A stem is a cylinder

Almost every picture of phyllotaxis is a sunflower seen from the front, and almost every claim about phyllotaxis is a claim about that picture. It is an unfortunate choice of example, for a reason this site has already had to work around at length: the spiral counts in a seed head change with radius. One head gives 21 and 34 near the middle, 34 and 55 further out, 55 and 89 at the rim, and a caption naming one pair has silently named an annulus without saying which.

That is not a defect of the sunflower. It is a consequence of the disc, and it goes away on the geometry that most phyllotaxis actually happens on.

Leaves on a stem, scales on a pine cone, florets on a teasel: these are arrangements on a cylinder or on something close enough to one, and the botanical literature has treated them that way since the nineteenth century. A cylinder has an axis, the pattern repeats along it, and the parastichy numbers are the same at the bottom of the stem as at the top. That is the whole reason the classical theory was built on cylinders and the popular account was built on sunflowers.

Two numbers, and no more than two #

Unroll the cylinder. Cut it along a vertical line, flatten it, and use the circumference as the unit of length so that going once around is going a distance of 1.

Then node number $k$ sits at

$$x = \{k\delta\}, \qquad y = k h$$

where $\{\cdot\}$ means the fractional part, $\delta$ is the divergence angle expressed as a fraction of a turn, and $h$ is the rise — how far up the stem the pattern advances between one node and the next, measured in circumferences.

That is the entire model. Two numbers, and everything a cylindrical phyllotactic pattern can be is somewhere in that two-dimensional space.

The rise deserves its own sentence because it is the parameter with no counterpart in the popular account. It is a ratio of two lengths — the internode distance and the circumference — so it falls when the internodes shorten and it falls when the stem thickens. A young shoot with long internodes on a thin stem has a large rise. A pine cone, whose scales are packed tightly onto something fat, has a very small one.

In the botanical literature the same quantity turns up as the plastochron ratio, defined slightly differently and measuring the same thing: how far the pattern moves between successive elements, relative to the size of the thing it is moving on.

What “one lattice” buys #

The disc has no equivalent of the rise, and that is exactly its problem.

In Vogel’s model the $n$th primordium sits at radius $c\sqrt{n}$, so between consecutive primordia the pattern advances $c^2/2r$ outwards on a circle whose circumference is $2\pi r$. The rise, expressed the same way, is

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

which is not a constant. It falls as the square of the radius, so a seed head is not one lattice at all — it is a continuous family of lattices, each annulus at a slightly smaller rise than the one inside it. That is the mechanism behind the changing counts, and it is worked out in its own essay.

A stem has a fixed rise. The lattice at the base of the stem and the lattice at the tip are the same lattice, so the counts are the same, so the question “how many parastichies does this plant have” has an answer that does not need a qualifying clause.

What the flattening does and does not change #

Unrolling a cylinder is an exact operation. Distances measured on the flattened strip, with the wrap taken into account, are distances on the surface, because a cylinder is a developable surface and has no intrinsic curvature to lose.

That is worth saying plainly because a great deal of intuition about this subject is built on projections that are not exact. A photograph of a sunflower is a genuine flattening of a genuinely flat thing. A photograph of a pine cone is not: the cone tapers, the scales at the base sit on a larger circumference than those at the tip, and the projection onto the page compresses the far side to nothing.

The one thing the flattened picture hides is the wrap, which is why the figures here draw the strip three times: once solid and once faintly on each side. A parastichy is a chain of nodes that runs off the right edge and continues from the left, and until the repeats are drawn the picture is of a lattice on a rectangle, which has different properties.

The nineteenth century got here first, and then it was forgotten #

The cylindrical formulation is not a modern convenience. Schimper and Braun set the divergence angle up as a fraction of a turn in the 1830s, the Bravais brothers wrote the spiral lattice down properly in 1837, and by 1907 Iterson had drawn the entire two-parameter plane with its regions labelled by parastichy pairs.

What happened next is instructive about how a subject decays into captions. The cylinder is the natural setting and the seed head is the memorable image, and over a century the memorable image displaced the natural setting almost completely. The result is a popular account built on the one geometry where the central quantity is not well defined, and a technical literature that mostly still uses cylinders and is mostly not read.

This site spent its foundation phase on the seed head, discovered the radius dependence the hard way, and only then went looking for the geometry where the problem does not arise. That is the wrong order and it is worth admitting: the awkwardness of the disc was a signal, and the signal had a hundred-year-old answer attached to it.

Opposed families, and why they come in pairs #

A spiral family on a stem is what it is on a disc: a chain of near-neighbours, labelled by the index offset that lands on one. Join every node to the node $m$ places earlier and, if $m$ is well chosen, the result is $m$ chains winding up the stem.

On a cylinder the chains have an unambiguous handedness. The offset $m$ moves the pattern $\{m\delta\}$ around, and whether the shorter way round is clockwise or anticlockwise is a fact about $m$ and $\delta$, not a matter of how the picture is oriented.

The two shortest offsets almost always wind opposite ways. This is the classical condition of an opposed parastichy pair, and it is not imposed here — the counter measures the handedness from the coordinates and reports it, which means it can report that a pattern fails the condition. That happens, and where it happens is interesting: right at a transition, when three families are competing, the two shortest can briefly wind the same way.

Whorls are the other case, and the counts say which #

Not every arrangement on a stem is a spiral one. Many plants put leaves on in whorls: two, three or four at the same level, with the next whorl rotated.

In this model a whorl is what happens when the divergence is a simple fraction of a turn. At exactly one third, node 0, node 3 and node 6 all sit at the same angular position, so the pattern falls onto three vertical rows — and the two shortest offsets share a factor of three.

That shared factor is the signature and it is checkable without looking at the picture. A pair of counts with a common divisor means the lattice is really some number of rows of something, and no single divergence angle can be recovered from it — the machinery refuses, which is the correct answer rather than a limitation.

What changes when the rise changes #

The rise is a knob, and turning it is the most instructive thing one can do to a cylindrical lattice.

At a large rise — long internodes, thin stem — successive nodes are far apart vertically, the pattern is loose, and the shortest hops are at small offsets: 1 and 2, or 2 and 3. As the rise falls, offsets 1 and 2 stop being short, because a hop of one node still costs at least one rise vertically while the higher offsets get to trade vertical distance for angular closeness.

So the pair climbs. The figure’s slider walks it: 2 and 3 at a rise of 0.19, 3 and 5, 5 and 8, 8 and 13 by the time the rise is down to a fortieth of the circumference. Each step is the sum of the two before it, and none of that was put into the arithmetic.

That climb is the Fibonacci ladder, it happens for a reason that has nothing to do with plants, and it is the subject of its own essay. What matters here is that it is a consequence of one parameter moving, and the parameter is one a growing plant moves for reasons of its own: internodes shorten, stems thicken, and a shoot that starts with leaves far apart ends with scales packed tight.

A cone is a stem with a changing circumference #

Between the cylinder and the disc sits the cone, and it is where most of the photogenic examples live. A pine cone, a pineapple, an artichoke: these taper, so the circumference changes along the axis, so the rise changes with it.

That makes a cone an intermediate case in the exact sense: it is a cylinder whose rise varies slowly, and its parastichy numbers change along the axis in the same way that a disc’s change with radius, but more slowly. A pineapple with 8 and 13 at the base and 13 and 21 at the top is not misbehaving; it is a lattice moving through a transition, and the transition is at a computable place.

The disc is the limit of the taper — a cone opened out flat, where the rise falls as fast as it possibly can.

Two consequences are worth having in advance. A cone’s counts change along the axis, so a photograph of a pine cone is subject to the same objection as a photograph of a sunflower: a count without a position on the axis has an unstated free variable. And the direction of the change is fixed by the taper — counts rise towards the narrow end, because a smaller circumference at the same internode spacing is a larger rise, and a larger rise is lower down the ladder.

Anyone who has counted a pineapple has met this without naming it. The 8 and 13 that are easy to see near the middle become harder to hold at either end, and the reason is that at one end the pattern has already moved to a different pair.

What the rise is, in a plant #

It is easy to treat the rise as a number in a model. It is worth spending a paragraph on what it corresponds to, because the correspondence is unusually direct and it is what makes the ladder a statement about development rather than about arithmetic.

The rise is the internode length divided by the circumference of the shoot apex. Both are real lengths on a real plant and both change during growth, almost always in the same direction: apices broaden as a plant matures and internodes shorten as elongation slows. So the rise falls, monotonically, over the life of a shoot — and a falling rise is precisely the motion that climbs the ladder.

This is why the phenomenon called rising phyllotaxis exists at all. A shoot that begins with a pair of 2 and 3 and ends with 8 and 13 has not changed its divergence angle; it has changed the only other parameter it has, in the direction that growth always changes it.

It also sets a limit on what any static model can claim. A finished pine cone records the rise it had when each scale was laid down, not the rise it has now, and the two are different because the cone grew after the scales were placed. Reading a developmental history off a mature specimen requires that correction and rarely gets it.

Why the counting is easier here #

Something small but load-bearing shows up in that curve, and it is worth stating because it changed the machinery.

The disc counter requires a parastichy number to be a local minimum of the hop curve, not merely a short hop. It has to: on a disc, offset 1 always gives a shortish hop and offset 2 nearly always does, so a counter that simply took the two shortest would answer “1 and 2” for every pattern ever made, which is true and useless.

On a cylinder the situation is different, and the difference is structural rather than a matter of tuning. A hop of one node is at least $h$ tall, so offset 1 is genuinely long whenever the rise is not tiny, and it disqualifies itself. The two shortest hops are the parastichy numbers, with nothing extra required.

The first version of the cylinder counter inherited the local-minimum rule anyway, and it broke. At a rise of 0.09 the two shortest offsets are 2 and 3, but 3 is not a local minimum — 2 is shorter than it — so the counter skipped past and reported 2 and 5. That pair has no cylindrical lattice consistent with it at that spacing, and the recovery refused, which is how it was found.

The same refusal, on the same site, for the third time. It is becoming the most productive piece of machinery here.

What a stem is good for #

The argument of these essays runs: the disc is the famous case and the awkward one; the cylinder is the tractable one; and the cylinder turns out to explain the disc rather than merely to contrast with it.

Concretely, four things follow from putting the pattern on a stem, and each gets its own essay.

The counting behaves. One pair, everywhere on the stem, from a counter shown nothing but coordinates.

Both parameters come back out of the points, to the last digit, which the disc’s recovery cannot manage — it returns an interval because a range of angles gives the same counts.

The ladder is exact. The rises at which the pair changes have a closed form, and consecutive ones stand in the ratio $1/\varphi^2$.

The disc is predicted. Feed $h(r) = c^2/4\pi r^2$ into the cylinder’s ladder and it says where a sunflower’s counts should change. Compared against a blind counter run on the disc itself, it agrees in fifteen bands of sixteen, and the one disagreement sits on a transition.

None of those is available from the front-on photograph, and all of them were available in 1907.