What one exponent reports

Somebody with a fir cone and a ruler measures where the parastichy numbers change along it, and wants a shape exponent out of the result. The relation is available: under a rise falling as $s^{-p}$ the transitions are geometric with ratio $\varphi^{2/p}$, so

$$\log s_k = \text{const} + \frac{2\ln\varphi}{p}\,k$$

Fit that line to the ring positions, invert the slope, and read off $p$. It is the obvious thing to do and it is what anybody would do.

The organ has no single $p$. So the fit returns something — and what it returns is the question.

The answer #

It returns the harmonic mean of what the individual steps report.

Each step of the ladder multiplies the rise by $\varphi^{-2}$, so a step from $s$ to $s'$ contributes $2\ln\varphi/p_k$ to the total span in $\log s$, where $p_k$ is that step’s own exponent. The slope is therefore an average of reciprocals, and inverting an average of reciprocals is a harmonic mean.

Measured on the ogive: the four steps report 1.980, 1.968, 1.905 and 1.697. Their harmonic mean is 1.880 and their plain average is 1.887. What the fit returns is 1.891.

That the fitted value sits a thousandth above the harmonic mean rather than exactly on it is the least-squares weighting: an ordinary regression through $\log s$ against ring number does not weight the increments equally, where taking the slope from the two end rings would. The difference is a part in a thousand here and it is worth naming rather than smoothing over, because it is the sort of gap that gets reported as agreement when it is arithmetic.

Why a harmonic mean is worse than an average #

Harmonic means are always at most arithmetic means, with equality only when every value is the same. So a fit to an organ with a varying exponent understates the organ’s mean exponent, always, and by more the more the exponent varies.

Here the understatement is 1.880 against 1.887, which is four parts in a thousand. That is not a large error and it would be dishonest to present it as one.

What makes it worth reporting is not its size but its three properties:

• It is systematic. It does not average away across specimens. Measure a hundred fir cones and the mean of the fitted exponents is still below the mean of their true mean exponents.
• It has a known sign. Whatever the profile, the fit is low. That is a stronger statement than knowing the magnitude, because it means a correction has a direction even when it has no size.
• And it is invisible in the fit. The residual of the fitted line is 0.045 of one step, which on four rings is exactly the scatter one would expect from a good measurement of a straight relationship. Nothing about the fit announces that it is averaging over a varying quantity.

The last is the one that matters. A fit that was badly wrong would show a curved residual; a fit that is slightly and systematically wrong shows nothing at all.

The size of it, honestly #

Four parts in a thousand is small enough that it is worth asking whether this essay is about anything.

It is, but the thing it is about is not the correction. It is the interpretation of the number.

A measured exponent of 1.89 on a fir cone invites the reading this organ’s circumference grows as the 1.89 power of arc length, which is false about every part of the organ: the exponent is 1.98 near the tip and 1.70 at the shoulder and 1.89 nowhere in particular. The right reading is the harmonic mean of the exponents over the countable stretch is 1.89, which is a much weaker statement and a true one.

The two readings differ in what they license. The first licenses predicting where the next ring will be, on the assumption that the pattern continues; the second does not, because the next ring is further out, where the exponent is lower, and the prediction from a mean will be systematically early.

That is a checkable consequence and it is worth stating as one. Extrapolating a fitted exponent past the rings it was fitted to should place the next ring too close to the ones before it, and by an amount that grows the further the extrapolation runs. On the specimen here the gaps run 1.626, 1.631, 1.657, 1.763; a fit to the first three predicts the fourth ring at a gap of about 1.64 where the organ delivers 1.76.

The derivation, since it is three lines and worth having #

The claim above is not an empirical regularity that happened to come out of a sweep. It follows from the ladder’s geometry, and setting it out makes clear what would have to change for it to fail.

The ladder is geometric in the rise: consecutive transitions sit at rises a factor of $\varphi^{-2}$ apart. That is a fact about lattices on a cylinder, measured in an earlier phase — the transitions at 0.1246, 0.0476, 0.0182, 0.00695, 0.00266, with consecutive ratios 0.3817 to 0.3828 against $\varphi^{-2} = 0.38197$.

Now let the rise vary along an axis as some function $h(s)$, with local exponent $p(s) = -\mathrm{d}\log h/\mathrm{d}\log s$. Between two consecutive transitions the rise falls by $\varphi^{-2}$, so

$$2\ln\varphi = \int_{s_k}^{s_{k+1}} p(s)\,\frac{\mathrm{d}s}{s} \approx p_k \ln\frac{s_{k+1}}{s_k}$$

where the last step is the mean value theorem and $p_k$ is the local exponent at some point inside the interval. Rearranging gives the step’s reported exponent, and it is exactly what the interval measurement computes.

Summing over $K$ steps, the total span is $\sum_k 2\ln\varphi/p_k$, and a slope taken across the whole is that divided by $K$. Inverting gives

$$p_{\text{fit}} = \left(\frac{1}{K}\sum_k \frac{1}{p_k}\right)^{-1}$$

which is the harmonic mean, exactly and not approximately. The only approximation anywhere is the mean value theorem’s, and that is an equality with $p_k$ defined as it is.

What would break it: a ladder that is not geometric in the rise, or an organ whose exponent is not continuous. The first is a fact about lattices and holds; the second would need a shape with a kink in it, which a surface of revolution grown by a plant does not have.

Where the number really comes from #

There is a subtlety in what “the step’s exponent” means, and it is the difference between this essay’s claim being a measurement and being a definition.

A step reports $2\ln\varphi/\ln(s'/s)$, which is a number derived from two positions. The claim that it is an exponent the organ has needs the mean value theorem: if the organ’s local exponent varies continuously from one end of the step to the other, then somewhere inside the step it equals the value the step reports.

Checked on all four steps of the ogive, each reported exponent lies between the profile’s own values at the two ends. That check is what stops the interval exponents being an arithmetic rearrangement of the ring positions and makes them a measurement of the surface — and it uses two calculations with nothing in common, since the ladder does not know the profile and the profile does not know the ladder.

The check also failed the first time it was run, for a reason worth recording. Run against the counted ring positions rather than the predicted ones, one step reported 1.983 where the profile’s values at its ends were 1.847 and 1.942 — outside the bracket. Nothing was wrong with the organ or the theorem; the counted positions are quantised by the counting bands at a few per cent, and an exponent inferred from a ratio of two such positions inherits that error amplified by about a factor of three. The fine-grained arithmetic belongs on the predicted positions and the counter’s job is to agree with them, which it does to a per cent and a half.

What the residual is worth, which is less than it looks #

A fit reports a residual, and the temptation is to read it as the evidence about whether one exponent is enough. It is evidence, and it is weak evidence, and the reason is a counting argument rather than anything about organs.

On the ogive the fit’s residual is 0.045 of one step. Four rings give three gaps, a straight line through four points has two degrees of freedom left, and a residual computed from two degrees of freedom has a sampling distribution wide enough to accommodate almost anything. A residual of that size is consistent with a perfectly constant exponent measured slightly imprecisely, and equally consistent with the drift the organ actually has.

Two rings are worse than weak: they are silent. One gap determines one exponent exactly, and there is no residual at all — not a small one, none. A varying exponent is invisible in principle rather than hard to see, which is a distinction 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 measurement.

Three rings give two gaps that can disagree, and on this specimen they disagree by 0.32 per cent — the first two gaps are 1.626 and 1.631. That is far inside any plausible measuring error, so three rings on an ogive are consistent with a constant exponent no matter how carefully they are measured.

It is the fourth ring that carries the information, because the drift on this shape is concentrated at the shoulder: the last gap is 1.763 against the first’s 1.626, eight per cent away. Everything this essay is about lives in one gap out of four, and that is the arithmetic the next essay turns into a requirement on a survey.

What this says about the previous phase’s result #

The φ^(2/p) relation was measured on five surfaces and every one agreed. This does not disturb that and it does change what a measured ratio can be taken to mean.

The previous phase already established two degeneracies. The ratio does not identify the shape: a cone that elongates and a paraboloid that fills have the same $p$ and give the same ratio. And it does not identify the growth law: one surface under the two laws gives 1.900 and 1.378. Only the product $p$ is identified.

This adds a third and it is of a different kind. Even the product is not identified pointwise — what a specimen reports is an average of $p$ over the stretch that was counted, weighted harmonically. Two organs with quite different profiles, one with a high exponent throughout and one running from very high to very low, can report the same number.

So the chain of inference from a measured ring spacing back to a plant runs: spacing → harmonic mean of $p$ over the counted stretch → some average of shape and growth law together → nothing about either separately. Each arrow loses something, and the previous phase had documented the last two.

What a measurer should do instead #

The fit is not the enemy, and the alternative is not more machinery. It is reporting one more number.

Report the gaps, not only the fitted exponent. Four ring positions give three gaps, and the gaps are the raw measurement; the exponent is a summary of them. Anybody who has the gaps can compute the exponent, can see whether it drifts, and can compare the drift against their own measuring error. Anybody who has only the exponent has thrown that away irreversibly.

That is a familiar shape on this site. The counting radius, the rotational symmetry, the handedness — this collection has spent three phases asking published counts for fields that cost nothing at the moment of measurement and cannot be recovered afterwards. The gaps are one more, and they are the cheapest of the lot: they are what was measured, before anything was done to them.

Whether the drift in those gaps could be seen, on a real specimen with a real ruler, is a separate question and it is the next essay’s.

A case where the effect is not small #

The ogive’s four parts in a thousand is small because the ogive’s exponent is nearly constant over its countable stretch. Nothing in the derivation says it has to be, and it is worth knowing what shape of organ would make the effect large, if only to recognise one.

The gap between the harmonic and arithmetic means grows with the variance of the values, roughly as the variance divided by the mean cubed. So the effect is large when an organ’s exponent varies a lot over the region where its counts can be followed — which needs two things at once: a strongly curved profile, and enough of the ladder visible on the curved part.

An ogive fails the second condition. Its exponent does run from 2 to nearly 0, which is as much variation as any surface offers, but the fall happens at the shoulder where the parastichy families are nearly parallel and the counting is hardest. Four of its five rings are on the nearly-constant part.

The organ that would show it is one whose curvature is in the middle of its countable range rather than at the end — a shape that widens sharply somewhere along its length rather than at its tip or its base. A pineapple, whose profile is close to a cylinder with a shoulder at each end, or a compound inflorescence whose axis broadens partway up, are the sort of thing. Neither is built here and neither is a claim; they are the answer to “what would make this matter”, written down so that the smallness of the effect on the specimen that was built is not read as smallness in general.

There is also a purely negative case, which the previous essay measured: an organ whose exponent varies by one per cent over its rings, as a convex head does, has a harmonic-arithmetic gap of order one part in a hundred thousand. On that organ the effect is not small — it is absent, and the fit means exactly what it appears to.

The general shape of the finding #

Strip out the phyllotaxis and what is left is a statement about fitting a one-parameter model to a system whose parameter varies, and it has three parts that are worth separating because they behave differently.

The fit returns a specific average, and which average is derivable. It is not “roughly the mean” or “some kind of average”: the geometry of what is being fitted decides it exactly, and here it is harmonic because each step contributes to the fitted span in proportion to the reciprocal of the local parameter. A different relation between the parameter and the observable would give a different average, and the way to find out which is to write down what each observation contributes.

The bias therefore has a sign that is known without knowing the profile. Harmonic means are below arithmetic ones for every set of positive values. Nothing about the organ has to be known to say which way the fit is wrong — only that its parameter varies at all.

And its magnitude depends on something the fit cannot see. The gap scales with the variance of the local values over the fitted range, which is precisely what a single-parameter fit has thrown away. So the direction of the correction is free and the size of it is not, which is an uncomfortable combination: it means a measurer can know they are low without being able to say by how much.

The way out is the one the last section recommends, and it is not statistical. Report the individual gaps. The variance is in them, it is what the fit averaged over, and it costs nothing to keep.