A cone has a rise that falls

Two essays on this site end with a sentence about cones and neither of them computes one.

A stem is a cylinder says that the parastichy pair on a stem is constant because the rise is constant, and adds that a cone is the case in between. A disc is a cylinder derives the rise of a seed head, $h(r) = c^2/4\pi r^2$, and observes in passing that on a cone the same reasoning gives one power rather than two. Both sentences are correct. Neither was checked, and on a site whose whole premise is that the confident sentences about this subject are the ones nobody checked, that is not a comfortable position to leave them in.

This essay builds the cone.

What a cone is, to a lattice #

A cone of half-angle $\alpha$ has one number in it, and choosing which number to call the parameter decides how much of the arithmetic is bookkeeping.

The choice used throughout here is the flare, $\sin\alpha$. A point at slant distance $z$ from the apex sits at $z\sin\alpha$ from the axis, so its local circumference is $2\pi z \sin\alpha$; and when the cone is unrolled the sector it becomes spans $2\pi\sin\alpha$ radians. So the flare is at once the fraction of a full turn the unrolled surface occupies and the constant relating slant distance to circumference. A flare approaching 1 is a flat disc; a flare approaching 0 is a needle. A stem is not in this family at all, because a cylinder has no apex.

Nodes are placed the way they are placed on a stem, with the two parameters a stem has:

$$\theta_i = i\delta \pmod{2\pi}, \qquad z_i = z_0 + i\,\Delta$$

The azimuth advances by the divergence angle and the slant distance by a constant step. That is a shoot that elongates at a steady rate on a surface that widens at a steady rate — which is what a young conifer cone, a pineapple or a spike of Plantago approximately is.

Unrolled, node $i$ lands in the plane at polar coordinates $(z_i,\ \theta_i\sin\alpha)$. Distances between nodes are measured there, and that is not an approximation: a cone is developable, so the geometry of its surface is the geometry of that sector. The only care needed is the seam, since the sector’s two edges are the same line on the cone, so angular differences are taken modulo the sector angle and to the nearer side. A parastichy that leaves one edge comes back in at the other, exactly as on the unrolled cylinder.

The rise, which is the whole of it #

The cylinder’s second parameter is the rise: the advance along the axis between one node and the next, measured in local circumferences. It is the only dimensionless thing a cylinder offers and it is what the ladder is indexed by.

A cone has one too, and it is not constant. The advance along the meridian is $\Delta$ and the local circumference is $2\pi z\sin\alpha$, so

$$h(z) = \frac{\Delta}{2\pi z \sin\alpha}$$

Three lines and the essay is decided. The rise falls as one over the distance from the apex. Everything the site knows about what happens to a lattice when its rise falls — the ladder, the transitions, the fork tree — now applies to a cone, indexed by position along its axis.

There is a small surprise sitting in that formula, and it is worth taking now rather than later. Substitute $z_i = z_0 + i\Delta$ and let $z_0$ be small compared with the length:

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

The step has cancelled. The rise at the $i$-th node of an elongating cone does not depend on how far apart the internodes are; it depends only on the node’s number and on the flare. Stretch a cone lengthwise and every rise is unchanged, because stretching moves the node outward and widens the circumference underneath it in the same proportion. So the transitions on a cone sit at node numbers, not at lengths — which is a considerably more useful statement for anyone holding one, since counting scales is easy and measuring a cone’s slant distance from a vanished apex is not.

What the approximation costs #

The sentence “a cone is a cylinder whose rise falls” is an approximation, and the honest thing is to measure it rather than to wave at it.

The rise formulation treats a band of the cone at slant $z$ as a cylindrical patch of circumference $2\pi z\sin\alpha$, so that two nodes separated by $\Delta\theta$ around and $\Delta z$ along are

$$\sqrt{(C\,\Delta\theta/2\pi)^2 + (\Delta z)^2}$$

apart. That is exact for a cylinder. On a cone the exact answer is the unrolled straight line, $\sqrt{z_a^2 + z_b^2 - 2z_az_b\cos\Delta\phi}$, and the two disagree because the local patch has parallel sides and the cone’s does not.

Over every parastichy hop of every offset in a four-hundred-node cone at a flare of 0.35, the worst disagreement between them is half a per cent, and it occurs at the shortest offsets, where the two nodes are furthest apart relative to their distance from the apex.

Half a per cent is far below anything that matters here, and it matters that the reason is structural rather than lucky. What the ladder needs is not distances but an ordering — which offset gives the shortest hop — and an ordering survives a good deal of distortion before it changes. That robustness is the same one that makes parastichy counts usable on real material at all, and it is why the counts are hard to pin down near a transition, where the ordering is nearly a tie.

The counter, shown a cone and nothing else #

Every count on this site is taken by machinery that has not been told the divergence angle, and the cone gets the same treatment. coneCount() receives an array of unrolled coordinates and the flare of the surface they live on. It does not receive the divergence, the slant step, or the fact that Fibonacci numbers exist.

Its method is the one the cylinder counter uses, with the cone’s metric substituted: for each candidate index offset $m$, take the median hop length from node $i$ to node $i+m$ over the band, and keep the two offsets whose median is shortest. No local-minimum requirement, for the reason the cylinder counter records at length — on a stem or a cone, offset 1 climbs a whole internode and disqualifies itself, so the failure mode the disc counter has to guard against cannot arise here, and imposing a local minimum actively breaks real counts.

Point it at a band near the apex of a cone at a flare of 0.35 and it returns 5 and 8. Point it at a band near the base of the same cone and it returns 13 and 21. This never happens on a stem. Counting up the stem is the essay whose whole result is that three disjoint bands up a cylinder give one answer, and the control that made that a measurement rather than an insensitivity was running the same counter on a disc, where it gives three. A cone is the third case, and it behaves like the disc.

The test, band by band #

The interesting question is not whether a cone’s counts change — the rise falls, so of course they do — but whether they change where the cylinder says they should.

That is a genuine prediction, because the two calculations share nothing but the divergence angle. The ladder comes from comparing $\sqrt{\langle j\delta\rangle^2 + (jh)^2}$ across $j$ on a periodic strip; the cone’s counts come from median distances between unrolled points on a widening sector. There is no wrap in the second and no apex in the first.

A nine-hundred-node cone is cut into geometric bands and each is handed to the blind counter. Beside each band’s answer sits the pair the cylinder’s ladder gives at $h(z)$ for that band’s mid-slant, with the flare and the step being the cone’s own description and nothing fitted.

Every usable band agrees. The answers climb 5/8 → 8/13 → 13/21 → 21/34 as the bands move outward, and each is the pair the ladder puts at that rise.

The bands are geometric rather than equal, and that is forced rather than chosen. Equal bands near the base all sit on one rung and crowd every transition into the first band, because the structure is geometric in the slant distance. Geometric bands sample it evenly.

What cannot be measured, and why #

Five bands survive out of ten, and the five that do not are the five nearest the apex. That is not a defect in the sampling; it is a real property of cones and it is worth stating plainly.

Transitions on a cone sit at node numbers $i_k = 1/(2\pi h_k \sin\alpha)$. At a flare of 0.35 the ladder’s rises put them at nodes 3.6, 9.5, 25.0, 65.3, 171.8 and 449.5. The first three transitions of this cone happen among its first twenty-five nodes, where there are not enough nodes in any band to count parastichies in. A cone’s early history is not countable, on the cone or on the model of it, and no amount of care with the bands changes that.

Real cones have the same problem in a worse form. The tip of a conifer cone is a cluster of small undifferentiated scales, and the first several are usually bracts rather than fertile scales at all. So the region where the model says the interesting changes happen is the region a real specimen also refuses to show.

What the cone does offer, and the disc does not, is the other end. Because the transitions sit at node numbers a factor of $\varphi^2 \approx 2.618$ apart, a cone with a few hundred scales has passed only a handful of them — and most of its length sits comfortably inside one rung. That is why a pineapple can be described as “8 and 13” without immediately being wrong, in a way that a sunflower cannot.

Where the flare goes #

The flare enters in exactly one place — as a multiplier on the node number at which each transition falls — and that gives it a clean interpretation.

A narrow cone has its transitions late. At a flare of 0.12 they sit at nodes 10.6, 27.8, 72.8, 190.5, 501 and 1311; at a flare of 0.35 they are at 3.6, 9.5, 25.0, 65.3, 171.8 and 449.5. The narrow cone spends nearly three times as many nodes on each rung, because a narrow cone’s circumference grows slowly and its rise therefore falls slowly in node number.

This has a consequence for what a specimen looks like. A slender spike shows a low pair over most of its length; a broad cone climbs faster and shows higher counts sooner. Both are the same lattice family at the same divergence, sampled at different rates.

It also means the flare is recoverable from a count and a node number, which is the sort of thing this site likes: if a cone’s counts change between scale 60 and scale 80, then $h$ passed through a transition rise there, and $\sin\alpha \approx 1/(2\pi \cdot 70 \cdot h_k)$. That is a measurement of the cone’s shape made by counting rather than by measuring, and it does not require finding the apex.

What the model is not being asked #

It is worth being explicit about what the divergence angle is doing here, because a reader arriving from the popular literature will expect it to be the answer to something.

It is an input. The cone is built at 137.508° because that is the angle the site’s other lattices are built at, and every result above would hold at 99.5° or at 151.1° with a different ladder underneath it. Nothing on this page bears on why plants are at one angle rather than another; that question belongs to the dynamical model and to the branch tree, both of which produce the angle rather than assume it.

What the cone contributes to that argument is negative and useful. If the parastichy pair on an organ is set by the rise, and the rise is set by the organ’s shape and the rate it adds material, then the counts on a plant are not a direct readout of the divergence angle at all. Two plants at the same divergence can be counted differently because one is a spike and the other a head; two plants at different divergences can be counted the same because their rises put them on the same rung. Every inference from “this plant has 8 and 13” to “this plant is at the golden angle” runs through the ladder, and the ladder needs the shape.

That is the sense in which this essay is about counting rather than about cones. The instrument is the same one the site has used since the founding essays; what changes is the surface it is pointed at, and what is learned is how much the surface contributes to the reading.

What a real cone does that this one does not #

The model here is a right circular cone with a constant slant step, and real organs depart from it in three ways that are worth naming.

A cone is not a cone. A conifer cone is closer to an ogive — pointed at the tip, nearly cylindrical through the middle, tapering again at the base. The local circumference therefore does not grow linearly with slant distance, and where it stops growing the rise stops falling and the counts stop climbing. That is not a failure of the argument; it is the argument applied to a different $C(z)$, and the essay after next does exactly that for a general surface of revolution.

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

The step is not constant. Internodes lengthen and shorten through a cone’s development, so the slant advance is a function of time as much as of position. That objection is the largest of the three and it is the subject of the rising-phyllotaxis essays, which put a rate into the model and ask whether the pattern keeps up with it.

None of the three moves the counts, and the reason is the one already given: what is being predicted is an ordering rather than a distance.

What this adds #

Three things, in increasing order of how much they change.

The smallest is that the sentence two essays ended with is now computed rather than asserted, and it came out right. That is the least interesting possible outcome and it is still worth having on a site whose subject is claims that were never tested.

The middle one is that a cone is a genuine third case rather than a rhetorical one. A stem does not move along the ladder, a disc moves along it at one rate, a cone at another — and the same instrument, pointed at all three, gives one answer, three answers and four answers respectively. Any account of phyllotaxis that treats “the parastichy pair” as a property of a plant rather than of a place on a plant has to explain all three.

The largest is the one the next essay is about. The rate at which a geometry moves along the ladder is a number, it differs between the disc and the cone by exactly one factor, and that factor is $\varphi$.

One qualification belongs with that, and it is about the organs rather than the arithmetic. A cone in this sense is a surface with a single exponent, and no conifer cone has one: its meridian is an arc, so the exponent runs from two near the tip towards nothing at the shoulder. The factor of $\varphi$ is therefore a statement about two idealised surfaces, and what a real cone supplies is a local exponent that drifts by about eight per cent across the rings a counter can follow.