How much of a cone to measure

The previous two essays established that an organ’s shape exponent varies along its axis and that a single fitted exponent returns the harmonic mean of what its steps report. Both are statements about a specimen built in this repository, with five rings on it and no measuring error anywhere.

This one asks the question that decides whether any of it can leave the repository: what would a real specimen have to give up before its exponent could be shown to vary at all?

The answer is a small piece of arithmetic and it is more discouraging than the previous essays imply.

Two rings say nothing, and not because they are few #

Start with the extreme case, because it sets the shape of the whole problem.

A specimen measured at two rings gives one ratio. One ratio determines one exponent, exactly: $p = 2\ln\varphi/\ln(s'/s)$, and there it is. There is no residual, no second estimate to compare against, nothing left over.

That is not a small amount of evidence about whether the exponent varies. It is no evidence, of a kind that no improvement in the measurement can remedy. A perfect measurement of two rings on an organ whose exponent runs from 2 to 0 returns one number and reports perfect agreement with a constant-exponent model, because a constant-exponent model has exactly one parameter and the data has exactly one degree of freedom.

The distinction between hard to see and invisible in principle is worth keeping because the two call for different responses. Something hard to see calls for a better measurement. Something invisible in principle calls for a different one — here, for another ring.

Three rings can disagree, and on an ogive they do not #

Three rings give two gaps, and two gaps can differ. That is the first configuration in which the question has an answer at all.

On the specimen here, they differ by 0.32 per cent: the first two gaps are 1.626 and 1.631. A ring position measured to within $\varepsilon$ makes a ratio uncertain by about $\sqrt{2}\,\varepsilon$, and two ratios can differ by $2\sqrt{2}\,\varepsilon$ before the difference means anything — so at one per cent per ring, two gaps may differ by 2.8 per cent for free.

Three rings on an ogive are therefore consistent with a constant exponent no matter how carefully they are measured, short of a tenth of a per cent per ring.

The reason is the shape rather than the arithmetic. An ogive’s exponent is near 2 for most of its length and falls steeply only at the shoulder, so the early gaps are nearly identical and all the drift is in the last one. Measured across the four gaps: 1.626, 1.631, 1.657, 1.763. The information is almost entirely in the fourth gap, which needs five rings.

The numbers #

At one per cent on each ring position — a ring located to within a per cent of its height up the cone — five rings suffice: the spread among the four gaps reaches 8.2 per cent against a bound of 2.8.

At two per cent, still five, with the bound at 5.7.

At three per cent the bound is 8.5 per cent and the organ supplies 8.2. No number of rings on this specimen is enough, because the total drift the organ has across every ring it possesses is smaller than what three per cent per ring allows.

That is the sentence the essay is really about. It is not that the measurement is hard; it is that at a fairly ordinary standard of measurement the question has no answer on this organ, and adding rings does not help because there are no more rings.

Where the bound comes from #

The bound used above is worth deriving rather than asserting, because it is the whole of the arithmetic and it has one assumption in it that is doing work.

A ring position $s_k$ is measured with a relative error $\varepsilon$. A gap is a ratio $s_{k+1}/s_k$, and the relative error on a ratio of two independently measured quantities is $\sqrt{2}\,\varepsilon$. Comparing two gaps means comparing two such ratios, and the difference between them has a relative error of $2\varepsilon$ — but the two gaps share a ring, since the second gap starts where the first ends, so the errors are not independent and the combination is $\sqrt{6}\,\varepsilon$ rather than $2\varepsilon$.

The figure uses $2\sqrt{2}\,\varepsilon$, which is between the two and slightly conservative. That choice is a judgement rather than a derivation and it moves the answers by less than one ring at every error tried, which is why it is stated here rather than hidden.

The assumption doing the work is that the errors are independent between rings. If the same observer systematically places every ring a little high, the gaps are unaffected — a common multiplicative bias cancels in a ratio — which is a piece of good fortune worth noticing. If the bias varies along the cone, because the scales are harder to read near the base, it does not cancel and the bound above is too optimistic. That second case is the likely one on a real specimen and there is no way to correct for it without a second observer.

What a per cent means on a fir cone #

It is worth converting, because “one per cent” is an abstraction and a cone is not.

A ring is where the counted parastichy pair changes — where, walking up the cone, 5/8 gives way to 8/13. It is not a line. It is a band several scales deep in which both readings are defensible, because the two families are of nearly equal length there and which pair is “the two shortest” is exactly the question in the balance.

Locating that band’s centre to within one per cent of its distance from the apex means, on a cone whose countable stretch runs from two to twenty centimetres, placing it to within a millimetre or two at the near end. On a real specimen, with real scales at real angles, that is optimistic. Three per cent — half a centimetre — is a fair description of what careful work would achieve, and three per cent is where the answer runs out.

So the honest summary is that the varying exponent of a conifer cone is at or just beyond the edge of what a ruler can establish, on one specimen.

What the counting itself contributes #

There is a second error in this problem which is not the ruler’s, and on a real specimen it may be the larger of the two.

Finding a ring means deciding where the counted pair changes, and the counter used in this collection does that by running in narrow bands up the axis and recording where its answer differs. The bands have a width — about three per cent of position each, geometrically spaced — so the counter’s own positions are quantised at roughly that scale before any measuring error is added.

That is why every piece of fine-grained arithmetic in these three essays is done on predicted ring positions, computed from the profile and the ladder with no points involved, and the blind counter’s job is only to agree with them. It does, to 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.

Reading that the other way round gives the sobering version. The counter, run on a perfect lattice with no noise and no measuring error at all, locates a ring to about a per cent and a half. A person with a real cone is not going to do better than a routine with exact coordinates. So the one and two per cent columns in the table above are not a description of achievable field work; they are a description of what would be needed, and they are already below what the instrument used to define the rings can deliver on synthetic data.

This is a familiar shape and this collection has hit it before. The resolution of the definition bounds the resolution of the measurement, and a study can spend a great deal of effort improving the measurement past a definition that was never that sharp.

Which is what a population is for #

One specimen is not the only option, and the arithmetic changes when there are many.

The measurement error on a ring position is not systematic — a ring found slightly high on one cone is found slightly low on the next — so averaging the gaps over $N$ specimens shrinks the bound by $\sqrt{N}$ while leaving the organ’s drift where it is. At three per cent per ring and the bound at 8.5, twelve specimens bring the bound to 2.5, and the 8.2 per cent drift clears it comfortably.

Twelve fir cones from one tree, each measured at five rings. That is an afternoon’s work and it is the shape of the answer.

Two conditions attach and both are real.

The specimens have to be comparable. The gaps are dimensionless ratios, so they do not need the cones to be the same size — which is the one thing that makes this feasible. They do need the cones to be the same shape, since a population of ogives with different arc radii has a spread in its true gaps as well as in its measured ones, and that spread does not shrink with $N$.

And the rings have to be the same rings. Averaging the third gap across specimens requires knowing which gap is the third, which requires each specimen to show the same sequence of pairs. A cone that starts at 3/5 rather than 5/8 contributes its gaps to the wrong slots. That is a bookkeeping requirement rather than a measurement one, and it needs the pairs recorded alongside the positions — which is the field that published counts most reliably do record.

The other organ, where the answer is no #

A capitulum’s exponent varies by 1.15 per cent across every ring it can be counted at.

No error, no number of rings, and no number of specimens fixes that, because averaging over specimens shrinks the error and not the effect, and the effect here is smaller than the smallest error anybody would claim. It would take a ring position measured to a tenth of a per cent — a hundred micrometres on a sunflower head — before 1.15 per cent of drift cleared the bound on one specimen, and a population would need hundreds.

So the two organs come out on opposite sides of a line drawn by measurement rather than by biology, and that is the useful output of this essay. A slightly convex head is, for every purpose this collection can reach, a flat disc. Not approximately — indistinguishably, in a sense that has a number attached.

What this contributes to the survey question #

This collection has spent three phases asking for a dataset, and this phase is where the asking is replaced by a specification. This essay supplies one line of it and it is the most demanding line.

For the census questions — how often are plants Fibonacci, are multijugate patterns real — a specimen contributes one observation and the sample sizes are tens. For the shape question, a specimen contributes a sequence of ring positions, and the requirements are sharper: five rings per specimen, each to within about three per cent, from a dozen specimens of one species.

That is more than any published count offers, and it is not more than a person with a bag of fir cones could produce. What makes it worth stating precisely is that the alternative — one specimen, three rings, carefully measured — is a measurement that cannot answer the question no matter how well it is done, and that fact is not visible from inside the measurement.

The general form, which is the reason this essay exists #

Everything above is an instance of one question, and the question is worth separating from the cone.

How many observations does it take before a model with $k+1$ parameters can be preferred to one with $k$? The answer is never fewer than $k+2$, and it is usually a great deal more, and the gap between those two numbers is where most of the difficulty lives.

Two rings and a constant exponent is $k = 1$ with one observation: the model fits perfectly and says nothing. Three rings is one degree of freedom left over, which is enough for a residual to exist and not enough for it to be informative. Five rings is three degrees of freedom, and on this organ that is where the drift finally exceeds what noise explains — but only because the drift happens to be concentrated in the last gap, which is a property of ogives rather than of counting.

The uncomfortable half is that the number of observations available is a property of the organ, not of the observer’s diligence. A fir cone has the rings it has. Working harder produces a better measurement of the same four gaps, and better measurements run into a floor that no amount of care crosses: the drift is 8.2 per cent and a three per cent ruler allows 8.5.

That is why the answer, when it comes, comes from a population rather than from a specimen. The single-specimen limit is set by the organ; the population limit is set by the observer, and observers can be persuaded to do twelve of something.

This is the same shape as the sample-size arithmetic in the wrong field of this collection, arrived at from the other direction. There the question is how many plants it takes to separate two shares, and the answer is tens. Here the question is how many rings it takes to separate two models of one plant, and the answer is more rings than a plant has — so the count moves back to plants again, and the two calculations meet.

A prediction, so this is not only a counsel of despair #

The arithmetic above says what cannot be established. It also says one thing that can, and it is worth writing down as a prediction because a specification with no falsifiable content in it is a wish list.

If a conifer cone is an ogive and its lattice follows the ladder, then its consecutive ring gaps must increase monotonically from tip to base. Not by a specific amount — the amount depends on the arc radius, which varies between species and between individuals — but the direction is fixed by the shape, because the exponent falls along the axis and a falling exponent widens the gaps.

That is a much weaker claim than a measured exponent and correspondingly much easier to test. Monotonicity in three gaps is a sign test with a one-in-six null probability; across twelve specimens it is decisive at any measuring error that resolves the gaps at all, because the direction of an error is independent from specimen to specimen while the direction of the effect is not.

And it has a clean failure mode. If the gaps come out decreasing, the organ’s exponent rises along its axis, which no ogive does — that would be a shape widening faster the further out it goes, and it would mean the profile assumed here is wrong rather than the ladder. If they come out unordered, the drift is below the measurement and the situation is the one this essay describes.

So the smallest useful experiment is not a measurement of an exponent at all. It is: take a dozen cones of one species, find four rings on each, and ask whether the gaps go up.