together. A factor of $\varphi^2$ between consecutive transitions in rise, or organ, belongs to the angle: on the Lucas branch it is the same, and it would be different on an angle with a different tail. And the exponent of a half belongs to a growth law. A head at 0.6 is exactly as golden as a head at 0.5, its transitions are 1.78 apart in radius, and that number has no special standing at all. The fact that the ratio of radii equals the same $\varphi$ that sets the divergence is a coincidence between the angle’s arithmetic and the assumption of a constant density, since $\varphi^{2 \times 1/2} = \varphi$. Any statement that the golden ratio appears twice in a head should bear in mind that it appears only once in the angle, and the second appearance is the square root.

What this does not establish #

It is a statement about heads that follow a single power law from centre to rim, and no head need do so. A law that changes with radius gives ratios that change with radius, and a set of three ratios that differ might mean either a changing law or noise. Whether a real capitulum’s transitions fall on a straight line in the logarithm has not been checked here; nothing in the published record that these essays have found reports the radii of more than one transition, which is why no specimen appears in this essay.

The angle matters too, in one respect only. The factor of $\varphi^2$ in rise holds at noble angles, whose quotients are all ones from some point on, and an angle with a large quotient has flips in a different rhythm, so its transitions are not evenly spaced in anything. A reading of this kind on an angle like 137.3° would not return one ratio. And the bias of about one per cent has been measured only with this window and this head size.

How this could be wrong #

If the arrival organ of a pair departed from the proportion to the exponent by more than a window’s step, at exponents where the transitions are well resolved, the claim that rise is all that matters would be wrong. If a head’s transition radii did not lie on a straight line in the logarithm at a stated exponent, the second prediction would be, and if a modest displacement moved the reading by more than the effect it is measuring, the claim to robustness would go with it.

Still open: a law that changes as the head grows #

Every head here has one exponent from centre to rim. The cases that matter are not like that. A real disc’s growth changes with developmental stage, and a head whose exponent falls from 0.55 at the centre to 0.45 near the rim would have ratios falling across its transitions. The next measurement builds heads with a stated varying exponent, reads the local ratio between consecutive transitions, and asks how much change between neighbouring ratios can be told from noise, whether three or four ratios are enough to see a change of law at all, and whether the band a window needs to read a transition sets a limit on how sharply a law can change and still be seen.