An organ has no single exponent

The scale phase produced a clean result about surfaces of revolution. Write the local circumference as $C(s) = k\,s^{a}$ in terms of arc length along the meridian, let the elements arrive either by elongation or by area filling, and the parastichy transitions sit at positions a fixed factor apart:

$$\varphi^{2/p}, \qquad p = a \ \text{(elongating)} \quad \text{or} \quad 2a \ \text{(filling)}$$

Five surfaces were built, walked with a blind counter, and every one agreed — 2.617 against 2.618, 1.914 against 1.900, 1.631 against 1.618, 1.380 against 1.378.

All five had a single exponent, because a power law is what having one means. No organ does.

What the organs actually are #

A conifer cone is not a cone. Its meridian is much closer to a circular arc than to a straight line — blunt at the tip, nearly parallel-sided at the shoulder — which is the shape a gunsmith or a naval architect calls an ogive. Write the meridian as an arc of radius $R$ tangent to the axis at the tip and the radius at arc length $s$ is $R(1 - \cos(s/R))$. Near the tip that is $s^2/2R$, an exponent of 2; at the shoulder the radius stops growing and the exponent goes to 0.

A capitulum is not flat. A sunflower head is slightly convex, which is a spherical cap rather than a disc: $r = R\sin(s/R)$, an exponent of 1 at the centre falling slowly outward.

Neither of these is an approximation being criticised for being one. They are the shapes, and the power-law family the previous phase measured does not contain either of them.

The local exponent is the primitive #

Everything here is built from one quantity:

$$a(s) = \frac{\mathrm{d}\log r}{\mathrm{d}\log s}$$

the exponent the surface has at one place. A cone has $a(s) = 1$ everywhere and is in the family as the control. The ogive and the cap are in it because their $a(s)$ varies, and how it varies is a consequence of their shape rather than a parameter chosen to make a point.

It is computed by differencing the profile rather than by looking up a formula, even where a closed form exists — because the closed form is what it gets checked against. All three profiles have an analytic $a(s)$ and the differencing reproduces each to better than two parts in a hundred at every position tried. Without that check every number below would be a statement about a difference operator.

What happens to the transitions #

Build a lattice on an ogive at the golden angle, walk a blind counter up it, and record where its answer changes. The counter is the site’s own: it receives arc length, position around, and local circumference, and never the profile, the step, or the divergence angle.

The rings come at 2.64, 4.28, 7.00, 11.59 and 20.43 along the meridian. The gaps between them are

$$1.626,\quad 1.631,\quad 1.657,\quad 1.763$$

which is not one number repeated. On a true cone under elongation every gap would be $\varphi^2 = 2.618$; on a disc under filling every gap would be $\varphi = 1.618$. This organ starts near the disc’s value and drifts upward, by eight per cent across four gaps.

The direction is right for the shape. A lower exponent spaces the transitions further apart — the factor is $\varphi^{2/p}$, which grows as $p$ falls — and the ogive’s exponent falls along its axis, so the gaps should widen along it. They do.

Predicted and counted, kept apart #

The figure above shows two sets of marks and the separation between them is deliberate.

The upper marks are computed from the profile and the cylinder’s ladder: the rise at arc length $s$ follows from the shape, the ladder says at which rises the pair changes, and inverting gives the positions. No points are involved anywhere.

The lower marks come from running the blind counter in ninety narrow bands up the built lattice and recording every place its answer changes. No profile is involved anywhere.

They agree to about a per cent and a half — 4.33 against 4.28, 7.03 against 6.99, 11.42 against 11.58, 20.33 against 20.42 — which is the width of the counter’s own bands.

That agreement is what makes the fine-grained claims in the next essay possible, and it also fixes which set of positions those claims are made about. The counted positions are quantised by the band grid, at a few per cent, and an exponent inferred from the ratio of two such positions inherits that error multiplied by $1/\ln(\text{ratio})$ — about nine per cent, which is wider than the whole variation being looked for. Two consecutive gaps came out as 1.625 and 1.625 to four figures at one point, which is the grid speaking rather than the organ.

So the arithmetic about exponents is done on the predicted positions, and the counter’s job is to agree with them to the resolution it has. Keeping those two roles separate is the difference between a measurement and a coincidence.

Where the meridian has to stop #

Both varying profiles have an end, and it is not the end of the organ — it is where the coordinate stops meaning what the family assumes.

The whole construction is in terms of arc length $s$ running outward along the meridian, with a circumference $C(s)$ that grows. On an ogive that holds until the tangent has turned a quarter turn; past that the surface has turned over, the radius stops growing and begins to shrink, and the exponent goes through zero and becomes negative. On a spherical cap the same thing happens at the equator.

A lattice built past that point is not a lattice on a widening surface, and every statement in this collection about rises falling and ladders climbing assumes a widening surface. So the constructor refuses to build one: the meridian is capped short of where the exponent would go negative, and asking for a specimen long enough to run past it fails rather than quietly producing a shape whose second half is upside down.

That refusal is doing real work rather than being a formality. The natural way to specify a cap is by its radius, and a small radius makes a short meridian rather than a small organ — a cap of radius 3 turns over after less than five arc-length units, which is a fifth of what a countable lattice needs. The failure without the guard is not an error but a plausible-looking lattice on a shape that closes on itself, with a counter reporting numbers about it.

The variation is real and it is not large #

Eight per cent across four gaps is the honest size of this. It is worth saying plainly, because the section headings above promise a bigger effect than the numbers deliver.

The reason is that the interesting part of an ogive is at the ends. The exponent is near 2 for most of the tip half and falls steeply only as the meridian turns towards parallel — and the region where the counts can actually be followed sits mostly before that. Measured gap by gap: the first two agree to 0.32 per cent, and almost the whole of the drift is in the last one.

That has a consequence which is more useful than the drift itself. An ogive looks exactly like a power-law surface over most of its length, and departs only where the counting gets hard anyway. Anybody fitting a single exponent to a fir cone is not making a large error, and the next essay works out exactly what error they are making and in which direction.

A convex head is a different case, and the difference is decisive #

The capitulum has a varying exponent too, and over the rings a head can actually be counted at, it varies by 1.15 per cent.

That is not a measurable quantity. A ring position found to within three per cent — a generous standard for finding where the counts change on a real head — allows the gaps to disagree by eight and a half per cent before the disagreement means anything, which is seven times what the head supplies.

So the two organs are in different situations and it is arithmetic rather than taste that separates them:

• An ogive’s variation is 15 per cent of its mean exponent across its rings. That is within reach of a careful measurement, and the next two essays work out what it would take.
• A convex head’s is 1.15 per cent. No ruler separates it from a flat disc, and treating a capitulum as flat is not an approximation with a stated error — it is a choice that no measurement on that organ could ever criticise.

What a step of the ladder reports, and why it is a real exponent #

There is a step between “the gaps are not equal” and “the exponent varies”, and it needs an argument rather than an assumption.

One step of the ladder multiplies the rise by $\varphi^{-2}$. A step running from arc length $s$ to $s'$ therefore reports

$$p = \frac{2\ln\varphi}{\ln(s'/s)}$$

and the question is what that number is a measurement of, given that the organ’s exponent is different at the two ends of the step.

It is the local exponent somewhere inside the interval, by the mean value theorem, and that is checked rather than assumed. For each step the profile’s own $a(s)$ is evaluated at both ends and the step’s reported exponent is required to lie between them. On the ogive: the step from 2.64 to 4.28 reports 1.980 with ends at 1.979 and 1.992; the step from 4.28 to 7.00 reports 1.968 with ends at 1.943 and 1.979; the step from 7.00 to 11.59 reports 1.905 with ends at 1.842 and 1.943; and the step from 11.59 to 20.43 reports 1.697 with ends at 1.492 and 1.842. Every one is bracketed.

That check is what makes the interval exponents a measurement of the surface rather than an arithmetic rearrangement of four positions. Nothing in the ladder calculation knows the profile, and nothing in the profile knows the ladder.

It also fixes an error the first version of this made. The obvious thing is to evaluate the organ’s exponent at each transition and average those, which answers a slightly different question and misses the fitted value by more than it should. A transition is a position; an exponent is a rate of change; so an exponent belongs to the gap between two transitions rather than to either of them.

Why this does not undo the previous phase #

The φ^(2/p) result stands, and it is worth being clear about what has and has not changed.

The result is a statement about power-law surfaces, it was checked on five of them, and every check still passes — the profile family here contains a straight meridian as a special case, and on it the transitions come out a factor of 2.617 apart with a fitted exponent of 0.995. If the machinery in this essay contradicted that, the machinery would be wrong.

What has changed is the reading. A measured transition spacing on a real specimen is not a measurement of “the” exponent, because the specimen does not have one; it is a measurement of some average, and which average it is turns out to be predictable. That is the next essay.

And the qualitative claim the previous phase drew from the formula is untouched and slightly strengthened. The spacing identifies neither the shape nor the growth law, only their product — two surfaces with the same $p$ give the same ratio, and one surface under the two laws gives two ratios. An organ whose exponent varies adds a third degeneracy on top: the number it reports is an average, and many different profiles average to the same thing.

The specimens are built, not fitted, and here is what that cost #

One more thing about method, since the figures show organs that look like plants and are not.

The ogive here has an arc radius of 12 and its lattice is 4,469 nodes long, which puts about six rungs of the ladder on it. The cap has a radius of 40 under the filling law and carries 6,000 nodes. Both were chosen so that the countable part of the organ holds enough transitions to space — four gaps on the ogive, four on the cap — and not because any conifer or composite has those proportions.

That is a limitation with a direction. A real specimen has fewer rings than these, not more. A fir cone carries perhaps two or three places where the parastichy numbers change over its countable length; the four here are on the generous side, and the essay after next is about exactly how much difference that makes. So every statement of the form “the drift is visible with this many rings” is an upper bound on what a plant would offer, and the honest reading of all of it is that the measurement is harder on a specimen than in this file.

The other cost is that the scale constant is fixed by the rise at the first node rather than given. Every specimen therefore starts on the same rung of the same ladder, which is what makes a comparison between two of them a comparison of shapes rather than of arbitrary units. The previous phase learned that the hard way: fixing the constant directly instead put two surfaces on rungs a hundred apart, and the one that had run past the counter’s order limit reported no transitions at all rather than reporting a disagreement.

What this changes about reading a cone #

The practical upshot, for anybody holding a fir cone and counting.

The pair changes along it, and where it changes is not evenly spaced. That was already known for a cone — the previous phase established that a cone’s transitions are a factor of φ² apart along its axis, against a disc’s φ — and what is added here is that on the real shape the factor is not constant. On the specimen built here it runs from 1.63 near the tip to 1.76 at the shoulder.

The pair near the tip is the informative one about shape. The exponent there is close to 2 and nearly constant, so the tip half of an ogive behaves like a clean power-law surface and its rings are where a fitted exponent is a description rather than an average.

And the shoulder is where the departure lives and the counting is worst. The exponent falls fastest exactly where the surface is closest to a cylinder, which is where the parastichy families are most nearly parallel and hardest to separate by eye. The information and the difficulty are in the same place, which is a thoroughly ordinary situation in measurement and worth naming when it occurs.