Counting up the stem

The claim that a stem’s parastichy numbers are constant along its axis is easy to state and easy to believe, which makes it exactly the kind of claim this site does not accept without a measurement.

There is a specific way it could be wrong, too, and it is not far-fetched. The counting is done on a finite band of nodes, and a band near the bottom of a stem has different edge effects from a band near the top. If the counter’s answer depended on how many nodes it happened to see, or on where the band’s edges fell relative to the lattice, then the constancy would be an artefact of asking three times in a favourable way.

So the check is arranged to be able to fail: three disjoint bands, the same machinery, the same claim run on a disc as a control.

What the counter is given #

The arrangement is the same one the disc counter uses, and the separation is the point of it.

cylinder() builds a lattice from a divergence angle and a rise. cylCount() is handed an array of coordinates, a circumference, and a range of heights. It receives neither parameter. It does not know that Fibonacci numbers are the expected answer and it would return 47 and 76 as readily as 3 and 5.

The method: for each candidate index offset $m$, measure the distance from every node in the band to the node $m$ places earlier — taking the shorter way round the cylinder — and take the median. The offsets with the shortest medians are the parastichy numbers.

Two details in that sentence are doing work. The shorter way round is what makes the flattened strip a cylinder rather than a rectangle; without it a family that wraps would be measured as though it crossed the whole stem. And the median rather than the mean is what keeps a single misbehaving node at the edge of the band from swamping the statistic.

The rule that had to be dropped #

The cylinder counter began as a copy of the disc counter, and one line of it was wrong here for a reason worth setting out, because it is a case where a correct piece of machinery becomes incorrect when the geometry changes under it.

The disc counter does not take the shortest hops. It takes the shortest hops that are local minima of the hop-against-offset curve, and it needs that qualification badly. On a disc, offset 1 always gives a shortish hop, because consecutive primordia are genuinely close together near the centre. Offset 2 usually does too. A counter that took the two shortest without qualification would answer “1 and 2” for every pattern ever built, which is true and carries no information.

On a cylinder, offset 1 disqualifies itself. A hop of one node is at least $h$ tall no matter what the divergence is, so at any rise that is not vanishing, the one-step hop is long. The two shortest hops are the parastichy pair, with nothing extra needed.

Keeping the local-minimum rule anyway did active damage. At a rise of 0.09 the two shortest offsets are 2 and 3 — clearly and by a good margin — but 3 is not a local minimum of the curve, because 2 is shorter than it and 4 is longer. The rule skipped straight past 3 to the next dip, which is 5, and reported “2 and 5”.

How the wrong answer was caught #

Nothing in the picture showed it. A lattice at a rise of 0.09 does have five parastichies of one family, and drawing five spirals over a pattern with five spirals looks entirely correct.

What caught it was the recovery: given a pair of counts and the two hop lengths that go with them, solve for the divergence and the rise. Fed 2 and 5 at those lengths, the solver found no cylindrical lattice at all and refused.

The refusal was right. There is no stem with two families of 2 and 5 at those two lengths, because the lengths were measured from a lattice whose real pair is 2 and 3. The counting error was not detectable from the counting; it was detectable from the fact that the counts were inconsistent with the geometry they claimed to describe.

This is the third time on this site that a counting bug has been found by a downstream refusal rather than by a check on the count itself — the same thing happened when the disc counter returned the two smallest offsets instead of the two shortest. The pattern is consistent enough to state as a working rule: a measurement is worth much more when something downstream can reject it, and a pipeline whose stages only ever agree with each other has not been tested.

Handedness, measured rather than assumed #

A parastichy family on a cylinder winds one way or the other, and the classical requirement is that the two families of a dominant pair are opposed — one clockwise, one anticlockwise.

The counter measures this. For each offset it records the mean signed displacement around the stem, so the sign says which way the family leans, and the report says whether the pair is opposed rather than assuming it.

That is a small change with a real consequence: the machinery can say no. And it does, in one specific place. Right at a transition, when three families have nearly equal hop lengths, the two shortest can briefly be the two that wind the same way. A pattern in that state is genuinely ambiguous — it is between two arrangements — and a counter that had “opposed” built into it as a filter would have quietly reported the second-best pair instead and given no indication that anything was unusual.

The measurement, band by band #

With that settled, the actual result.

A stem of 240 nodes at the golden divergence and a rise of 0.05 is cut into three equal bands by height, and each band is counted separately. All three return 3 and 5. The nodes in the bands are disjoint, the band edges fall at unrelated places in the lattice, and the counts do not move.

The same counter, run on a disc of 1,600 points at the same divergence, is given three annuli — around a quarter, a half and nine tenths of the radius. It returns 21 and 34, then 34 and 55, then 55 and 89: three different pairs, one per band.

Both halves come from machinery that has never seen a divergence angle. The stem’s constancy is therefore a measurement of the stem, not an artefact of asking gently, and the disc’s variation is a measurement of the disc rather than noise in the counter.

Reading the hop curve #

The curve of median hop against offset is the counter’s raw evidence, and it repays being read rather than summarised, because everything the count can and cannot resolve is visible in its shape.

At any rise the curve begins high. Offset 1 costs a full rise vertically and $\{\delta\}$ around, and neither is small. It then falls to the first dip, rises, falls to a second, and so on — the dips at the offsets that trade well, the peaks at the offsets that land the pattern nowhere useful.

Three features matter.

The depth of the two lowest dips relative to the rest is how confident the count is. When the third-shortest hop is twice the second-shortest, the pair is unambiguous. When it is within a few per cent, the pattern is at or near a transition and the pair is genuinely uncertain — not badly measured, but uncertain in the object.

The gap between the two lowest dips is the handedness structure. An opposed pair sits at two offsets whose angular displacements have opposite signs, and the sum of those two offsets is the next dip along, which is the arithmetic the ladder is made of.

The floor of the curve is the nearest-neighbour distance, and it is a number the recovery needs. It is not a free parameter of the drawing; it is measured from the same coordinates as everything else.

The counter has to be able to be wrong #

An assertion that has never rejected anything proves nothing, and on a subject this loaded with expected answers the point is sharper than usual: a spiral counter that always returned Fibonacci numbers would confirm every claim in the popular literature and would be worthless.

So the gate feeds the machinery three things it must refuse.

A band with too few nodes. Twenty is the floor and the reason is in the previous section: below it the median hop for a high offset rests on a handful of pairs.

A non-coprime pair. Two counts sharing a factor describe a whorled stem — rows rather than spirals — and no single divergence angle can be recovered from them. The solver refuses rather than returning the nearest thing it can find, which is the honest answer and not a limitation.

A rise of zero. With no rise every node sits on one ring, the lattice degenerates, and there is nothing to count. The constructor refuses at the point of building rather than producing a degenerate object for something downstream to trip over.

The third is the newest and it earned its place immediately: an early sweep of the rise ran its lower limit to zero and produced a stack of identical rings that the counter dutifully described.

Why constancy is the useful property #

It is tempting to treat this as a tidiness argument: the cylinder gives one number and the disc gives three, so the cylinder is nicer. That undersells it.

The real consequence is that on a cylinder, the parastichy pair is a property of the object. It can be compared between two plants, tabulated across a species, and treated as data. On a disc it is a property of the object and the annulus, and since the annulus is almost never reported, most published disc counts cannot be compared with each other at all.

The botanical literature that works with cylinders has been able to build a comparative record for a century for exactly this reason. The popular literature, which works with sunflower photographs, has produced a set of numbers that look like data and are not.

A second consequence is that the pair on a cylinder is enough to reconstruct the lattice. Two counts and two hop lengths determine the divergence and the rise exactly, which the disc’s counts do not — there the recovery returns an interval, because a whole range of angles makes the same two offsets shortest at a given radius.

What a band needs to contain #

One practical limit, stated because it constrains what can be measured on a real plant.

The counter refuses a band with fewer than about twenty nodes, and the refusal is not conservatism. With too few nodes the median hop for a high offset is computed from a handful of pairs and the curve becomes lumpy enough that the ordering of the two shortest offsets is decided by which nodes happened to fall inside the band.

For a pair of 3 and 5 that limit is easy to satisfy. For a pair of 21 and 34 it is not: an offset of 34 needs at least 34 nodes above the bottom of the band before it contributes anything at all, so a band that resolves such a pair must be several times that. This is the cylindrical version of a constraint the disc has too, and on both geometries it means the same thing — a high count needs a large sample, and a high count taken from a small one is a guess.

It also explains a practical fact about counting real plants: parastichies are easy to count on a pine cone with a hundred scales and hard on a shoot with twelve leaves, and the difficulty is not about eyesight.

What this changes about the disc essays #

The disc essays on this site are careful, and one of the things they are careful about now looks different.

The counts change with radius was written as a finding about seed heads: a structural feature of every spiral lattice on a disc, visible in every sunflower, and absent from almost every account of them. That stands. What it did not say — because the cylinder had not been built — is why the disc has this property and the stem does not.

The answer is that the disc has no constant rise. In Vogel’s model the pattern’s advance per node, measured against the local circumference, falls as $1/r^2$, so each annulus is a slightly different lattice from the one inside it. The changing counts are not a property of discs as such; they are a property of a rise that changes, and a cone shows the same behaviour more slowly.

That reframing has a payoff, and it is the strongest result in this field: because the disc’s rise is a known function of radius, the cylinder’s ladder predicts where a sunflower’s counts change with nothing fitted.

What the counting is for #

Everything downstream in these essays depends on this being trustworthy, so it is worth listing what has been established and what has not.

Established. The pair is constant up a stem, measured on disjoint bands. The measurement is made without access to the parameters. The counter can be shown to give different answers on a geometry where the answer genuinely differs. The pair’s handedness is measured rather than assumed, and the machinery can report a pattern that fails the classical opposed condition.

Not established, and not claimed. That any of this describes how a plant places its leaves. A lattice that matches a shoot is a description of the shoot’s form, and reproducing a pattern is not explaining it — a point this site makes about every model it draws, and one that gets its own field rather than a footnote.

What the counting does establish is the ground the rest stands on. With a pair that is a genuine property of a lattice, the next question is whether the lattice can be reconstructed from the counts alone — and on a cylinder, unlike a disc, it can be.

One last thing about the shape of the argument, because it recurs.

The interesting comparison in this essay was not between a good counter and a bad one. Both halves of the stem-against-disc figure use the same code; the difference is entirely in what it is pointed at. That is the arrangement to aim for whenever a claim is about one situation being better behaved than another, because it removes the most common way such claims go wrong — a method tuned on the case it is meant to favour, and applied to the other case in whatever form it happened to be in.

Here the disc column is not a straw man. It is the same measurement this site has been making since its foundation phase, unchanged, and it produces the same three pairs it always has. What is new is having something to hold it against.