The rate decides the branch
The previous essay found that a growing pattern hits the rungs of the ladder on schedule at every rate it can be run at, and closed by noting that this says nothing about which ladder.
This one is that question, and the answer is the opposite shape. The rungs do not depend on the rate at all. The branch depends on almost nothing else.
The experiment
A stem is seeded with forty nodes of ideal lattice at the Lucas angle, 99.502°, at a rise of 0.12. The rise then falls to 0.004 — about three and a half rungs — at a stated rate, with every node after the seed placed by the repulsion rule and nothing else.
At 65 nodes per rung the pattern walks
which is the Lucas ladder, at a divergence that stays at 99.5° the whole way.
At 131 nodes per rung the same seed walks
which is the Fibonacci ladder, at a divergence that leaves 99.5° at the first fork it reaches and settles at 137.7°.
Nothing differs between the two runs except how many nodes the stem spends getting there.
The control is the third run. A stem seeded at the golden angle ends on the Fibonacci ladder at both rates, so the slow run is not simply forgetting its seed — it is changing branch, and it changes onto the branch the rule would have chosen anyway.
Metastable, not stable
The reading is that the Lucas branch is metastable.
The Fibonacci branch is where this rule goes. The bifurcation diagram found that the Douady–Couder rule settles on the golden angle over a broad range of its parameter, from starting conditions that know nothing about it; the branch tree found that always keeping the larger count at every fork converges on 137.508° while one different choice at the first fork converges on 99.502°. Both routes make Fibonacci the default and Lucas an alternative a pattern has to be put onto.
Being put onto it is what the seed does. Once there, the pattern is at a genuine arrangement — the Lucas ladder is a real sequence of dominant pairs at real rises — but it is not the one with the lowest repulsion energy, and a rule that keeps placing nodes will find the lower one if it is given enough placements at nearly one rise to find it in.
A slow decline is many placements per rung. That is the whole mechanism. Hurry the stem and it passes each fork before the arrangement has been probed enough times to fall out of it; give it room and it falls out.
That is the reverse of the intuition a lag would have suggested, and it is worth saying why the two are not in conflict. A lag would be the pattern failing to keep up with a moving equilibrium. This is the pattern failing to fall into a stationary lower state. Slowness helps the second and would have hurt the first.
Where the threshold is
The outcome is not gradual. Bisecting on the rate, with everything else held fixed, the Lucas seed keeps its ladder up to 87 nodes per rung and loses it from 91.
Between those two the answer flips, and either side of them it is stable: at 46, 58, 65, 75 and 85 nodes per rung the pattern ends on 7/11 at 99.5°, and at 92, 108 and 131 it ends on 8/13 at 137.5°. The transition is monotone in the rate — there is no interleaving — which is what makes it a threshold rather than a scatter.
What the threshold depends on
The number 90 is not universal and it is worth being clear about what it is a property of.
It depends on the starting rise. Seeded at a rise of 0.12 the threshold is between 87 and 91 nodes per rung; seeded at 0.15 it is lower, somewhere between 46 and 65. The reason is visible in the ladder: the Lucas branch separates from the Fibonacci one at a rise of 0.1849, where 1/2 gives way to 1/3. A pattern seeded just below that fork is barely on its branch and leaves easily; one seeded well below it is established and holds.
So a branch has to be entered with room to spare. Being past the fork is necessary and is not sufficient, and the gap between “the branch exists” and “the branch can be held” is a real interval — here, roughly from 0.185 down to 0.13 in rise.
It depends on the seed length. Forty nodes is two or three rows of parastichies, which is enough to be a lattice. Eight nodes is not, and a run seeded with eight is on whatever branch the rule prefers within a few placements regardless of the angle it was given.
And it depends on the rule. Inverse-cube repulsion against a truncated neighbourhood is one way to place a node. A rule with a different falloff would probe the arrangement differently and would find the lower state at a different rate. What is robust is the shape — a threshold, monotone, with fast keeping and slow losing — rather than the number.
The confound, and the control for it
There is an obvious objection to the experiment and it has to be met with a measurement rather than an argument.
The seed is forty nodes in every run. A slow run is long — 463 nodes at 131 per rung — and a fast one is short — 232 at 65. So the seed is a larger fraction of the fast run than of the slow one, and “the fast run keeps its branch” could just be “the fast run is mostly seed”.
Two things rule that out.
The first is arithmetic. Forty nodes at 65 per rung spans 0.61 of a rung; at 131 per rung it spans 0.31. In both cases the seed occupies well under one rung and the pattern crosses three more forks after it ends. Whatever the seed is doing, it is not doing it by being most of the pattern.
The second is the control run. Seeded at the golden angle, the same forty nodes, the pattern ends Fibonacci at both rates — which it would do with no seed at all, since Fibonacci is where the rule goes. If the fast runs were simply preserving their seeds, the golden-seeded fast run and the Lucas-seeded fast run would both be preserving them, and they are; but the slow runs would then both be preserving them too, and only one of them is. The asymmetry is between branches, not between seed lengths.
There is a third check that was run and is worth reporting because it failed in an instructive way. Scaling the seed length with the rate — so that the seed covers the same stretch of rise rather than the same number of nodes — makes every run lose the branch, at every rate. That looks at first like a refutation and is not: scaling the seed with the rate makes the slow runs’ seeds longer and the fast runs’ seeds shorter, and at 24 nodes per rung a scaled seed is eight nodes, which is not a lattice at all. The comparison it makes is between a well-seeded slow run and an unseeded fast one, which is a different experiment.
What that failure does show is that the effect needs both ingredients — an established initial arrangement, and few placements per fork — and that neither alone is enough. Which is the correct reading of a metastable state — being in it is necessary, and so is not being given time to leave.
Which way round hysteresis goes
The result is a hysteresis result and the direction is worth pinning down, because “hysteresis” is used loosely enough that the sign is often left ambiguous.
In the usual driven-system version, sweeping a parameter fast makes a system overshoot — it stays in a state past the point where that state stops being the best one, and the faster the sweep the further it overshoots. That is what a lag is, and the previous essay found none of it.
What happens here is the same phenomenon applied to a discrete choice rather than to a continuous state. The pattern’s arrangement is not part-way between Lucas and Fibonacci at any point; it is on one or the other. Sweeping fast means arriving at each fork with less exploration behind it, and less exploration means the current arrangement survives. Sweeping slowly means more exploration, and more exploration means the lower arrangement is found.
So both results are hysteresis and both are about exploration rather than about relaxation:
- the rungs are found by local comparison at every node, so exploration is irrelevant and they never lag;
- the branch is a global arrangement that is only left when something knocks the pattern out of it, so exploration is everything and the rate controls it.
What this settles about the Lucas angle
The site has had two independent routes to 99.502° and no account of when a plant would be there.
The bifurcation diagram reaches it as a branch of the dynamical model outside the golden range. The branch tree reaches it by taking the smaller count at the first fork and the larger thereafter — a single different choice, made once, at the very beginning. Two routes whose failure modes cannot produce each other’s answer, which is the strongest form of agreement this site has.
What neither said is why a plant would make that choice, or having made it, why it would stick.
This essay supplies the second half. A pattern on the Lucas branch stays there if it is hurried and leaves if it is not, and the threshold is a number. Lucas phyllotaxis being rare is then not a statement about the Lucas angle being worse — it is a statement about how many shoots enter the branch with room to spare and then pass their forks quickly enough to keep it.
That is a testable-in-principle claim about development rather than about geometry, and it is the first thing on this site that is.
The first fork is where it is decided
One detail of the losing runs is worth pulling out because it says where the action is.
When the Lucas seed loses its branch, it does not drift away. It walks 1/3, and then at the first fork it reaches it goes to 2/3 rather than to 3/4 — and everything after that is the Fibonacci ladder. The change happens at a fork and nowhere else.
That is what the static picture predicts and it is satisfying to see it. Between forks there is one dominant pair and no choice to make; the arrangement is determined and the pattern has nowhere else to be. At a fork three families are equally short, the repulsion landscape is nearly flat between two arrangements, and a pattern with any exploration at all will find the lower one.
A branch is not lost gradually. It is lost at a fork, or not at all.
Which also means the threshold is really a statement about a single event — how many placements the pattern makes while it is in the neighbourhood of one fork — rather than about the run as a whole. A stem could in principle be hurried past three forks and given room at the fourth, and would keep its branch through the first three and lose it at the last. That is a prediction the machinery can make and this phase did not test.
What a plant would have to do
The claim has a shape that could in principle be checked, and it is worth writing out even though nothing here checks it.
Lucas phyllotaxis is reported in a small minority of specimens across a range of families — the usual figure quoted is a per cent or two, though this site is careful to note that its own census gives 1.5% at a fine rise from the geometry alone, so a survey finding one or two per cent would be finding what the geometry hands over and nothing more.
If the account here is right, the specimens that are on the Lucas branch should share two properties.
They should have entered the branch early — that is, the pattern should have been established at a rise well below the 1/2 → 1/3 fork rather than just below it. On a real shoot that means the first few nodes were formed on an apex already narrow enough to be two rungs down, which is a statement about apex size at initiation.
And they should have passed their forks quickly — few nodes per rung, which on a real shoot means the internodes shortened fast relative to the plastochron.
Both are measurable on material that exists, and neither is usually recorded. A survey that reported the node number at which each specimen’s count changed, alongside the count, would give the nodes-per-rung directly, and it would give it for the Fibonacci specimens too — which is the comparison that matters, since the claim is that the Lucas ones are hurried and the Fibonacci ones are not.
That is a smaller and more checkable claim than “why is phyllotaxis Fibonacci”, and it is the sort of thing this site would rather leave behind than an explanation.
What is not shown
Two limits, both about how far the mechanism generalises.
Only one alternative branch was tried. Lucas is the natural second choice because the site has two routes to it, but the tree has many branches — taking the smaller count at the second fork instead of the first gives another, and so on. Whether they all behave this way, or whether some are stable rather than metastable, is not known here.
And no plant is in this. A rule that places nodes by repulsion is a model of form and not a mechanism, and the exploration doing the work here is a property of the rule’s discreteness rather than of any biological process. A real apex has noise, and noise would knock a pattern out of a metastable arrangement whether or not the rule explored — which would make the threshold a statement about noise amplitude rather than about rate. That is the obvious next thing to build and it is not built.