Seven rises and two seeds

The most useful thing this collection has found about the ablation experiment is also the one hardest to be sure of. Remove one organ from a settled stem, and whether the next organ moves depends on how far back the removal was: it moves for every offset out to a boundary, and for none past it. The boundary is the larger of the two spiral counts — which means a spiral count can be obtained from a sequence of yes-or-no answers, without counting anything.

The difficulty is that every measurement of it so far has changed two things at once. On a single branch, the parastichy pair is a function of the rise: make the rise smaller and the stem walks up its ladder from 3/5 to 5/8 to 8/13, and every other property of the stem changes with it — the spacing between organs, how many of them lie within any distance of the tip, how finely the pattern is divided. An account that tied the front to the rise, or to the spacing, or to some loose notion of how fine the pattern is, would have survived all of it.

The way out is to find two stems that carry different lattices at the same rise. There is one, and this collection has grown it before: the Lucas branch.

Two branches at one rise #

A placement rule has more than one settled state. Started from a stretch of lattice near the golden angle it walks the Fibonacci ladder; started near 99.5° it walks the Lucas one — 3/4, 4/7, 7/11 — and stays there. Nothing about the rule differs between the two runs. Nothing about the rise, the heights, the azimuth grid, the size of the neighbourhood, or the number of organs differs either. The only difference is the eight organs the run was started from.

So at a rise of 0.013 there are two stems with lattices 5/8 and 4/7. If the front is the larger parastichy number, they should be felt out to eight and seven organs respectively; if the front is a property of the rise, they should agree.

They do not agree. The golden stem’s run of felt offsets ends at eight and the Lucas stem’s ends at seven.

That is the matched pair, and it has been made once before, in passing. It is worth being clear about what it settles and what it does not. It settles that the front is not a function of the rise alone, because two stems at one rise give two answers. It does not settle that the front is the parastichy number, because at this rise the golden lattice has both the larger count and the finer pattern, and an account tying the front to fineness survives.

Seven pairs, and the ordering reverses #

One pair is an anecdote. The remedy is not a better argument but more pairs, and the ladders of the two branches happen to be arranged so that the pairs are not all alike.

Both branches hold a settled lattice from a rise of 0.030 down to about 0.006 — below that the Lucas branch scatters by fifty degrees and stops being a lattice at all, and the golden one follows shortly after. Inside that range the golden branch walks 3/5 → 5/8 → 8/13 and the Lucas branch walks 3/4 → 4/7 → 7/11, and because the two ladders have different rungs in different places, the ordering of the two fronts is not constant.

At a rise of 0.024 the golden stem carries 3/5 and the Lucas one 3/4, and the golden front is the wider. At 0.020 the Lucas stem has moved to 4/7 while the golden one is still on 3/5, and now the Lucas front is wider. Through 0.016, 0.013 and 0.010 the golden stem is on 5/8 against the Lucas 4/7 and the golden front is wider again. At 0.008 the Lucas stem reaches 7/11 and takes the lead back. At 0.006 the golden stem reaches 8/13 and takes it again.

Four reversals. The gap between the two fronts, golden minus Lucas, runs +1, −1, +1, +1, +1, −2, +1.

No function of the rise can produce that column, and neither can the rise with a constant added per branch, because the gap takes both signs. That is what the seven pairs buy over the one.

The other half of the dissociation #

The design has a second half, and it is the one that turns the front is not the rise into the front is the lattice.

The golden branch carries 5/8 at four of these seven rises — 0.016, 0.013, 0.010 and 0.008. That is a factor of two in the rise, which is a factor of 1.41 in the spacing between organs and rather more than that in how many organs sit within a fixed distance of the tip. Its front is eight at all four.

So: two lattices at one rise give two fronts, and one lattice at four rises gives one front. That is a double dissociation, and between them the two statements leave the rise with nothing to do. The front follows the lattice.

The Lucas branch supplies the same thing on its own account. It carries 4/7 at four rises in the design and its front is seven at three of them and six at the fourth — which is a caveat rather than a counterexample, and is the subject of the next essay in this ladder.

Reading one row in full #

The design is a table of fourteen numbers and a table hides the measurement it came from, so it is worth opening one row.

At a rise of 0.016 the golden stem settles at 136.55° and the Lucas stem at 100.78°. A blind counter shown the golden stem’s positions returns 5 and 8; shown the Lucas stem’s, 4 and 7. Neither counter is told the angle.

Now remove one organ. On the golden stem, offsets one through eight each move the next organ by something between four and a hundred and seventy degrees. Offset nine moves it by 0.70°, offset ten by less, and so on out to twelve. On the Lucas stem, offsets one through seven move it, offset eight moves it by 1.41°, and nothing past that moves it either.

The contrast between the two sides of the boundary is the part that makes this usable. Inside the front the displacements are tens of degrees; outside it they are under one and a half. There is no intermediate regime a threshold has to be tuned to, which is why the same threshold — two and a half degrees, a tenth of the local spacing — works at every cell of the design without adjustment.

That sharpness is itself a consequence of what the rule does. It places the next organ at the least of a profile summed over the neighbourhood, and the position of a minimum is not a linear function of the terms that make it. Removing an organ outside the front lifts the profile slightly everywhere and changes which point is lowest by nothing at all. A loser that is lifted a little is still a loser.

What a matched design controls for, item by item #

The value of a design of this kind is that the controls are structural rather than argued, so it is worth listing what is held fixed at each row and by what.

The rule. Both stems are grown by the same function with the same parameters: the same repulsion exponent, the same neighbourhood, the same argmin over the same number of sampled azimuths. Not a re-implementation — the same code path.

The geometry. The heights are prescribed by the rise, so both stems have their organs at identical heights. The circumference is the same. The spacing between neighbouring organs is the same to the extent that the lattice allows.

The resolution. Both are computed on a grid of 1,536 azimuths, which is the grid every measurement in this thread uses, so a displacement of a tenth of a degree means the same thing in both rows.

The intervention. The same organ index is removed, the same number of organs are placed afterwards, and the same threshold decides whether the next organ moved.

And the seed is not a free parameter tuned to taste. The two seed angles are the golden angle and the Lucas angle, both of which this collection derives from the arithmetic rather than choosing, and each stem’s settled divergence is measured rather than assumed — the golden branch drifts to within 0.9° of its seed and the Lucas branch to within 2.3°, and both are reported.

Why the Lucas branch is worth the trouble #

There is a version of this experiment that does not need a second branch: sweep the rise finely, find the rise where the golden stem changes rung, and compare just below with just above. It has been done and it is much weaker, for a reason worth naming.

Near a rung boundary the lattice is ambiguous — the two candidate second families have nearly equal step lengths — so a stem there is a poor instrument for anything and its front is measured against a boundary that is itself in doubt. The pairs in this design sit in the middle of their rungs by construction, and they get their contrast from the branch rather than from proximity to a transition.

The Lucas branch also happens to be the commonest real exception. Plants that are not Fibonacci are, more often than anything else, Lucas — 4 and 7 rather than 5 and 8 — so a result that has been checked on both branches is a result checked on the two arrangements a census would actually encounter.

What it would take to do this on plants #

The design is cheap here because both stems are grown to order. On plants it is not, and the price is worth stating because the result is one a census could in principle check.

The comparison needs pairs of specimens matched on everything a rise decides and differing in their lattice. In a plant, the rise is not a knob; it is set by how fast the stem elongates against how fast organs are made, and it is not usually measured at all. So a matched pair on real material means two plants whose internode spacing, organ size and apex radius agree closely enough that the geometry is common, one of them Fibonacci and one Lucas.

Lucas plants are uncommon but not rare — they are the commonest exception by a wide margin — so the limiting quantity is not how many exist but how many can be matched. This collection has priced its other survey questions in tens of specimens rather than thousands, and a matched-pair design is worse than that, because the matching throws specimens away.

The consolation is that the measurement asked of each specimen is coarse. It is not a divergence to a tenth of a degree; it is whether a removal at a stated offset was felt, which is a yes-or-no answer about a displacement of tens of degrees against a background of one. That is the property that made the front result worth pursuing in the first place, and the design does not spend it.

What this does not say #

It does not say the front is exactly the larger parastichy number at every cell. It is at eleven of the fourteen. Three read one offset short, and the reason they do is the subject of the next essay: at each of them the offset that would have completed the run is felt weakly rather than not at all.

It does not say the branch is the only thing that matters. The design holds the rise fixed across each pair and varies it down the column, and the front does change down the column — because the lattice changes down the column. The claim is about which of the two the front follows, not that the rise is irrelevant to anything.

It does not say either branch is a plant. Both are stems grown by a rule from a stated seed. What makes the Lucas branch interesting is that real plants supply its counts often enough for the comparison to be worth making, not that this particular run is a specimen.

And it does not extend below a rise of 0.006. Both branches lose their lattices there — the Lucas one first — so the design has an edge, and the edge is where the comparison stops existing rather than where the result stops holding.

The check that would refuse it #

Four assertions run with the figures, and they are separable on purpose.

The first is that both branches have settled at every rise in the design: a divergence spread under half a degree over the last sixty organs, and at least three distinct parastichy pairs on each branch across the range. A row where one stem had lost its pattern would be a comparison between a front and a number read off noise, and the second half stops the design being one comparison repeated seven times.

The second is that the two fronts differ at every rise. This is the claim a single pair makes, and asserting it at all seven rises is what turns it from an observation into a property of the design.

The third is the reversal: the ordering must change hands at least three times, and the gap between the two fronts must take both signs. An account tying the front to the rise, or to the rise plus a per-branch constant, fails here and nowhere else.

The fourth is the dissociation’s other half: any lattice appearing at three or more rises must give the same front at all of them, to within the one offset the threshold can move, across a range of rises spanning at least a factor of 1.5. It is the assertion that would catch the front drifting slowly with the rise inside a rung — which is the last way the rise could still be doing the work, and it is not.