Why a cone can be counted once

There is a real asymmetry in how the two most-photographed phyllotactic objects are described, and it survives contact with the specimens.

A pine cone is “5 and 8”. A pineapple is “8, 13 and 21”. Nobody feels the need to say at what height, and a count taken at the base and again halfway up generally agrees.

A sunflower is “34 and 55”, and it is not. The founding essays’s first real finding was that a single head gives 13/21, 21/34, 34/55 and 55/89 in four bands, so that every published count is a statement about an annulus and hardly any of them say which.

The obvious explanation is that a head is a harder object. It is not the explanation, and this essay is about what is.

The two rise laws, in element number #

Write each geometry’s rise as a function of the element’s number rather than of its position, and something falls out.

On an elongating cone, element $i$ sits at slant distance $z_0 + i\Delta$ on a surface of local circumference $2\pi z\sin\alpha$, so the rise is $\Delta / (2\pi z \sin\alpha)$. Substituting and dropping $z_0$:

$$h_i = \frac{1}{2\pi\, i\, \sin\alpha}$$

The internode step has cancelled.

On a Vogel disc, element $i$ sits at radius $c\sqrt i$ and the rise is $c^2/4\pi r^2$, so

$$h_i = \frac{1}{4\pi\, i}$$

The scale constant has cancelled too.

Both fall as one over the element number, and they differ by the single factor $2\sin\alpha$. In element number a Vogel disc is exactly a cone of flare one half. Not approximately, not in some limit: the same function of $i$, with the same constant, when $\sin\alpha = 1/2$.

So the ladder is the same ladder, the transitions happen at the same element numbers up to a constant, and there is nothing about a head that makes it intrinsically harder to count.

Where the elements are #

What differs is the map from element number to position, and it differs by one power.

A cone’s elements are laid along a length: $i \propto z$. A disc’s are spread over an area: $i \propto r^2$.

Take the outer fraction $f$ of an organ’s linear extent — the part anyone actually counts, since the tip of a cone and the middle of a head are both unreadable. On a cone that region holds the elements from $fN$ to $N$, a factor $1/f$ in element number. On a disc it holds the elements from $f^2 N$ to $N$, a factor $1/f^2$.

Since transitions are a factor of $\varphi^2$ apart in element number on both, the number of changes visible in that region is

$$\log_{\varphi^2}(1/f) \quad\text{on a cone}, \qquad 2\log_{\varphi^2}(1/f) \quad\text{on a disc}$$

Exactly twice as many, at every $f$. Ignore the inner half and a cone shows 0.72 of a transition and a disc 1.44; ignore the inner third and it is 1.14 against 2.28.

That is the whole asymmetry. It is not that a head has a worse pattern; it is that a head packs four times as many elements into the outer half of its radius as a cone packs into the outer half of its length, and every one of those extra elements is another step down the ladder.

What a triple count means #

The pineapple’s description is not a pair. It is three numbers — 8, 13 and 21 — and three numbers is a different phenomenon from two, with an exact position in the theory.

The forks are exact established that at particular lattices three families are equally short, that these are van Iterson’s branch points, and that they have a closed form: for families $m$ and $n$ with $q = m^2 + mn + n^2$, the rise is $\sqrt3/2q$ and the nearest-neighbour spacing is $1/\sqrt q$.

Three families being equally short is precisely the condition under which an eye traces three sets of spirals and finds all of them convincing. So an organ described by three consecutive Fibonacci numbers is an organ sitting near a fork, and the fork’s position is computable in element number.

For the 8/13/21 fork, $q = 337$ and the rise is $2.5698 \times 10^{-3}$. Converting:

• on a disc, that is element 31;
• on a cone of flare 0.35, node 177;
• on a cone of flare 0.2, node 310.

A pineapple has on the order of a hundred to two hundred fruitlets and a flare somewhere near a third. The fork sits inside that range, and it sits there for a range of flares wide enough that the agreement is a consistency rather than a coincidence to be impressed by.

On a sunflower the same fork is at element 31, which is inside the undifferentiated middle that Vogel’s model does not describe and that no photograph resolves. The head’s visible forks are the next ones up — 13/21/34 at element 81, 21/34/55 at element 212, 34/55/89 at element 556 — and a large head passes several of them across its face, which is why a careful description of a sunflower ends up naming four or five numbers rather than three.

The pineapple is describable in three numbers because it has one fork in view. The sunflower is not because it has three.

The countable region, and why it is what it is #

The argument above turns on “the outer fraction of an organ”, and that phrase is doing enough work to deserve a paragraph of its own.

Both organs have an inner region where counting is impossible, and the two reasons are different.

On a cone the reason is developmental. The tip is a cluster of small undifferentiated scales, and on a conifer cone the first several are sterile bracts rather than fertile scales at all. Beyond that there is a geometric reason as well: the model says the first three transitions of a cone at a flare near a third happen among its first twenty-five nodes, and twenty-five nodes is not a population one can count two families in. The unreadable region and the busy region are the same region, and they are the same region on the model as on the specimen.

On a disc the reason is different and more interesting. Vogel’s $\sqrt n$ puts infinite density at the centre, so the model’s middle is not merely hard to read — it is not a description of anything. A real head has a finite disc of primordia there that have not differentiated. And the geometric fact runs the same way: a quarter of a thousand-element head’s elements sit inside a quarter of its radius, so a large fraction of its ladder is compressed into a region nobody resolves.

The consequence is that both organs hide their early rungs, and the visible part of each is the tail. What differs is how much tail there is: a cone’s outer half is half its elements, a head’s outer half is three quarters. Everything in this essay is that sentence with the arithmetic attached.

The pineapple, counted properly #

The fork calculation gives more than a position, and the rest of it is checkable on a fruit.

At the 8/13/21 fork the lattice is equilateral: every element has six equidistant neighbours, which is what “three families equally short” means. The closed form gives the nearest-neighbour spacing as $1/\sqrt q$ with $q = 337$, so

$$s = \frac{1}{\sqrt{337}} = 0.0545$$

of a circumference. On a pineapple of circumference 40 cm that is a spacing of 2.2 cm between adjacent fruitlets — which is about right for a fruit, and is a number arrived at from a spiral count and a fork position with nothing measured.

The fork’s divergence is more striking and is the part worth stopping on. Every fork sits at a rational divergence with denominator $2q$, and for this one the closed form gives $257/674$ of a turn, which is 137.2700°. So an organ sitting exactly at the 8/13/21 fork is at a rational divergence, not at the golden angle — the golden angle is the limit the fork angles approach and is at none of them.

That is the same finding the forks essay records, arriving here attached to an object. A pineapple described as “8, 13 and 21” is, if the description is exact, a pineapple at 137.27° rather than at 137.51°. The difference is a quarter of a degree, which no measurement on a fruit will ever resolve, and that is the honest end of it: the theory distinguishes two angles that the specimen cannot.

One organ that does neither #

The third case is the one the site started the cylinder field with, and it is worth putting beside the other two now that the arithmetic makes the comparison exact.

A stem’s rise does not depend on the element number at all. There is no $1/i$; there is a constant. So a stem never reaches a fork, never passes a transition, and shows one pair from its base to its tip however many nodes it has. Counting up the stem measured exactly that — three disjoint bands, one answer — with the disc as the control that made it a measurement rather than an insensitivity.

Three organs, then, and one relation between them:

• a stem holds its pair for ever, because its rise is constant;
• a cone changes once or twice over a countable length, because its rise falls as one over element number and its elements are spread along a line;
• a head changes three or four times over a countable radius, because its rise falls the same way and its elements are spread over an area.

The first difference is a difference in the rise law. The second is not: it is a difference in geometry alone, with the rise law identical. That distinction is what the essay is for, and it is not visible until the rise is written as a function of element number rather than of position.

The prediction this makes about specimens #

Three statements follow that could be checked against real material by anyone with a ruler, and they are stated here as claims rather than as results, because this site has no dataset and says so.

A cone’s counted pair should be stable over most of its length and should change once, low down. With a flare near a third, the transitions sit at nodes 3.6, 9.5, 25, 65, 172 and 450. A cone of a hundred and fifty scales passes the fifth of those somewhere around its middle and nothing else. The change should be findable — count at the base, count at the tip, and the answers should differ by one rung, not two.

A wider cone should climb faster. The flare enters as a multiplier on every transition node number, so doubling it halves them. Two cones with the same number of scales and different taper should not be on the same rung.

A head’s counted pair should change on a geometric radial schedule with ratio $\varphi$. This is the one the site has already measured against its own model and cannot measure against a plant, and it is the one a published survey with stated counting radii would settle. That absence is the standing gap in this site’s wrong field: every frequency statement here is about the geometry rather than about plants, and says so.

The instrument the asymmetry hands over #

There is a use for the factor of two beyond explaining a description, and it is worth setting down because it is the sort of measurement that needs no equipment.

Suppose one has a specimen and wants to know its rise exponent — whether it elongates or fills, and how fast its circumference grows. The direct route is to measure the shape and watch the growth, which requires the plant to be alive and the observer to be patient.

The indirect route is to count. Find two places along the organ where the parastichy pair changes, and record their positions as fractions of the organ’s extent. The ratio of those two fractions is $\varphi^{2/p}$, so

$$p = \frac{2\ln\varphi}{\ln(z_2/z_1)}$$

and $p$ is the shape exponent multiplied by one if the organ elongates or by two if it fills. A cone that elongates gives $p=1$ and a ratio of 2.618; a head that fills gives $p=2$ and a ratio of 1.618; a dome that fills gives $p=1$ again and the same 2.618 as the cone, which is the degeneracy the previous essay is about.

Two counts, two positions, one division. The imprecision is honest and it is large — a transition is fuzzy, because near one three offsets are nearly equally short and the answer depends on where the counting band’s edges happen to fall — but the quantity being estimated is a developmental one, and it is being estimated from a finished object with no history attached.

This is the same shape of argument as recovering a growth factor from a drawn shell and as recovering a divergence from a spiral count. A rule that ran leaves a geometric signature; the signature is readable; and the reading is a measurement rather than a restatement, because the machinery that does it is never shown the rule.

What would refute this #

The argument has one load-bearing assumption and it is worth naming, because it is the assumption most likely to be wrong about a real organ.

It assumes the element-addition law is constant over the organ. A cone that elongates at a steady rate has $i \propto z$; a head that fills at a steady rate per unit area has $i \propto r^2$. Both are idealisations of a process that starts, runs, and stops.

If a cone’s internodes lengthen through its development — which they do — then $i$ is not proportional to $z$, and the map from element number to position is not the one used above. The correction is in the same direction on both organs and is smaller on the cone, so the factor of two is robust, but the individual transition positions are not.

The sharper version of that objection is that there is no time axis in any of this. Every surface here is a static lattice, and a plant is a process. Whether the pattern on a growing organ actually tracks the equilibrium lattice at each stage — or lags behind it, or overshoots — is not something a static calculation can answer, and it is where the rising-phyllotaxis essays go next.

Why this is worth having #

The observation this essay started from — that a pineapple can be counted once and a sunflower cannot — is the sort of thing that gets explained by adjective. Heads are complicated; cones are simple; sunflowers are famously intricate.

None of that is what is going on. The two organs run the same ladder at the same rate in element number, differing by a factor that vanishes for a cone of flare one half. What separates them is a single power in how they lay their elements out in space, and the consequence of that power is a factor of exactly two in how many changes fall inside a countable region.

That is the sort of explanation this site is for. It replaces a description with an arithmetic relation; the relation has a number in it; the number is 2, and it is 2 for a reason that could have come out otherwise.

And it turns the asymmetry into an instrument. If a cone and a head at the same divergence show a different number of changes over comparable extents, the ratio of those counts measures the ratio of the exponents in their rise laws — which is a measurement of how each organ adds material, made by counting spirals in two places on each of them.

How many rings a specimen would have to give up #

This page is about how much of an organ is countable. The work after it turned that into a requirement, by asking what a specimen would have to show before its exponent could be established to vary along it.

The answer is discouraging in a specific way. Two rings cannot show it at all — one gap determines one exponent exactly, with no residual and nothing left to disagree — which is invisibility in principle rather than difficulty. Three rings give two gaps that on an ogive differ by 0.32 per cent, far inside any measuring error. It is the fourth gap that carries the drift, and it needs five rings.

So the countable stretch this page measures is not only a limit on how much pattern is visible; it is a limit on how many models can be told apart using it. A fir cone offers about four gaps, an ogive’s drift is 8.2 per cent across them, and a three per cent ruler allows 8.5 — which puts the question at the edge of a single specimen and moves it to a population of a dozen.