How many plants would it take

This collection has said, in four phases and about a dozen essays, that its frequency statements are about geometry rather than about plants. At a rise of 0.008, 14.7 per cent of divergence angles give a consecutive Fibonacci pair. Grown from a coarse start with a rule that mentions no angle, better than ninety per cent do. Neither of those is a statement about a meadow.

The obvious question is how much of a meadow it would take to make one, and it has an answer that is easy to compute and easy to avoid computing.

The census question #

The sharpest form is this. Take specimens, count each one, and record whether the pair is consecutive Fibonacci. Under the geometry’s census the share is 14.7 per cent; under the reading this collection favours — that plants follow a continuous history from a coarse start — it is over ninety.

Separating 14.7 per cent from 90 per cent at ninety per cent power and a five per cent false-positive rate takes four specimens.

Four. The weaker and more interesting comparison — separating the geometry’s share from a coin weighted to a half, which is what one would need if plants were only often Fibonacci rather than nearly always — takes fourteen.

Why the arithmetic is exact rather than approximated #

At sample sizes of tens the usual normal approximation is doing real work at exactly the point the answer is quoted, so the calculation here sums binomial terms directly.

For each $n$, the cut is placed at the smallest number of Fibonacci specimens whose probability under the null falls below five per cent; the power is then the chance of reaching that cut under the alternative. At $n = 14$ the cut is 5 and the power is 91.0 per cent. The normal approximation for the same comparison returns 13.

One specimen out of fourteen is a seven per cent error in the answer, which would be unimportant if the answer were four hundred and is not when it is fourteen. It also matters in a way the approximation cannot express: the exact calculation reports the power it achieves — 91.0 per cent, not the 90 asked for — because a whole number of specimens cannot deliver exactly the level requested. The staircase is real and the smooth curve is the fiction.

The other questions #

Three more comparisons this collection could put to a survey, with the same arithmetic.

Are multijugate patterns a real minority or a miscount? The previous phase established that the counts cannot decide jugacy — an ordinary lattice at 180.5° counts 2 and 4 with rotational symmetry of order 1, and a genuine bijugate stem counts the same pair with order 2 — so the field needed is the symmetry of the whorl rather than another count. Separating a two per cent rate, which is what miscounting would leave, from a fifteen per cent real minority takes 34 specimens.

Is a conifer cone’s ladder spaced as a cone’s or an ogive’s? The ratios are $\varphi^2 = 2.618$ and $\varphi^{2/1.88} = 1.878$, which differ by a third in the log. At three per cent on each ring position, that takes one specimen measured at three rings.

And how often is a pattern on the Lucas branch rather than the Fibonacci one? That is the same shape as the census question with a smaller effect, and it is the one where the sample sizes start to climb — separating a two per cent Lucas rate from a five per cent one needs several hundred.

Why so few, when the shares are shares #

Fourteen specimens sounds too few for a question about a proportion, and the intuition that says so is calibrated on a different kind of comparison.

Most sample-size arithmetic in practice is about detecting small differences — a treatment that moves a rate from 40 per cent to 46, an effect of a few points. Those need hundreds or thousands, because the sampling noise on a proportion at $n$ specimens is about $1/(2\sqrt{n})$ and a six-point difference needs that to be smaller than three points.

The comparisons here are not small. The geometry says 14.7 per cent and the alternative says 90; the gap is seventy-five points on a scale that is a hundred points wide. At four specimens, all four being Fibonacci has a probability of 0.0005 under the geometry’s census and 0.66 under the alternative, and that is already a decisive experiment.

The reason the gap is so large is worth naming, because it is a result rather than a convenience. The geometry is not generous with Fibonacci. Pick a divergence angle at random at a fine rise and the odds of a consecutive Fibonacci pair are about one in seven; the reason plants show them almost always is continuity from a coarse start, which is a claim about history rather than about which arrangements are good. Two mechanisms that differ that much in their predictions are cheap to tell apart, and the cheapness is a consequence of the earlier finding rather than an accident.

The corollary is that the interesting survey questions are the expensive ones. Whether plants are 90 per cent Fibonacci or 97 per cent — which is where a real dataset would land, and which bears on how often a real apex loses its branch — is a seven-point difference and needs a few hundred specimens. This collection’s questions are cheap because they are still coarse.

The uncomfortable part #

If these numbers were in the thousands, the absence of the dataset would explain itself. Botanical surveys of that size are expensive, and a collection of illustrated essays would have no business asking for one.

They are in the tens. Four plants. Fourteen plants. One cone.

So the reason these questions are unsettled is not that settling them is expensive. It is that the fields that would settle them are not the fields anybody records, and the fields in question are free at the moment of counting and unrecoverable afterwards.

• The census needs the pair and the rung it was counted at. The pair is always recorded; the rung — near the middle, at the rim — usually is not, and without it a collection of counts from unstated places is a mixture of rungs rather than a sample of one.
• The jugacy question needs the rotational symmetry of the whorl, which is a glance at the specimen and a note of whether the primordia come singly or in pairs. It is not recorded because the quantity everyone reports is the pair, and the pair cannot express it.
• The shape question needs three ring positions along one axis rather than one count on it, which is a different measurement of the same specimen rather than a harder one.

None of these costs money. Each costs a line in a notebook, at a moment when a person is already holding the plant.

The test is one-sided, and that is a choice #

A conventional design would test two-sided: is the plant share different from the geometry’s, in either direction. This one asks whether it is larger, and the difference is worth defending rather than assuming.

The question being put is not “do plants agree with the geometry”. Nobody expects them to, and a survey finding plants less Fibonacci than a random divergence angle would not refute anything in this collection — it would refute the counting, or the sampling, or the species chosen, long before it refuted the geometry.

The question is whether the excess this collection predicts is there. The grown-history reading says plants should be far more Fibonacci than the geometry, by a factor of six, and the survey is being run to see whether that excess exists. One-sided is what that question is.

The cost is stated: at a five per cent one-sided level, a result in the other direction has no interpretation within this design. That is acceptable here and it would not be if the two mechanisms made predictions on either side of the null.

What a small sample size does not mean #

There is a way of reading “four specimens” that this essay is not offering, and it is worth blocking.

It does not mean the census question is nearly settled by four plants anybody has already counted. The four have to be a sample — chosen without reference to their pattern, from a stated population — and almost every published count fails that in a specific and fatal way: the plants that get counted are the ones with countable patterns, which is to say the ones with clean high-order spirals, which is to say the ones most likely to be on the Fibonacci ladder.

That selection is not a small bias. If the reason a plant appears in the literature is that somebody could count it, then the observed Fibonacci share is a statement about what is countable rather than about what grows, and no sample size fixes it. Four specimens chosen at random from a field would settle the question; four thousand chosen because they photographed well would not.

This is why the sample sizes are the least interesting part of the specification, and why the essay after this one is about the fields and the sampling rather than about the counts.

There is a version of the selection problem that cannot be designed away, and it is worth separating from the version that can. Choosing the plants at random from a stated population fixes the bias introduced by an observer preferring photogenic specimens; that is a matter of protocol. What it does not fix is that some plants have no countable pattern at all — a rosette too tight to trace, a stem whose leaves have fallen — and those are not missing at random either. A protocol can record them as uncountable rather than dropping them silently, which turns an unknown bias into a stated exclusion, and that is the most a survey can do.

The share of specimens that are uncountable is, incidentally, a number this collection would find interesting for its own sake. The noise essays in the emergence field measure that a lattice fails abruptly rather than gradually — there is no such thing as a slightly disordered phyllotaxis, only a lattice or a scatter. A field survey recording what fraction of plants have no countable pattern would be measuring something this collection predicts should be either very small or very large, and has no way to guess which.

The design, written out #

The whole of the census experiment fits in a paragraph, and writing it out is the point of the arithmetic above.

Choose a species and a rung — say a sunflower head counted at eight tenths of its radius, which reaches 34/55 and pins a divergence angle to a tenth of a degree. Choose fourteen individuals from a named population without looking at their patterns first. Count each one, in two adjacent annuli rather than one, so that a miscount shows up as a pair of counts that are not consecutive rungs of the same ladder. Record for each: the two counts, the annulus, and whether the primordia in the youngest whorl arrive singly. Score a specimen as Fibonacci if its pair is consecutive Fibonacci.

If five or more of the fourteen are Fibonacci, the geometry’s census is rejected at five per cent, one-sided. Under the geometry alone that happens 4.6 per cent of the time; if plants are half Fibonacci it happens 91 per cent of the time; if they are nine tenths Fibonacci it is a formality.

That is a complete experiment, it takes an afternoon, and every number in it comes from computation done in this repository rather than from any prior expectation about plants.

What the arithmetic is good for #

Two things, and neither is planning a survey that nobody is going to run.

It sizes the ambition correctly. A question that needs fourteen specimens and a question that needs fourteen thousand are different kinds of open question, and this collection had not distinguished them. Every essay in the wrong field says the frequency statements are about geometry; none of them said the gap was four plants wide.

And it says which questions are worth the fields. The jugacy question needs 34 specimens and a field nobody records; the shape question needs one specimen and a different measurement. Those are very different asks, and a specification that treats them as one list of missing fields is asking for the wrong things with equal urgency.

The ordering that falls out is: the shape question first, because a single specimen carefully measured settles it; the census second, because the specimens are cheap and the field it needs is nearly free; and the jugacy question third, because it needs both a new field and a real sample.

What would change these numbers #

The sample sizes above assume the effect sizes this collection has computed, and those are geometry rather than observation, so they are the part most likely to move.

The census comparison uses 14.7 per cent as the null. That number is the share of divergence angles giving a consecutive Fibonacci pair at a rise of 0.008, and the previous phase measured that the share falls as the pattern gets finer — 67.5 per cent at a rise of 0.12, 14.6 per cent at 0.008. So the null depends on the rung, and a survey pooling counts from different rungs is comparing against a null it cannot state.

That is the same defect as the sampling problem, one level down: the arithmetic here is correct given a stated rung, and the rung is exactly the field the previous essay found is worth ten per cent for the angle question and is indispensable for this one. The counting radius comes back in a place it was not being justified for, which is what happens when a request is re-derived rather than repeated.

The other two effect sizes are softer still. The fifteen per cent multijugate rate is a guess at what a real minority would look like, chosen because it is the order of magnitude the botanical literature suggests for teasel and its relatives rather than because anything here computes it; the two per cent null is a guess at a miscounting rate. Both are the sort of number a survey would produce rather than test against, so the 34 specimens should be read as an order of magnitude and not as a design.

The shape comparison is the firm one, because both ends of it are computed here: 2.618 for a cone and 1.878 for the ogive built in the cylinder field of this collection, with the error model stated explicitly. That is the question whose sample size means what it says, and it is also the one whose answer is a single specimen.

The shape of the whole calculation #

Three quantities go into every line of this, and separating them is most of the value.

The effect size — how far apart the two hypotheses are — comes from geometry, is computed here, and is the part that moves if the geometry is refined.

The measurement error comes from the observer, is a stated assumption, and is the part that a better method improves.

And the sample size falls out of the first two and is the part that costs money.

Almost every discussion of a missing dataset runs those together, and the result is a request for “more data” that cannot be sized or prioritised. Pulling them apart gives a specification with three separable knobs: a survey too small to answer a question can be enlarged, a measurement too coarse can be sharpened, and a comparison whose two hypotheses are too close together cannot be rescued by either — which is the situation the convex head is in, and which is worth knowing before anybody starts measuring one.