What "whorled" was hiding

The expansion phase asked the frequency question of the geometry rather than of a field survey, and got a number that has been the headline of this field ever since.

Pick a divergence angle at random. Fix a rise. Ask what the dominant parastichy pair is. At a rise of 0.12, consecutive Fibonacci numbers come up 67.5% of the time; at 0.03, 31.1%; at 0.008, 14.7%. The share falls as the pattern gets finer, so what makes plants Fibonacci is not the geometry being generous.

That census sorted every result into four buckets — Fibonacci, Lucas, whorled, other — and “whorled” meant “the two counts share a factor”. At the finest rise it held 35% of the circle, and the essay said so and moved on.

This one opens it.

What is in the bucket #

At a rise of 0.008 the non-coprime pairs the census produces are

$$2/4,\quad 3/6,\quad 4/8,\quad 5/10,\quad 6/12,\quad 7/14,\quad 8/16$$

and nothing else. Seven pairs, seven different common factors, and every one of them is $k$ and $2k$.

Divide out the factor and each is $1/2$ — the bottom rung of the ladder, the coarsest pattern there is. So the bucket is not a residue of odd arrangements. It is one arrangement, repeated $k$ times around the stem, for $k$ from 2 to 8.

The reason is visible once the lattice is written down. A single-jugate lattice at a divergence near $p/k$ of a turn has its offset-$k$ vector nearly vertical, because $k$ steps of $p/k$ turns is $p$ whole turns plus a little. Its offset-$2k$ vector is twice that. At any small rise those two are the shortest, so the counted pair is $k$ and $2k$ — for every rational $p/k$ with small $k$, which is a lot of the circle.

That is the same phenomenon the foundation phase found from the other end: near a rational divergence the nodes fall into $k$ radial rows and the pattern collapses. The census sees the collapse as a pair sharing a factor.

Reading the census up to jugacy #

Now the question this raises. A botanist counting 42 and 68 on a bijugate sunflower does not say “not Fibonacci”. They say “twice 21 and 34”, and they are right to: the underlying lattice is the ordinary Fibonacci one, wrapped twice.

So the census can be read a second way. Divide out any common factor first, and classify what is left.

Done that way the Fibonacci share at a rise of 0.008 goes from 14.7% to 50.1%. At 0.03 it goes from 31.1% to 55.7%. At 0.12, from 67.5% to 72.4%.

More than triple, at the finest rise. And the whole of the increase is the “whorled” bucket moving across, because — as above — every pair in it is $k$ times a consecutive Fibonacci pair, namely $k$ times $1/2$.

Why the generous reading is not the better one #

A number that triples when the definition is loosened is a number to be suspicious of, and the suspicion is warranted here.

Every pair the loosening admits is $k$ and $2k$. Underneath, that is the lattice with one family of one spiral and one family of two — the very bottom of the ladder, a pattern with essentially no structure. Calling it “Fibonacci because 1 and 2 are consecutive Fibonacci numbers” is technically correct and describes nothing anybody would photograph.

No pair like 42 and 68 appears in the census at all, and it cannot, because the census sweeps ordinary single-jugate divergences and a single-jugate lattice never has a deep multijugate-looking pair. To get 42 and 68 the pattern has to be genuinely bijugate, which is a fact about the point set and not about the divergence — exactly the distinction the two-at-a-time essay is built on.

So the generous reading admits 35% of the circle on a technicality and admits none of the cases it was invented for. That is the opposite of a useful widening.

The honest version of the question #

There are two separate questions here and the census was conflating them.

How often is an ordinary lattice Fibonacci? 14.7% at a rise of 0.008. That is the number the expansion phase reported and it stands.

How often is a bijugate lattice twice-Fibonacci? Also 14.7%, at half that rise. Sweeping a bijugate pattern’s divergence over its whole period of 180° sweeps the underlying ordinary divergence over the full circle, so the bijugate census is the ordinary census at twice the rise with the counts doubled. Same shares, relabelled.

The two questions have the same answer because they are the same question, and neither of them is answered by dividing out common factors in a sweep over single-jugate divergences. That third procedure answers a question nobody asked: how much of the circle produces a pattern that looks like several copies of something coarse.

How the census is taken #

The procedure is worth stating plainly, because a census is only as good as what it sweeps and how finely.

Fix a rise. Step the divergence across half a circle in 3,600 equal steps — half, because $\delta$ and $360-\delta$ are mirror images and produce identical lattices. At each step, compute the two shortest lattice vectors up to an order limit of 140 and record their offsets as the dominant pair. Tally.

Three things about that are limitations rather than choices.

The grid is rational. Every sampled divergence is a rational multiple of a turn, which is the observation the foundation phase’s essay on the surviving claim is built on: the sweep can never land on the golden angle, because the golden angle is irrational and every grid point is not. What the census measures is therefore the share of an interval around each outcome, which is the right quantity for a frequency question and is not the same as evaluating the outcome at any particular angle.

The order limit truncates. Past offset 140 the calculation stops looking, so at a rise fine enough that the true dominant pair exceeds it, the census reports the shortest offset it is allowed to consider. At a rise of 0.008 the deepest Fibonacci pair in play is 21 and 34, comfortably inside; at a rise ten times finer it would not be, and the census would silently answer a smaller question. This is the same failure the cone essays record from a figure that drew the opposite of the truth when its order limit was left at its default.

And the rise is fixed. Every number here is a statement about lattices at one rise, and a plant is not at one rise — it passes through many. That is not a defect that a better census could repair; it is a different question, and it is the one the last group of essays in this phase asks.

The Lucas bucket falls faster than anything #

One column of the decomposition behaves differently from the rest and it is worth a paragraph, because it bears on how much of the geometry is “named” at all.

The Lucas share — pairs of consecutive Lucas numbers, 1/3, 3/4, 4/7, 7/11, 11/18 — is 27.1% at a rise of 0.12, 8.4% at 0.03, and 1.5% at 0.008. It falls by a factor of eighteen where the Fibonacci share falls by a factor of five.

That asymmetry has a reason and it is the same one that decides the whole census. Both sequences grow geometrically, Fibonacci by $\varphi$ and Lucas by $\varphi$ as well, so both ladders have rungs at the same spacing. What differs is how much of the circle each rung’s region occupies, and that is set by how well the corresponding angle resists approximation. The Lucas angle is noble too, but it converges to its limit from further away, and its early rungs are wide while its late ones are narrow.

The practical consequence is that at fine rises the geometry is overwhelmingly unnamed. Fibonacci 14.7%, Lucas 1.5%, whorled 35.4%, and 48.5% belonging to no sequence anyone has a word for. A field survey finding Lucas phyllotaxis in a per cent or two of specimens would be finding exactly what the geometry hands over at that rise — which would be an argument that the plant is doing nothing at all, rather than evidence of a preference.

What the decomposition does say #

Three things survive, and all three are about the shape of the census rather than about its headline.

The whorled share grows as the rise falls. It is 4.9% at a rise of 0.12, 24.6% at 0.03, and 35.4% at 0.008. That is not an artefact: as the rise falls, a divergence has to be closer to a rational to produce a row pattern, but there are more rationals with small denominators available at the resolution the counter can see, and the second effect wins. The finest patterns are the ones most likely to look whorled.

The number of distinct factors grows too. One factor at a rise of 0.12, three at 0.03, seven at 0.008. So the bucket is not one population growing but a family of populations appearing one after another as the rise falls, each keyed to a rational denominator.

And “other” is the bucket that actually matters. At a rise of 0.008 the coprime, non-Fibonacci, non-Lucas share is 48.5% — nearly half the circle — and none of it is touched by any reading of jugacy. These are the genuinely unnamed pairs: 4 and 11, 7 and 18, whatever the geometry happens to hand over. They were the largest bucket in the original census and they remain the largest.

That last point is the one to carry away. The expansion phase’s finding was that Fibonacci is a minority outcome of the geometry, and the minority is not rescued by reading it generously — it is 14.7% strictly, 50.1% under a reading that admits only the trivial cases, and the untouched half of the circle is neither.

Why the bucket grew unnoticed #

It is worth asking how a bucket holding a third of the circle went two phases without being opened, because the answer is a general one about how a classification hides things.

The census reports four numbers and the essay quoted one of them. The Fibonacci share was the interesting quantity, “whorled” was the name of a known botanical arrangement, and a bucket with a familiar name reads as a bucket that has been understood. Nobody looks inside a category they can already name.

The second reason is that the name was nearly right. Non-coprime pairs really do come from lattices whose nodes fall into rows, and rows really are what a whorled plant looks like. What the name gets wrong is not the appearance but the mechanism: a row pattern from a single-jugate lattice at a rational divergence is not the same object as a genuinely multijugate one, and the difference is invisible in the counts.

The third is that the bucket’s contents are uniform in a way that makes them look deliberate. Seven pairs, seven factors, every one of the form $k$ and $2k$: it has the shape of a designed category rather than of a residue, so opening it looks unnecessary until it is opened.

This is the same shape as the eighteen captions written for other figures that the standard pass found, and as the counter that returned the two smallest offsets rather than the two shortest. The symptom is agreement, not error. A category with a plausible name, holding plausible members, in a table whose other columns are being argued about, is exactly where an unexamined thing survives.

What a classifier ought to return #

The site’s classifyPair returns one of four words and the word “whorled” for any non-coprime pair. That is now visibly the wrong shape, and it is worth saying what the right one is even though changing it would ripple through three phases of recorded numbers.

A pair should be classified in two independent parts: its jugacy, which is the common factor, and the family of the pair underneath, which is Fibonacci, Lucas or other. So $42/68$ is “bijugate Fibonacci” and $5/10$ is “quinquejugate, bottom rung”, and neither is a single word.

And a pattern needs a third part that a pair cannot supply: whether its jugacy is real. A counted pair of $2/4$ is consistent with a genuinely bijugate stem and with an ordinary one near half a turn, and only the rotational symmetry of the point set decides. A classifier that reports jugacy from a pair alone is reporting a possibility as a fact.

None of that changes any number already published on this site. It changes what the numbers are called, and on a subject where the arguments are mostly about what things are called, that is not nothing.

What a survey would have to record #

Every number on this page is about the geometry. The comparison this field most wants is with a survey of plants, and the reason it cannot be made is not that surveys do not exist — it is that they do not record the things the comparison needs.

Four fields would be enough.

The counting radius, or the counting height. Without it a reported pair is a statement about an unstated annulus, and the whole of the counts-change-with-radius result says that different annuli give different answers on the same specimen. A survey of a thousand heads reporting “34 and 55” with no radius is a survey of where the observers happened to look.

The element count. The rise at element $i$ is $1/(4\pi i)$ on a head and $1/(2\pi i \sin\alpha)$ on a cone, so the element number is the position on the ladder. Reporting how many florets or scales the specimen had converts a count into a rung.

The rotational symmetry, not the inferred jugacy. A pair sharing a factor is consistent with a multijugate plant and with an ordinary one near a rational divergence, and the two are told apart by whether rotating the pattern maps it onto itself. That is observable on a photograph and it is not usually recorded.

And the handedness. Not because it bears on any of this, but because it is free, it is the one quantity a mirror-image ambiguity makes genuinely undetermined in the recovery, and a survey that recorded it would let someone else ask a question this site cannot.

None of those is expensive. The reason they are absent is that the quantity everyone reports — the pair — is the one that looks like the answer, and the fields that turn it into a measurement look like context.

The claim that is left standing #

Strip the reading question away and the expansion phase’s finding is unchanged and slightly sharper.

At a fine rise, a divergence angle picked at random gives a consecutive Fibonacci pair about one time in seven. Almost half the circle gives a pair with no name at all. The share falls monotonically as the pattern gets finer, which is the opposite of what a story about the geometry preferring Fibonacci numbers would need.

So the preference has to come from somewhere else, and the somewhere else is the thing this phase’s last group of essays is about: a plant does not pick a divergence at random at its final rise. It starts coarse, where the Fibonacci share is 67.5% rather than 14.7%, and it carries whatever it has downward as the rise falls. Whether that carrying actually happens — whether a growing pattern keeps the branch it started on — was an interpretation with nothing behind it until the rising essays put a rate into the model and measured it.

The census cannot answer that, and no amount of re-reading it will. A census is a statement about a fixed rise; continuity is a statement about a history; and the two are different objects however carefully the buckets are labelled.