A pattern with a rate

Everything this site has built is a family of static lattices indexed by a parameter.

The ladder walks the rise downward and reports which pair is dominant at each value. The van Iterson diagram labels the whole plane. The branch tree gives the forks in closed form. Each of those is a statement about lattices, and none of them contains a plant, because none of them contains a rate.

A shoot does not visit the rungs of the ladder. It moves through them, at a speed, while placing nodes one at a time — and each node is placed among the ones already there rather than among the ones the equilibrium lattice would have.

That distinction has been available since the expansion phase and has never been tested. This essay builds the object that makes it testable.

The rule #

The placement rule is Douady and Couder’s, moved from the disc to the cylinder.

Node $i$ sits at height $y_i = \sum_{j<i} h_j$, and at whichever azimuth minimises

$$E(x) = \sum_{j} \frac{1}{d(x, \mathbf p_j)^{3}}$$

over the nodes already placed, with $d$ the distance on the unrolled cylinder taking the shorter way round. Inverse cube, which is what the original paper’s magnetised droplets actually obey, and the exponent barely matters.

Nothing in that rule mentions Fibonacci, the golden angle, spirals, or the ladder. It is the same rule the emergence field is built on, with a cylinder underneath it instead of a growing disc.

The neighbourhood is the previous $W$ nodes, and $W$ cannot be a constant. The nodes within a fixed distance of the growing tip number about $1/\sqrt h$, which climbs as the rise falls, so a fixed window starves the fine end of every run. That is the same mistake the disc implementation records — a fixed window of twenty-six that was fine at large growth parameters and left the slow end of the sweep with far too few neighbours to order itself, so that the model “never converged there”, which looked like a property of the model and was a property of the truncation. Here the window is $\lceil 6/\sqrt h \rceil$, capped, and the reason is written next to it.

The rate, and what to call it #

The rise falls exponentially in node number:

$$h(i) = h_0 \, e^{-i/T}$$

$T$ is the whole of the time axis — the number of nodes over which the rise falls by a factor of $e$.

Exponential rather than linear, and the reason is that the ladder is geometric. Its rungs are a factor of $1/\varphi^2$ apart, so a constant $T$ means a constant number of nodes per rung:

$$T \cdot 2\ln\varphi \approx 0.9624\,T$$

That is the quantity the whole subject turns on: how many nodes the pattern is given between one transition and the next. A shoot at 20 nodes per rung is being hurried; one at 130 is being given room. Reporting $T$ alone would be reporting an arbitrary unit; reporting nodes per rung is reporting the thing that could matter.

It is also the quantity a plant has. A shoot whose internodes shorten quickly relative to its plastochron passes rungs fast; one that thickens slowly passes them slowly. The number is dimensionless and it is in principle measurable on a specimen by counting nodes between two transitions — which is exactly the measurement the cone essays propose from the other direction.

The seed, and why there has to be one #

The rule needs something to repel from, so a run begins with a stretch of ideal lattice at a stated divergence.

That is not a technicality to be minimised. It is the initial condition, and the initial condition is what makes “can a pattern change branch” a question with an answer. A run seeded at the golden angle starts on the Fibonacci branch; one seeded at the Lucas angle starts on the Lucas branch; and what happens to each as the rise falls is the experiment.

The seed has two parameters and both matter. Its divergence picks the branch. Its length in nodes decides how firmly: eight nodes at a coarse rise is barely a lattice and the rule reorganises it immediately, while forty nodes is two or three rows of parastichies and the rule has something to preserve.

Where the seed does not matter is the case this essay is about. Seeded at the golden angle, at any rate tried, the pattern walks the Fibonacci ladder — because the Fibonacci branch is where this rule goes anyway. The seed’s influence only shows on branches the rule would not have chosen, which is the next essay but one.

What comes out #

Grown at 92 nodes per rung with the rise falling from 0.2 to 0.0045, the pattern is 365 nodes long and a counter shown only its coordinates walks it through

$$1/2 \ \to\ 2/3 \ \to\ 3/5 \ \to\ 5/8 \ \to\ 8/13$$

which is the ladder. Not the ladder to within something: the ladder, in order, with no rung skipped and none repeated.

Two things about that are worth separating.

That it is Fibonacci is not surprising and is not the finding. The rule is known to settle on the golden angle over a range of its parameter, and a pattern at the golden divergence has that ladder by construction.

That it is the ladder at all is not obvious. The ladder is a statement about equilibrium lattices — which pair of vectors is shortest at each rise — and a grown pattern has no obligation to be at equilibrium. It is built out of neighbours placed earlier, at coarser rises, by a rule that optimises one node at a time against a truncated neighbourhood. That it lands on the equilibrium sequence is a result, and measuring how closely is the next essay.

Measuring a pattern that is changing #

Counting a grown stem needs more care than counting a static one, and two choices deserve stating because both were got wrong first.

A sliding window, not bands. The obvious method is to cut the run into equal bands and count in each. It loses the experiment: a fast shoot has fewer nodes in total — that is what being fast is — so twelve bands of forty leave nothing to measure at the rates the question is about. A window of thirty-six nodes slid along in steps of four gives one count per step however short the run.

Looking only downward. The window counts using the nodes at or below its own top. A count that could reach upward would be using nodes placed after the ones it is describing, and the whole point of asking when something happened is spoiled by an instrument that can see the future.

Neither choice changes the answer on a slow run. Both decide whether there is an answer on a fast one.

What the divergence does #

A static lattice has one divergence by construction. A grown one has a sequence, and what it does between transitions is something no static picture can show.

Over the second half of a run at 92 nodes per rung the node-to-node divergence stays within a few degrees of 137.5°, wandering up and back rather than sitting still. The wander is largest around the transitions — which makes sense, since a transition is where three families are nearly equally short and the repulsion landscape is nearly flat between two arrangements.

That is a small observation and it is the first thing on this site that is a property of the process rather than of the lattice. A static account has nothing to say about it, because in a static account the divergence is an input.

Holding the right thing fixed #

Comparing two rates sounds like it needs no thought and it needs a good deal, because there are two obvious ways to do it and one of them is useless.

The first is to fix the number of nodes and vary $T$. That is what a naive sweep does, and it fails immediately: at 480 nodes and a fast decline the rise runs down to $10^{-18}$, where the parastichy numbers are in the billions, nothing is countable, and the answer is about floating-point arithmetic rather than about plants.

The second is to fix the range of rise and let the node count follow. A run covers $h_0$ down to $h_{\text{end}}$ and therefore needs $T\ln(h_0/h_{\text{end}})$ nodes to do it. A slow shoot needs many; a fast one needs few.

The second is right and the reason is not merely practical. A fast shoot really does have fewer nodes in which to reorganise — that is not an artefact of the comparison, it is the mechanism by which a rate could matter at all. Holding the node count fixed and letting the rise range vary hides exactly the thing being looked for.

So every run compared in these essays covers the same rise, and the node count is the dependent variable. A run at 135 nodes per rung is 814 nodes long; one at nine is 53. Both start at the same rung and end at the same rung.

What a lag would look like #

It is worth saying in advance what the measurement of the next essay is, because a negative result is only interesting if the positive one was well defined first.

The static ladder puts each transition at a definite rise — 1/2 gives way to 2/3 at $h = 0.12472$, 2/3 to 3/5 at 0.04767, and so on down. A grown pattern makes the same transitions at whatever rise it makes them at. The gap between the two is the lag, and the natural unit is a rung, because the ladder is geometric:

$$\text{lag} = \frac{\ln(h_{\text{static}} / h_{\text{measured}})}{\ln \varphi^2}$$

Positive means the pattern changed late — it held its old pair past the rise at which the new one became shorter, which is what a pattern built out of its own history should do. A lag of one rung would mean the pattern was a whole transition behind, arriving at 5/8 when the equilibrium lattice was already at 8/13.

The measure is dimensionless and does not depend on where in the run a transition happens, which is what makes lags from different rates comparable at all.

What the model cannot be asked #

Three limits, all of which bound what the next three essays can claim.

It is a discrete rule on a sampled circle. The azimuth is chosen from 512 sample points, so the divergence is quantised to about 0.7°, and asking the model for better than that is asking the arithmetic for something it does not have.

It is truncated. The window is finite and capped, and at very fine rises the capped window spans a vertical extent much smaller than a circumference. Below about six nodes per rung a counting window no longer contains a rung and the measurement stops meaning anything — so every claim made from this model is a claim about the rates it can be run at, not about all rates.

And it is a model of form. A stem that places nodes by repulsion is not a stem that computes repulsion. This site is careful about that everywhere, because the same patterns come out of magnetised droplets with no biology in them at all, and adding a time axis does not change it. What has been added is a history, not a mechanism.

What a plant’s rate would be #

The rate is dimensionless and it is worth asking what value a real shoot has, because the answer decides which end of the sweep is the relevant one.

Two quantities set it. The plastochron is the time between one primordium and the next; the rate at which the rise falls is set by how fast the internodes shorten or the apex widens. Their ratio is the number of nodes per rung.

A sunflower head is the extreme slow case. Its rise falls as one over the element number, so the transitions sit at elements 31, 81, 212, 556 and so on — each a factor of $\varphi^2$ beyond the last, which means the number of elements spent on a rung is itself growing. Out at the rim a head is spending three hundred elements crossing a rung, and a head is therefore at the far slow end of anything measurable here.

A stem is the opposite extreme and it is a degenerate one: a cylinder’s rise does not fall at all, so it never crosses a rung and the rate is zero.

The interesting middle is a shoot whose apex is changing shape — a vegetative stem becoming an inflorescence, or a cone forming at the tip of a branch — and there the rise can fall over a handful of nodes. That is the regime where a rate could matter, and it is the fast end of the sweep in the next essay.

So the honest summary is that most of the material this site talks about sits at the slow end, and the question of whether a hurried pattern behaves differently is a question about transitions rather than about mature organs. Which is where the interesting cases in development actually are.

What is not new here #

Rising phyllotaxis is not an invention of this site and it is worth saying whose it is, and what is being added.

The idea that a phyllotactic pattern moves through a family of lattices as the apex changes shape is old — it is the reading van Iterson’s diagram was drawn for in 1907, and the reason the diagram is a plane rather than a curve. The idea that a pattern arriving at a fork chooses a branch, and that the choice is the whole question, is the standard modern account. Douady and Couder’s own experiments swept their parameter and watched the pattern change, which is a rate applied to a physical system rather than to a model.

What is added here is narrow. It is a measurement of the tracking: how closely the pattern a growing rule produces sits on the equilibrium sequence a static calculation gives, expressed in rungs, across a stated range of rate, by a counter that sees only coordinates. That number has been assumed by everything on this site and by a good deal of the literature, and assuming it is cheap.

The measurement can come out three ways, and only one of them is boring. It could show a lag that grows with rate, which would mean the static ladder is a limit rather than a description. It could show no lag, which would license the static work. Or it could show the pattern doing something else entirely — skipping rungs, or leaving the branch — which would be the most interesting outcome and is the one the third essay in this group finds a version of.

What this makes askable #

Three questions become well posed, and each is an essay.

Does the pattern keep up? The static ladder says the pair at rise $h$ is whichever lattice vectors are shortest there. A pattern with a history has no obligation to agree, and the amount by which it disagrees is a lag, measurable in rungs. That is the next essay, and the answer is not the one expected.

Can it change branch? The tree forks, and in the static picture nothing chooses which way. Here something does: the pattern arrives at a fork on a branch and leaves on one. Which one, and what decides.

And does continuity explain Fibonacci? The expansion phase found that only 14.7% of divergences give a Fibonacci pair at a fine rise, and wrote that what makes plants Fibonacci must be continuity from a coarse start. That was an interpretation with nothing behind it, because the site had no way to start coarse and continue. Now it does, and the measurement is the last essay of this phase.