A disc is a cylinder

The strongest thing a figure on this site can do is arrive at a number twice, by arithmetic that shares nothing. This essay is the clearest instance of it the site has.

On one side is a lattice on a cylinder: a periodic strip, two parameters, and a ladder of parastichy pairs as one of them falls. On the other is Vogel’s disc: the $n$-th primordium at angle $n\delta$ and radius $c\sqrt n$, which is where the seed-head figures come from and which has no rise in it anywhere.

The two share the divergence angle and nothing else. No constant is fitted between them. And the cylinder’s ladder predicts, to a few per cent, the radii at which a blind counter run on the disc changes its answer.

The rise of a seed head #

The derivation is three lines and it is the whole essay.

In Vogel’s model primordium $i$ sits at radius $r = c\sqrt i$. Differentiate: between one primordium and the next, the radius advances by

$$\frac{dr}{di} = \frac{c}{2\sqrt i} = \frac{c^2}{2r}$$

The local circumference is $2\pi r$. The rise — the vertical advance per element, measured in circumferences, which is the cylinder’s second parameter — is therefore

$$h(r) = \frac{1}{2\pi r} \cdot \frac{c^2}{2r} = \frac{c^2}{4\pi r^2}$$

That is it. A disc has a rise, it is not constant, and it falls as the square of the radius.

What that means about seed heads #

Three consequences follow immediately, and together they replace an observation with an explanation.

A disc is not one lattice. Each annulus is a cylindrical lattice at its own rise, slightly smaller than the one inside it. The head is a continuous family, indexed by radius.

The counts must change. Which is what the founding essays found by counting, without being able to say why. A falling rise climbs the ladder, so the parastichy pair at the rim is higher than the pair near the centre. This is not a special property of seed heads; it is what any cylinder does when its rise falls, and the disc is simply a geometry in which the rise falls very fast.

The changes are at computable radii. Inverting the relation gives

$$r = \frac{c}{2\sqrt{\pi h}}$$

so each transition rise on the cylinder’s ladder maps to a transition radius on the disc, with $c$ the only quantity involved and $c$ being the disc’s own scale rather than a fitted parameter.

The foundation essay The counts change with radius established the phenomenon by counting. It did not have this, and could only say that the transitions happen. Now they can be placed.

The prediction, and the test #

The cylinder’s transitions at the golden divergence sit at rises of 0.1246, 0.0476, 0.0182, 0.00695, 0.00266 and so on downwards, each 0.382 of the one before.

Convert with $r = c/2\sqrt{\pi h}$, at $c = 1$: transition radii of 0.80, 1.29, 2.09, 3.38, 5.48, and continuing upward by a factor of $\varphi$ each time — 8.87, 14.4, 23.2, 37.6.

Now the test. A disc of 3,000 points at the same divergence is cut into sixteen radial bands, and each is handed to the disc’s own parastichy counter — the one that receives coordinates and a radius window and nothing else. Its answers are compared with the pair the cylinder says should hold at the rise corresponding to that band’s mid-radius.

Fifteen bands of sixteen agree. The pairs run 13/21, 21/34, 34/55, 55/89, 89/144 as the radius grows, and the predicted and measured sequences match band for band.

How much of that agreement survives on a smaller head is a separate question, and it costs nothing to ask. The prediction is the same curve whatever the head holds, because it is computed from the divergence and the radius and knows nothing about how many points there are; what the size decides is how far out a blind counter can still find chains, and therefore how many of the predicted transitions the test gets to reach at all. So the same test is made at six head sizes, from three thousand points down to nine hundred, and the six differ only in how many rungs of the ladder they are able to put a dot on.

The one that disagrees #

The disagreement is at a mid-radius of 37.1, where the counter returns 89/144 and the cylinder says 55/89.

The predicted transition between those two pairs is at 37.6. The band that disagrees straddles it.

This is the least surprising possible failure and it is worth not smoothing over, because a prediction that agreed in sixteen bands of sixteen would be less informative. Right at a transition three offsets have nearly equal hops; the counter’s answer there is decided by where the band edges fell and how many points landed inside them; and the model’s answer is decided by whether the mid-radius is a hair above or below a threshold. Both are correct and they are answering a question that has no sharp answer at that radius.

Any measurement that never disagreed with its prediction anywhere would be a measurement with no resolution.

What c is, and why it is not a knob #

The conversion carries one symbol from the disc into the cylinder’s world, and it is worth being clear that it is not a free parameter dressed up as a constant.

In Vogel’s model, $c$ is the only scale there is: the $n$-th primordium sits at $c\sqrt n$, so $c$ fixes how much area each element gets. Since the area of the disc out to primordium $n$ is $\pi c^2 n$, the area per element is exactly $\pi c^2$ — which makes $c$ a statement about seed size relative to head size and nothing else.

It is therefore measurable on a head without reference to any of this. Count the elements inside a radius, divide the area by the count, and $c = \sqrt{A/\pi n}$. The disc recovery in the round-trip essay already does exactly that, estimating $c$ from the density in the counted band rather than taking it from the model, precisely so that the recovery is not handed its own answer.

So the prediction here uses a quantity that comes out of the point positions. If it had to be fitted — if the transition radii could be moved to match the measurements by adjusting $c$ — the agreement would mean very little. It cannot: change $c$ and every predicted radius moves together, by the same factor, so the spacing of the transitions is untouched and only their absolute position shifts. The spacing is a factor of $\varphi$ per rung and it is not adjustable at all.

Running it backwards #

If a rise predicts a pair, a pair constrains a rise, and that turns the relation into a measuring instrument.

Suppose one has a photograph of a head, counts 34 and 55 at a stated fraction of the radius, and wants the head’s scale. The pair pins the rise to the interval between two transitions — for 34 and 55 that is 0.000149 to 0.000390 — and the radius at which it was counted then gives $c$ to within a factor of $\sqrt{0.000390/0.000149} = 1.6$.

That is not a precise measurement, and the imprecision is honest: a pair of counts is a coarse observation, and a range of rises produces it. But it is an independent one. The usual way to get $c$ from a photograph is to count elements and measure area, which requires resolving every element; this way requires only tracing two families of spirals, which is what a person can do on a bad photograph of a distant flower.

Two routes to a scale, one of them cheap and coarse and the other expensive and precise, disagreeing by less than the coarse one’s stated range. That is the same arrangement as everything else in this field, run in the direction that is actually useful for measuring things.

Why this is not circular #

It is worth being precise about what would make this trivial, because the arrangement invites suspicion.

It would be circular if the disc counter used the cylinder’s arithmetic. It does not: the disc counter measures hop lengths between points in an annulus, normalises by the local spacing, and takes local minima of the resulting curve. There is no wrap, no rise, and no lattice vector anywhere in it.

It would be circular if $c$ were fitted. It is not: $c$ is the disc’s own scale factor, fixed when the point set is built, and the conversion uses it as it stands.

It would be circular if the two models shared a derivation. They do not. The cylinder’s ladder comes from comparing $\sqrt{\langle j\delta\rangle^2 + (jh)^2}$ across $j$. The disc’s counts come from medians of Euclidean distances between points at $c\sqrt i$ and $c\sqrt{i+m}$. The $\sqrt n$ is in one and not the other; the wrap is in the other and not the one.

What they share is the divergence angle, which both are given, and which by itself determines neither answer.

What the agreement buys #

Two things, one about the models and one about the subject.

About the models: the cylinder is not a separate topic that happens to sit next to the disc in this collection. It is the general case, and the disc is a particular trajectory through it — the trajectory $h \propto r^{-2}$. A cone is a slower trajectory. A stem is a stationary point. Everything the site has said about seed heads is a statement about one path through the two-parameter plane, which is why it kept needing qualifications.

About the subject: this is the cleanest available answer to why sunflower counts are Fibonacci numbers at every radius rather than only at one. The head does not settle on 34 and 55; it passes through 13/21, 21/34, 34/55 and 89/144 on the way out, and it does so because the rise falls monotonically and the ladder has rungs at the convergent denominators. The famous pair is whichever rung the counting band happened to land on.

How fast a head climbs #

The geometric spacing of the transitions has a consequence specific to discs, and it is a number.

Consecutive transitions are a factor of $\varphi^2$ apart in rise, and $h \propto r^{-2}$, so they are a factor of $\varphi \approx 1.618$ apart in radius. Every time a head’s radius grows by 62%, its counts advance one rung.

That is a slow climb and it explains something about how the pattern reads. The outer half of a large head spans a factor of two in radius, which is one rung and a bit — so most of the visible surface of a sunflower shows a single pair, with the changes squeezed into the crowded middle where nobody counts. Hence the impression that a head has a pair, and hence the surprise when someone counts carefully at two radii.

It also puts a bound on how many transitions a head can show. A countable region spanning a factor of four in radius holds about three, which is what the three-regime figure finds directly.

The cone in between #

A cone is the intermediate case and the relation makes it precise.

On a cone the circumference grows linearly with distance from the apex, so with fixed internode spacing the rise falls as $1/z$ rather than as $1/r^2$ — one power rather than two. Transitions are therefore a factor of $\varphi^2 \approx 2.618$ apart along the axis rather than a factor of $\varphi$, and a cone of ordinary proportions shows fewer of them than a disc of comparable extent.

Which matches what one sees. A pineapple holds 8 and 13 over most of its length; a sunflower changes pairs two or three times across its face; and a stem holds one pair for ever. The two objects are the same lattice family sampled along two different paths, and the difference in how often they change is the difference between one power and two.

What a real head does that neither model does #

The agreement is between two models. Real heads depart from both, in ways that are known and that the prediction should be read against.

The centre is wrong. Vogel’s $\sqrt n$ gives infinite density at $r = 0$, and a real head has a finite disc of tiny undifferentiated primordia there. Every count taken inside about a tenth of the radius is a count of something the model does not describe, which is why the bands in the test start at 0.10 of the radius rather than at zero.

The rim is wrong. Real seeds are packed against a boundary, are often flattened against it, and the outermost ring is frequently incomplete. The last band in the test is the one most likely to be measuring an artefact of the model’s edge rather than a lattice.

Seeds are not points. They have size, and they compete for space, so a real head is closer to a packing than to a lattice of centres. The effect is small where the pattern is regular and is not small where it is not.

None of these disturbs the prediction, and there is a reason: the prediction is about which offsets give the shortest hops, which is an ordering rather than a distance, and an ordering survives a good deal of distortion before it changes. That robustness is why parastichy counts are a usable measurement on real material at all, and it is the same reason the counts are hard to make precise near a transition, where the ordering is nearly a tie.

What this does not settle #

The prediction says where the counts change. It does not say why the divergence is 137.5°, and nothing in this essay bears on that question — the angle is an input to both models.

It also does not say anything about mechanism, and what a mechanism would have to show is a longer list than either model touches. Both models describe form. A plant that produces a pattern matching Vogel’s disc is not thereby shown to compute $c\sqrt n$, and this site is careful about that everywhere because the same patterns come out of magnetised droplets with no biology in them.

What has been established is narrower and more useful than either: that two descriptions of phyllotactic form, built from different arithmetic, are descriptions of the same object. When a model’s prediction lands on a measurement made by machinery that never saw it, the number is not an artefact of either — and on a subject where almost every quoted figure is an artefact of how it was quoted, that is worth having.

One more thing the derivation gives away #

The relation $h \propto r^{-2}$ has a consequence that is easy to miss and that reframes a piece of the founding essays.

The rise falls without limit as the radius grows. There is no smallest rise, so there is no highest rung, so a sufficiently large head has arbitrarily high parastichy numbers — 233 and 377, and beyond, with no change in the model and no new physics. The Fibonacci numbers people quote are the ones that fit on a flower, not the ones the model favours.

The other direction is more interesting. As $r \to 0$ the rise grows without bound, and above a rise of about 0.125 the dominant pair is 1 and 2 — the lowest rung there is. So the very middle of a head is, in the model’s terms, a distichous pattern: two families, one of one spiral and one of two, which is to say almost no pattern at all.

That is not visible in a real flower, because the middle of a real flower is the undifferentiated disc the model does not describe. But it is a statement the model makes, it follows from arithmetic already on the page, and it is the sort of thing worth recording when a derivation turns out to say more than it was asked.

It also has a check attached, which is what stops it being a curiosity. If the disc really is a cylinder with $h \propto r^{-2}$, then the rises at which its counts change are the cylinder’s own transition rises — a discrete set with a closed form — mapped through that relation into radii. So the model predicts where on a head the counts should change, not merely that they should, and the prediction is a list of numbers rather than a trend. Run a blind counter outward across a head and compare: agreement in fifteen bands of sixteen, with the one disagreement sitting on a transition.