What a mechanism would have to show
Every field on this site ends up saying some version of the same sentence. A model that reproduces a pattern has not explained it. The spirals come out of magnetised droplets with no biology in them; a lattice that matches a sunflower does not show that the plant computes it; an L-system that produces something that looks like a plant explains nothing at all.
Repeated often enough, that sentence becomes a reflex rather than a claim, and a reflex is not checkable. So this essay does the work of making it precise: what, specifically, would an explanation have to supply that these models do not?
Four things a model can be doing
It helps to separate claims that get run together.
Description. The pattern has this form. “A sunflower head’s primordia lie near the points at angle ” is a description, and it is checkable by measuring a sunflower.
Reproduction. A rule with these parameters generates something with that form. Vogel’s model reproduces; so does Douady–Couder; so does a reaction–diffusion ring, for the whorled case.
Mechanistic claim. The system does this, with these parts. “Auxin is transported by PIN carriers polarised towards the richer neighbour” is a claim about objects that either exist or do not.
Explanation. The mechanism, operating as described, produces the form, and it does so for reasons that would have been different had the form been different.
The four are routinely presented as one, usually by a diagram that reproduces a pattern and a caption that describes a mechanism. The distance between the second and the third is where almost all of the confusion in this subject lives.
What reproduction is worth
Not nothing. Reproduction eliminates: a rule that cannot produce the observed form is not the rule, and a good deal of nineteenth-century phyllotaxis was killed off exactly this way.
Its weakness is that it eliminates very little, because the space of rules that produce spiral lattices is enormous. This site alone contains four: Vogel’s disc, the Douady–Couder repulsion rule, the cylindrical lattice family, and the ladder of forks. They have almost nothing in common as processes. They agree on the form.
That is not a defect of the models; it is a fact about the form. A spiral lattice is what a very wide class of spacing rules produces, which is why it appears in ferrofluid droplets, in vortices, in packed spheres and in plants. The pattern is robust, and robust patterns are poor evidence about mechanism, precisely because they would have appeared anyway.
The list
So: an account of phyllotaxis would have to establish the following. Each item is marked with whether anything on this site establishes it.
1. That the pattern has a stated form, measured rather than asserted. Established. The counting extracts parastichy numbers from coordinates without being shown the angle, and the recovery returns the parameters exactly.
2. That some rule generates that form. Established, several times over, which is the problem rather than the achievement.
3. That the rule’s parameters correspond to measurable quantities in the organism. Partly. The rise is internode length over apex circumference, both measurable. The Douady–Couder growth parameter is the same quantity. The reaction–diffusion constants are not measured in any plant.
4. That the parts the mechanism names exist and behave as claimed. Established for the transport account and not by anything here. PIN proteins are visible, their orientation points towards forming primordia, and the mutants behave as the account requires — none of which is a result this site produced, and all of which is the reason that account is the textbook one.
5. That interfering with the mechanism changes the form in the predicted way. Not established here, and this is the item that matters most. Ablation experiments on real apices exist and are the strongest evidence in the field. Nothing computed on this site bears on them.
6. That the form would have been different had the mechanism been different. Not established, and probably not establishable in the form usually wanted, because of the robustness above.
The underdetermination, with numbers
“Many rules produce the same form” is the kind of sentence that gets nodded at, so it is worth making it concrete on material this site actually has.
Four constructions here produce a spiral lattice with consecutive Fibonacci parastichy counts:
Vogel’s disc. The $n$th point at angle and radius . No dynamics, no interaction between elements, no time.
The Douady–Couder rule. Elements drift outward at rate ; each new one goes where the inverse-cube repulsion from the existing ones is least. Time, interaction, no lattice anywhere in it.
The cylindrical lattice family. Two parameters, a periodic strip, and a ladder of shortest-vector pairs as one parameter falls. No dynamics, no elements, no interaction.
The transport model on a growing apex. Cells, a hormone, carriers polarised up the gradient. Chemistry, no geometry stated anywhere.
Given a photograph of a sunflower, each of these accounts for it. Given a measurement of the sunflower — the counts, their change with radius, the recovered divergence — each still accounts for it, because they agree on all of those. The measurements this site is proudest of do not discriminate between them.
That is the underdetermination stated exactly: not “models are hard to distinguish” but “the observable form is a fixed point that all four constructions land on, so no observation of the form separates them.”
Where the models on this site actually sit
Items 1 and 2, comprehensively. Item 3, in part. Items 4 to 6, not at all.
That is a smaller claim than most illustrated accounts of phyllotaxis make, and it is worth being exact about what it still buys, because “these are models of form” is sometimes taken as a confession that the whole exercise is decorative.
It is not. The measurements here are measurements: the counts change with radius is a fact about spiral lattices that most accounts get wrong; no packing criterion singles out 137.5° is a refutation of a claim that is made constantly; the nautilus is out by a factor of 2.14 settles an argument with a number. None of those depends on a mechanism. They are statements about form, they are the statements the popular literature actually makes, and they are wrong in the popular literature.
So the division of labour is: this collection can say what the form is and what does and does not follow from it. It cannot say what the plant is doing, and the essays that would need to say so instead say what would have to be shown.
Why the pattern is bad evidence, stated once properly
The reason a robust form is weak evidence is worth one paragraph of care, because “robust” sounds like a compliment.
Evidence discriminates. An observation supports one hypothesis over another to the extent that it was more likely under the one than the other. If a spiral lattice is what almost any spacing rule produces on a growing surface, then observing a spiral lattice is nearly as likely under every candidate mechanism, and the observation moves nothing.
This is not a technicality about statistics; it is why the ferrofluid experiment matters so much. Douady and Couder’s droplets have no cells, no hormone, no genome and no growth in any biological sense, and they produce the pattern. That single result establishes that the pattern is available to systems with almost nothing in common with plants, which is exactly the condition under which the pattern stops being evidence about plants.
The counterpart is that departures from the form are informative. A plant whose counts do not fit any lattice, or whose divergence changes mid-shoot, or whose primordia appear in an order the geometry forbids, would discriminate — which is one reason the recovery is built to refuse rather than to fit. A method that always returns an answer cannot report the case that would be informative.
The specific gap in this field
Two essays of this field establish that a stationary ring, with either chemistry, selects a spacing and not an arrangement. That is a negative result and it is the most useful thing here.
It is useful because it identifies what the missing ingredient is: motion. The elements have to leave the competent region, and the rate at which they do relative to the rate at which they form is what decides whether the outcome is a whorl or a spiral.
And that rate is not a new quantity. It is the rise on a cylinder, it is the growth parameter in the dynamical model, and it is the plastochron ratio in the botanical literature. Three descriptions, one number.
So the shape of a complete account is visible even though the account is not here: chemistry fixes the spacing, growth fixes the rate, and the geometry of what those two admit fixes the arrangement. Every part of that except the first is on this site. The first is the part with the microscopy behind it.
The one case where form did discriminate
It would be too tidy to leave the impression that form never settles anything, because on this site it has, once, and the exception marks the boundary usefully.
The nautilus is claimed to be a golden spiral — one whose radius multiplies by per turn. Fitting the growth factor to actual shells gives about 3.2, a factor of 2.14 out, and the one free choice in the fit — where the centre is assumed to be — moves the answer by under a percent for a quarter-radius error.
That measurement does discriminate, and the reason is instructive: the claim was quantitative and wrong. It named a number, the number can be measured, and the measurement disagrees.
The claims that resist discrimination are the ones that are qualitative — “the pattern arises from an activator–inhibitor system”, “the arrangement optimises packing” — because they predict a kind of form rather than a value, and many mechanisms predict the same kind.
So the rule is not that form is useless. It is that form discriminates exactly to the extent that the competing accounts predict different numbers, and most accounts of phyllotactic mechanism do not.
What would falsify what
A checklist is worth more if each item comes with the observation that would break it, so:
The lattice description fails if a plant’s parastichy counts are inconsistent with any — which is a real test, since the recovery refuses rather than fitting when the numbers describe no lattice.
The rise account of rising phyllotaxis fails if counts rise on a shoot whose internode-to-circumference ratio does not fall. That is measurable and it has been measured; the ratio does fall.
The spacing account fails if a mechanism-level intervention changes the number of primordia without changing their spacing, or the reverse.
The claim that arrangement comes from motion fails if a demonstrably stationary system produces a reproducible divergence angle. This one is checked here in miniature, on the ring, and passes — the angles scatter over 129°.
None of these is a knockdown, and the last is the only one this site can run. Listing them is still worth doing, because a model with no stated failure condition is not a model, and this subject is full of accounts that have never been given one.
What an honest figure caption looks like
The practical version of all this is a rule about captions, since captions are where the four kinds of claim get run together.
A caption that says “the golden angle produces the most efficient packing” is making an explanatory claim on the strength of a reproduction, and on this site it would additionally be false — three criteria give three winners.
A caption that says “a model of auxin transport produces phyllotactic spirals” is making a mechanistic claim on the strength of a reproduction. It is the commonest form and it is the one worth watching for.
A caption that says “places each new element where the repulsion from the existing ones is least; the settled divergence is 137.4°, from a starting angle of 112°” states the rule, the measurement, and the input. It claims a reproduction and nothing more, and a reader can tell what was done.
Every figure on this site carries its parameters in a strip along the bottom for exactly that reason. It is not decoration and it is not a house style; it is the difference between a picture that reports what was computed and a picture that reports what the author believes.
Why the list is worth having written down
A checklist is a modest device and it is doing two specific jobs here.
It makes the site’s own position auditable. Anyone can check items 1 and 2 against the figures, and the claim that items 4 to 6 are not met is a claim that can be contradicted by pointing at a figure that meets one. Vague scepticism cannot be contradicted, which is why it is worth converting into a list that can.
And it constrains what future work here is allowed to say. A phase that adds an elaborate simulation of an apex will still be at item 2, however impressive the simulation, and the list is there to prevent the impressiveness from being mistaken for progress up it. Items 4 and 5 are not reachable by computation at all — they require a microscope and a plant.
That second job is the reason this essay sits in the mechanism field rather than in an appendix. The field exists because a collection that says “reproducing is not explaining” on every page owes a statement of what explaining would consist of, and having made that statement the collection is bound by it.
The sentence, made precise
The reflex sentence was: reproducing a pattern is not explaining it.
The precise version is: a form robust enough to arise from many rules is weak evidence for any of them, so an account of phyllotaxis has to be tested on its parts and its interventions rather than on its pictures. The pictures are how the account is communicated and they cannot be how it is checked, because a rule that produced the wrong pattern would have been abandoned before anyone drew it.
That is a demand this site does not meet and does not claim to. What it does instead is make the form itself checkable — count rather than admire, recover rather than assume, and state what a test would have to be. On a subject where the pictures have been doing the work of evidence for a century, that seems the right thing for a collection of pictures to do.