A stem on the other branch

The placement rule has more than one lattice available to it at a given rise, and which one a stem sits on is decided by what it was started from. The famous family is the Fibonacci one — 2/3, 3/5, 5/8, 8/13, walking down as the rise falls — and it is what a stem seeded near 137.5° stays on. The other family this collection has computed is the Lucas one: 1/3, 3/4, 4/7, 7/11, 11/18, reached from a seed near 99.5°, and known from real plants, if uncommonly.

The Lucas branch has been used here as a curiosity and as an example of metastability. This essay uses it as an instrument. Two accounts of what a wrecked stem settles into agree on every Fibonacci rung and disagree wherever the counts are not Fibonacci, and 4/7 is the cheapest place to look.

The control comes first #

An intervention on a Lucas stem is only a measurement about a Lucas lattice if the stem is still on one when the organ is removed. The rule is free to leave.

At a rise of 0.013 a golden-seeded stem settles at 136.781° and carries 5/8. A Lucas-seeded stem at the same rise, with the same neighbourhood, the same exponent and the same azimuth grid, settles at 99.785° and carries 4/7. Neither drifts towards the other: the golden stem is 0.73° from the angle it was seeded at after four hundred organs, and the Lucas one 0.28°, with scatters of 0.24° and 0.10°. Thirty-seven degrees separate them, which is the number the control actually turns on.

That is the whole control and it does two jobs. It says the Lucas stem is on the Lucas lattice, and it says the difference between the two runs is the lattice and nothing else — same rule, same rise, one seed.

What the two accounts predicted #

Both accounts of the block start from the same observation: a stem cut in the middle of its front never returns to its divergence, and settles into a repeating cycle of angles.

One says the cycle carries a count of the lattice that was cut — whichever count that is. On a 4/7 lattice it predicts four or seven.

The other says the cycle is a two-jugate arrangement whose repeat is the count of the ordinary lattice underneath it, which on a Fibonacci rung is the smaller number and which has no natural extension to 4/7 except by way of the smaller number again. It predicts four, and it predicts that the orbit precesses by one part in twice its block.

What the Lucas stem does #

Five offsets fail to repair, out of the nine tried, and four of the five settle on a block of seven.

The motif is a hundred and fifty-five degrees wide, so it is an orbit and not a grid artefact. The effective divergence is 151.14° at three of the four offsets and 202.57° at the fourth, precessing by −22.04° in every case. The fifth unrepaired offset, four places back, settles on a block of four at 189.96°.

So the block is seven at four offsets of five: a count of the lattice, the larger of the two, and not a Fibonacci number. The count-of-the-lattice account survives; the reading that the answer is the smaller number does not, and neither does anything that requires the block to belong to the Fibonacci sequence.

Why 4/7 is the right lattice to ask on #

The choice of arrangement is doing work and is worth defending, because a comparison between two lattices that differ in several ways at once decides nothing.

Four and seven are coprime, so the lattice is genuinely single-jugate: it is not an ordinary lattice seen twice over, and the repeat-unit account cannot appeal to a hidden smaller lattice underneath. That distinguishes this test from the bijugate one, which asks a different question with a different answer.

Four and seven are also not Fibonacci, and neither is their ratio anywhere near the golden one — 7/4 is 1.75 against 1.618. So an account that predicts a Fibonacci block, for whatever reason, has somewhere to fail here, and an account that predicts the smaller number has somewhere to be distinguished from an account that predicts a number of the lattice.

And 4/7 is available at the same rise as 5/8, which is the part that costs nothing and is worth the most. Rise sets the spacing between organs, the depth of the neighbourhood the rule sums over, and the number of organs in the front; a comparison between two lattices at two rises confounds all of that with the lattice. Here the only difference between the two runs is eight seed organs.

And the two-jugate relation misses #

The second half of the repeat-unit account is a relation between two independently computed numbers: the orbit should precess by 360 divided by twice its block. A block of seven should therefore turn by 25.71° and give fourteen rows.

It turns by 22.04°, which is 16.33 rows. That is a fifteen per cent miss on a quantity that had agreed to a part in a hundred at both Fibonacci measurements — 10.04 rows on a block of five, 16.00 on a block of eight.

The miss is not universal. The same Lucas stem cut at a coarser rise, 0.020, gives a block of four precessing by 44.07°, which is 8.17 rows against eight: the relation holds there. So twice the block is what these orbits usually are and not what they must be, and a relation that fails on a lattice chosen because it was different is a relation that was never being tested.

Why the larger number, here #

The honest answer is that this collection does not know, and the shape of the ignorance is worth setting out.

On the golden branch the block is the smaller number at three rises and the larger at one, and where both appear the offset decides. On the Lucas branch it is the larger at four offsets and the smaller at one. Pooling the two branches, eleven unrepaired offsets give the smaller number and seven give the larger, and there is no lattice property that separates them — the same lattice gives both.

What is left is the offset, and the offset is where an explanation would have to start. A vacancy k places back removes a member of the k-family, and the obvious guess is that the surviving family sets the period. It is refused by the arithmetic: at the Lucas arrangement the offsets giving seven are three, five, six and seven, and the offset giving four is four. Three and six are not multiples of four, and four is not a multiple of seven.

The front behaves, which is the reassuring part #

Everything above is about what happens after the pattern fails to repair, and it is worth separating from the result the intervention was built for.

On the Lucas stem the run of felt offsets ends at seven, which is the larger parastichy number of the lattice it carries. Removing the organ eight places back moves the next one by half a degree; nine places back by nothing measurable. The count-from-a-yes-or-no result therefore holds on a branch it was never tested on, and it holds with a different number, which is a much better test of it than a repetition on the Fibonacci branch would have been.

Where the branch came from #

The Lucas lattice is not an invention for this experiment. It is one of the branches the placement rule’s own bifurcation structure supplies, reached by seeding a stem on a Lucas arrangement and lowering the rise slowly enough for the pattern to keep it — this collection measured the threshold at somewhere between eighty-seven and ninety-one organs per rung, with a golden-seeded control ending on Fibonacci at every rate.

It is also on the tree of forks the rule’s maximin criterion generates, where keeping the larger count at each fork converges on the golden angle and one different choice at the first fork converges on 99.502°.

That history matters for reading the result. The Lucas stem is not an artificial lattice built to order; it is a state the rule reaches on its own from a plausible starting condition, and a stem sitting on it behaves in every other respect like a stem sitting on the Fibonacci branch.

What an experiment on a 4/7 shoot would settle #

The measurement suggests a specific and unusually cheap addition to the ablation proposal this collection has been assembling.

The proposal so far is: find a shoot, count its parastichies, remove one primordium at a stated offset, and record whether the next primordium moves. Done across offsets it returns the larger count without a protractor. What the Lucas result adds is that the experiment is worth doing on a Lucas shoot as well as on a Fibonacci one, and that the two are distinguishable in advance by counting.

The reason is that the two branches make the same structural prediction with different numbers. A 4/7 shoot should be sensitive out to seven and quiet at eight; a 5/8 shoot sensitive out to eight and quiet at nine. Two shoots on the same plant at the same stage, differing only in which branch they are on, should give different boundaries — and that is a much harder pattern to produce by accident than a single boundary that happens to match a count.

Lucas phyllotaxis is uncommon and it is not rare. It is reported in a handful of genera, it is what the census of divergence angles puts in the second-largest bucket after the Fibonacci one, and a survey that recorded the pair rather than assuming it would find some.

What this does not say #

It does not say Lucas stems behave differently in general. The front is the larger count, the unrepaired band is in the middle, the orbits are exact — every structural statement this collection makes about ablation holds here. What differs is the number that comes out.

It does not say the block is always the larger number off the Fibonacci branch. One offset of five gives four, and at a coarser rise on the same branch both unrepaired offsets give four. The claim is that the smaller number is not the rule, not that the larger one is.

It does not say the two-jugate relation is dead. It holds at three of the five lattices measured, and misses at two. A relation with that record is a tendency in need of a condition, and this essay supplies neither.

And it does not say anything about a plant with Lucas phyllotaxis. These are stems grown by a rule. What the measurement licenses is that an ablation experiment on a 4/7 shoot is a different measurement from one on a 5/8 shoot, and worth doing for that reason.

One number worth keeping separate #

It is easy to let a result about what a wrecked stem becomes contaminate the result about what a healthy one reports, so the separation is worth restating at the end as well as the middle.

The Lucas stem’s front is seven. That is the larger of its two counts, it is obtained without a protractor, and it is the third branch-and-rise combination the boundary has been checked at. Nothing in the argument about blocks touches it, because the boundary is measured on the organ placed immediately after the cut and the block is measured three hundred organs later.

The blocks are the open question, and they are open in a way that is now much better specified than it was: the answer is one of two known numbers, both accounts that predicted which one have been refuted on some lattice, and the quantity that decides it is the offset. That is a smaller claim than the one this thread started with and it is a claim that has survived being taken somewhere new.

The check that would refuse it #

Three assertions run whenever these figures are drawn.

The first is the control: each stem settles within a few degrees of the lattice it was seeded on, the two are tens of degrees apart, and they carry different pairs. The “few degrees” is deliberately loose and the separation is not — a branch approaches its own limit angle only as the rise falls, so the Lucas stem sits 0.28° from 99.50° at a rise of 0.013 and 2.26° from it at 0.020, while being thirty-seven degrees from the golden branch at both. A Lucas seed that drifted onto the Fibonacci branch would fail it, and would turn every number in the essay into a number about 5/8.

The second requires every block the Lucas stem settles on to be one of its own two parastichy numbers, and requires the larger to be at least as common as the smaller. That second half is the essay’s claim; the first half is the account that survives, and a block of six or eleven would refuse both.

The third requires at least one orbit whose precession is not one part in twice its block. It exists because the alternative — quietly reporting the ones that fit — is the failure mode of a relation that holds most of the time, and because the sixteen-and-a-third rows are the most specific thing in the essay.