Divergence angle — where it appears
A head is a set of points
The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work.
A stem is a cylinder
The sunflower is the photograph, and it is the hard case. Nearly all real phyllotaxis happens on a stem, where the geometry is a lattice on a cylinder with two parameters — and where the spiral counts, which on a disc change with radius, are the same the whole way up.
Fibonacci is a branch, not a law
Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.
The angle is an output
137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart.
Why the average cell has six sides
Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.
A ring cannot make a spiral
The peaks on a Turing ring do not all appear at once — there is a first and a second. But which two lead is decided by the starting disorder, so the angle between them comes out at 177°, then 47°, then 109°, then 151°. A divergence angle is a relationship that repeats, and this one does not.
Counting the spirals
Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by machinery that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.
Counting up the stem
The same counting machinery, pointed at a stem instead of a seed head, returns one answer three times where the head returned three answers. That contrast is a measurement rather than a preference, and it is the one the whole cylindrical argument rests on.
Packing, measured four ways
The claim is that the golden angle packs best. It is measurable, and the measurement gives three different winners on three criteria — all near 137.5° and none of them it. That does not make the claim wrong; it makes the usual statement of it wrong.
The bifurcation diagram
Sweep the one parameter of the rule and the settled angle traces a diagram — a broad golden branch, a transition, and a two-whorl regime at exactly half a turn. The famous constant is one branch of it, which is a more useful thing to know than the constant.
The claim that survives
Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.
The counts change with radius
The same head gives 13 and 21 near the centre, 21 and 34 further out, 34 and 55 beyond that, and 55 and 89 at the rim. The transitions are at computable radii, and a photograph captioned with one pair is a statement about one annulus rather than about a flower.
The gap that grows
A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of nearly four between two hundred primordia and sixteen hundred. An irrational one does not. That is the statement about the golden angle that survives measurement.
Two numbers out of the points
A seed head's divergence angle can be recovered from its spiral counts only to within an interval, because a range of angles gives the same counts. On a stem the counts come with lengths attached, two measurements pin two unknowns, and the lattice comes back to the last digit it was built with.
Recovering the angle from the counts
Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates.
Where the model stops
Below a growth parameter of about 0.18 this implementation does not converge — the settled angle wanders over a hundred degrees however long the run. That is the range where the literature says the interesting behaviour lives, and it is worth a figure rather than a quietly chosen axis.
A disc is a cylinder
Vogel's seed head makes the rise fall as one over radius squared, so a disc is not one lattice but a family of them. Feed that into the cylinder's ladder and it predicts where a sunflower's spiral counts change — with nothing fitted, and against a counter that never sees either model.
How often is it Fibonacci
The claim that plant spirals come in consecutive Fibonacci numbers is stated as a near-universal. Asked of the geometry, the answer collapses with scale — at a coarse rise 67% of divergences give Fibonacci pairs, and at a fine one 15%, with whorled and unnamed pairs taking the rest.
A cone has a rise that falls
A stem holds one parastichy pair for ever and a seed head changes its pair with radius. A cone does both — it is a cylinder whose rise falls as one over the distance from the apex, and the same blind counter that finds one answer up a stem finds four up a cone.
The angle is not the object
Every popular account of phyllotaxis is organised around a number. After a phase spent on stems, forks and frequencies, the number looks like the wrong thing to organise an account around — it is the limit of one path through a branching structure, it is at no fork, and a plant that has it got there by not jumping.
What a count is worth
A reported parastichy pair pins the divergence angle to a band 221°/mn wide — so 2 and 3 says almost nothing and 34 and 55 fixes it to a tenth of a degree. Every rung of the ladder is worth a factor of φ², and the counting radius this collection has asked published counts for since its foundation adds ten per cent.
How many plants would it take
Fourteen specimens separate the geometry's Fibonacci share from a coin weighted to a half. Four separate it from what a grown history gives. One fir cone measured at three rings settles whether its ladder is spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.
The survey this site cannot do
Four phases of asking for a dataset, and it is still not here. What this phase can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all.
Noise is not a slow rate
A stem seeded on the Lucas branch keeps it below ninety nodes per rung and abandons it above — which invites the objection that a real apex's fluctuations would knock it off regardless. Measured across a hundred and sixty runs of two independent kinds of noise, one escapes, at the amplitude where the pattern is already coming apart.
Two degrees of scatter
A lattice tolerates about two degrees of wander in its divergence angle, and two kinds of noise sharing no code agree on the number to within a third of a degree. It is not a constant: carried finer, the same stem survives 0.8°, and the tolerance tracks the band of angles that produce its pair at all.
Where the noise gets in
Ninety runs of noise applied after the rule has chosen, and not one changes branch. Fifty-six of noise applied to the choice itself, and one does. Only a disturbance upstream of the decision can restructure which nodes are neighbours of which — which is what a branch is.
The exponent that barely matters
A code comment on this site claimed since its foundation phase that the repulsion's falloff exponent hardly changes the answer. It could not be tested, because the function that would have taken it never passed one down. Tested at last, it is true on a disc — by three and a half degrees across a sixteenfold range — and on a stem it decides whether there is a pattern at all.
A window that makes a pattern
A rule whose energy has no well-defined minimum produces a clean 8/13 lattice at 137.62°, with half a degree of scatter, when its neighbourhood is cut at three node spacings. Let it see twelve and the pattern is gone. Every simulation of this kind truncates something, and truncation manufactures exactly the result it is used to look for.
Named alongside it
The objects these essays reach for when they reach for this one.
RiseFibonacciBranchCylinderLadderParastichyMeristemNoiseParastichy pairRateRepulsionSurvey