What a count is worth

Almost every published statement about phyllotaxis is a pair of numbers. Thirty-four and fifty-five. Eight and thirteen. Two and three. This collection has spent four phases arguing that the pair is the thing that can actually be measured, and that most claims about divergence angles are really claims about pairs.

So it is worth asking, quantitatively, what one of those pairs is worth. Given only the report — no radius, no photograph, no angle — how much does it narrow down the divergence angle of the plant it came from?

The answer is a clean law and it is not the one this site expected.

The measurement #

A report of “34 and 55” is a statement that at some rise, the 34th and 55th lattice vectors are the two shortest. Sweeping the divergence angle from 20° to 180° and asking, at each angle, whether that pair is ever dominant at any rise gives the set of angles consistent with the report. The width of that set is what the report pins down.

Doing it needs care in two places and both are worth stating.

The edges are found by bisection, not by the sweep. The bands are narrow — a tenth of a degree at the top of the ladder — and on a grid alone every width comes out as a whole number of grid steps, which is a measurement of the grid. The scan locates each band and forty bisections find each edge to better than a ten-thousandth of a degree.

And the scan has to be fine enough to find the band at all. A band narrower than the scan step is not mismeasured, it is invisible: the routine reports that no divergence angle produces the pair. The gate requires the narrowest band measured to be several scan steps wide, and it caught exactly that failure once — 34/55’s band came out 1.3 steps wide at the first resolution tried.

The law #

The widths, up the Fibonacci ladder:

Multiply each by the product of its two counts and the results are 233, 216, 221, 221, 221, 220, 221. The width is

$$\frac{221°}{mn}$$

and the constant is not fitted to this family: the same measurement on the Lucas pairs — 7/11, 11/18, 18/29, which share no members with the Fibonacci ones — gives 226, 220 and 223.

Consecutive Fibonacci pairs have $mn$ growing by $\varphi^2$ each rung, so the constraint tightens by a factor of 2.618 per rung. Measured, the ratios between consecutive widths are 2.608, 2.617, 2.629 and 2.609.

What a count is worth is set by how high the count is, and it climbs geometrically.

What that means for reading the literature #

Put concretely: a plant reported as showing 2 and 3 has a divergence angle somewhere in a thirty-nine-degree band. That is a quarter of the useful range, and it is consistent with the golden angle, with the Lucas angle, and with a great many angles that are neither.

A plant reported as showing 34 and 55 has a divergence angle within a tenth of a degree.

Three consequences follow and they are all practical.

A count taken further out is worth more, and by a lot. Counting a sunflower at the rim rather than near the middle is not a matter of convenience; it is worth a factor of two and a half in the resulting constraint for every rung it climbs. A count at 34/55 is a thousand times more informative than one at 2/3.

A collection of low counts does not substitute for one high count. Averaging over many plants reduces the noise in a share, but a bare 2/3 does not become informative about an angle by being repeated: every one of those reports allows the same thirty-nine degrees, and their intersection is that same thirty-nine degrees.

And every band contains the golden angle. That is not an accident of the pairs chosen — a consecutive Fibonacci pair is what a golden-angle lattice gives, so its band must contain it. It does mean that a report of a Fibonacci pair is not evidence against the golden angle at any rung, and only evidence for it in proportion to how narrow the band is.

Where 221 comes from, as far as this goes #

A constant that reproduces itself on two unrelated families is asking to be derived, and this collection can get part of the way.

The pair $(m, n)$ is the two shortest families when the divergence is close enough to a particular value that neither $\langle m\delta \rangle$ nor $\langle n\delta \rangle$ — the distance from $m\delta$ or $n\delta$ to the nearest whole turn — is large. Move the divergence by $\Delta$ and $\langle m\delta \rangle$ moves by $m\Delta$ while $\langle n\delta \rangle$ moves by $n\Delta$. The pair survives while both stay small, so the tolerance on $\Delta$ scales as $1/n$ from one condition and $1/m$ from the other; the band’s width is set by the tighter, and the product $mn$ appears because the two conditions have to hold at once with the rise free to compensate between them.

That gets the $1/mn$ and it does not get the 221. The constant depends on how much room the third family leaves — the pair stops being the two shortest when some other offset becomes shorter than one of them — and the offsets that compete are $m+n$ and $n-m$, whose behaviour under the same shift involves the same two quantities again. Working it out properly is a piece of lattice arithmetic this phase did not do.

What the measurement does establish is that the constant is family-independent, which is the part that matters for reading a report. A Lucas plant reported at 18/29 and a Fibonacci plant reported at 21/34 have their angles pinned to 0.427° and 0.308° respectively, and the difference between those is the difference between 522 and 714, not anything about which ladder they are on.

The bands are single intervals, which is not obvious #

One property of the measurement is worth pointing out because the opposite would change what a report means.

Each band is one contiguous interval, not a set of scattered candidates. That is checked on every pair measured, and it is why a report can be summarised by a width at all.

The alternative was quite possible a priori. The pair $(m,n)$ is dominant in a region of the van Iterson plane, and a region in two dimensions can project onto a disconnected set in one — which would mean a report of “34 and 55” was consistent with, say, an angle near 137.5° or one near 99°, with nothing between. A survey reading such a report would have to carry both, and averaging over specimens would be a mixture problem rather than an interval one.

It does not happen for any pair tried here, and the reason is that the region is a single lobe of the van Iterson diagram — one node of the branching tree, bounded below by the two forks it splits into and above by the fork that produced it. The tree’s structure is what makes the projection connected, which is a small piece of the previous phase’s work being useful somewhere it was not built for.

The correction: the counting radius adds ten per cent #

Here is where this essay revises its own collection.

The first item on this site’s list of missing fields, carried in _plan/STATE.json for two phases and stated in the plan for three, is the counting radius. The reasoning behind it is in the foundation phase and it is correct: the spiral counts change with radius, one head gives 21 and 34 near the middle and 55 and 89 at the rim, so every published count is a statement about an annulus and a count without its annulus is incomplete.

The conclusion drawn from that was that the radius is what a survey most needs to record. Measured, it is not.

Take a reported pair, and compare the divergence angles consistent with the pair alone against those consistent with the pair plus the rise its counting radius implies. The band narrows by a factor of 1.10, at every rung tried.

The reason is straightforward once seen. The ambiguity a radius resolves is which rung the counter was standing on — and a report that names the pair has already resolved it. Knowing the rise as well says where within the rung, and a rung is narrow.

So the request was for the right field for the wrong reason. The radius is worth recording, and what it is worth recording for is different:

• Where the transitions sit along an axis. That is the whole of the φ^(2/p) shape question, and there the position is the measurement rather than a label.
• Whether two specimens were counted in the same regime. Comparing counts across plants requires knowing that they were taken at comparable places, and without a radius they might not have been.

Neither of those is recovering an angle, and recovering an angle is what the request was justified by.

Why this kind of correction is worth publishing #

There is a temptation, on finding that a long-standing request was mis-justified, to quietly restate the justification and carry on. This collection’s habit runs the other way, and the reason is that the mis-justification was doing work.

For three phases the phase plan has led with the counting radius as the largest missing field, and every essay in the wrong field has said some version of these frequencies are about the geometry because the data does not record what would make them about plants. Had a dataset arrived with radii in it, the expectation was that it would unlock the angle question. It would not have, by a factor of ten, and the disappointment would have arrived after the data collection rather than before it.

The arithmetic that shows it takes about two minutes to run and was never run, because the reasoning behind the request was sound and the conclusion followed from it in the way that conclusions seem to. A correct argument for a field being missing is not an argument about how much the field is worth, and only the second is a reason to ask somebody for it.

A note on how the measurement was made fast enough to make #

The sweep is nine thousand angles by a hundred and seventy rises, which is a million and a half evaluations of “which two families are shortest”, and the site’s own routine for that allocates an array of objects and sorts it every time. Run that way the measurement takes a minute and a half, which is too long for a page that draws several of these figures.

It was rewritten to compute the offsets’ angular parts once per angle and scan the rises with nothing allocated, which took it to four seconds. That is a routine optimisation and it is mentioned for one reason: a rewrite for speed is a second implementation, and a second implementation is a second opinion unless it is checked against the first. It is checked, on a grid of angles and rises, and required to give the same answer everywhere — 4,800 of 4,800.

There was also a shortcut that looked exact and was not, and it cost an hour. The two families $m$ and $n$ are equally long at exactly one rise, and it is tempting to test only that one: if the pair is ever the two shortest, surely it is the two shortest there. It fails whenever $\langle m\delta \rangle < \langle n\delta \rangle$, where the two never cross at all and the pair can still be dominant throughout its interval. The symptom was a band that was narrower than one of its own slices — a union smaller than a subset — which is the kind of contradiction that announces itself, and it took a while to see because the numbers it produced were plausible.

What the pair is worth, restated #

The useful summary is short.

The most informative thing a published count can carry is how high the count is, and that is already in every report. A survey of plants counted at 34/55 constrains divergence angles a thousand times better than the same survey counted at 2/3, at no extra cost to anybody, by choosing where to count.

The second most useful is the pair’s jugacy — whether the pattern has two or three primordia arriving at a time — which the counts cannot decide and the rotational symmetry can. That is a separate field, it is cheap, and the previous phase established that the counts alone are genuinely unable to supply it.

The counting radius is third, it is worth about ten per cent for the angle question, and it is worth a great deal more for the two questions that are actually about position. It stays on the list, in third place, with its justification rewritten.

What this does not say #

Two limits, because a law with a clean constant invites more weight than it can carry.

It assumes the pattern is an ordinary lattice. A multijugate plant has $k$ primordia arriving at a time, its counts share a factor of $k$, and the whole calculation above is about coprime pairs. A report of “4 and 6” is not a report of a pair with $mn = 24$; it is a report of a bijugate pattern whose underlying pair is 2 and 3, and its angle is defined only modulo $180°$. The routine refuses a non-coprime pair rather than returning a number for it, which is the right behaviour and worth saying, because a survey full of whorled plants would otherwise be silently mis-analysed.

And it assumes the report is right. The width is what a correct count leaves open. A miscount — the two smallest offsets rather than the two shortest, which is a mistake this collection’s own machinery made and which produced 21 and 34 for a head whose real neighbours were 34 and 55 — produces a band that is narrow and in the wrong place. Precision without accuracy is worse at high counts than at low ones, exactly because the band is narrower: a wrong 34/55 excludes the truth confidently, where a wrong 2/3 excludes almost nothing.

That is an uncomfortable pairing with the recommendation to count further out. Counting at the rim is worth a thousandfold in precision and it is also where the families are most numerous, most nearly parallel, and hardest to trace by eye. The recommendation stands, with the caveat attached: the value of a high count is entirely conditional on it being right, and a survey that gathers high counts needs a check on them that a survey of low counts does not.

The check that this collection would suggest is the one its own machinery uses: count in two adjacent annuli rather than one. Consecutive rungs are consecutive Fibonacci pairs — 21/34 below and 34/55 above — and a pair of counts that are not consecutive in that sense is a miscount somewhere, detectable without knowing which of the two is wrong.