Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

Continuity from a coarse start

At a fine rise, one divergence in seven gives a Fibonacci pair. Grow a stem from a coarse start at a divergence nobody chose, down to that same rise, and sixteen runs out of sixteen end on 8/13. The expansion phase's interpretation was that continuity does the work; this is the measurement it never had.
15 min read 6 figures State the claim, state the testThe angle is an output

The expansion phase ended one of its essays with a sentence that has been the load-bearing interpretation of this whole site ever since, and it had nothing behind it.

How often is it Fibonacci swept the divergence angle at a fixed rise and counted what came out. At a rise of 0.12, 67.5% of divergences give a pair of consecutive Fibonacci numbers. At 0.008, 14.7%. The share falls as the pattern gets finer, which is the opposite of what a story about the geometry preferring Fibonacci numbers would need — and the essay concluded that “what makes plants Fibonacci is continuity from a coarse start rather than generosity in the geometry”.

That is a claim about histories. The site had no histories. It had a family of static lattices and a sweep over one parameter, and no way to start coarse and continue.

Now it has one.

Fibonacci at a rise of 4.8e-3, asked two waysChoose a divergence at random and a static lattice at this rise gives a consecutive Fibonacci pair 10.8 per cent of the time. Start coarse at a divergence nobody chose, grow the stem down to the same rise, and it is 100 per cent of 16 runs. The geometry is not generous; continuity is.grown from a coarse start100.0%divergence chosen at random10.8%share ending on a consecutive Fibonacci pair16 grown runs, starting divergences from 67° to 299°every one of them ended on 8/1367 nodes per rung · rise 0.4 → 4.8e-3100% against 10.8%
Fig. 1 The same question at the same final rise, asked two ways. On the left, sixteen stems grown from a coarse start at divergences nobody chose. On the right, the share of divergences that give a Fibonacci pair if you simply pick one at that rise.

The experiment

Sixteen runs. Each begins at a rise of 0.4 — the coarse end, where the dominant pair is 1 and 2 and there is barely a pattern — with a starting divergence drawn from a reproducible generator, anywhere between 60° and 300°. The rise then falls to 0.0048 at 67 nodes per rung, with every node after the seed placed by the repulsion rule.

At the end, a counter shown only the coordinates reports the pair.

Sixteen runs out of sixteen end on 8/13. Not sixteen out of sixteen Fibonacci, with a spread of pairs — the same pair, every time, from starting divergences spread over 231 degrees of the circle.

The static census at that same final rise gives a Fibonacci pair 10.8% of the time — a little lower than the 14.7% the expansion phase quoted, because that number was taken at a rise of 0.008 and this run ends finer, at 0.0048.

What the two numbers are

They are shares of the same outcome at the same rise, taken over different things, and the difference between them is the whole result.

The static number is a share over divergences. Fix the rise. Pick an angle. Ask what a periodic lattice at that angle and that rise has as its dominant pair. One in seven gives consecutive Fibonacci numbers.

The grown number is a share over histories. Fix the starting rise, the ending rise and the rate. Pick a starting angle. Grow. Ask what the pattern has at the end. All of them give consecutive Fibonacci numbers, and all of them give the same ones.

Neither is a wrong measurement of the other. They are measurements of two different quantities, and the sentence the expansion phase wrote is precisely the claim that the second is the relevant one for plants.

Why the answer is so clean

One hundred per cent of sixteen is a suspiciously tidy result, and it is worth saying why it is not surprising once the two previous essays are in place.

At a rise of 0.4 there is almost no pattern: the dominant pair is 1 and 2 for nearly every divergence, and the arrangement carries very little information about the angle it was built from. The repulsion rule, given a coarse arrangement and a few placements, does what the bifurcation diagram says it does — it settles on the golden angle, from starting conditions that know nothing about it.

Then the tracking result takes over: once on the Fibonacci branch, the pattern walks the rungs on schedule at any rate, so it arrives at the final rise on 8/13 rather than anywhere else.

And the branch result explains why nothing escapes: a branch has to be entered with room to spare to be held, and a start at a rise of 0.4 is above every fork, so there is no branch to be on yet. Everything funnels.

So the three results compose, and the composition is the mechanism the expansion phase’s sentence was gesturing at: coarse means no branch; the rule then chooses the default branch; and the default branch is walked down without lag.

What the coarse start has to be

The word “coarse” is doing real work and the experiment can be run to find out how much.

Seeded at a rise of 0.4 with an arbitrary angle, every run ends Fibonacci. Seeded at a rise of 0.05 — well down the ladder, with forty nodes of established lattice — the runs no longer agree: about four in five end Fibonacci and one in five ends on the Lucas ladder, at 7/11 and 99.5°.

That is the same phenomenon the previous essay measured from the other side. A pattern established below a fork can hold a branch; a pattern that begins above all the forks cannot be on one.

So the claim is not “plants are Fibonacci because they grow”. It is “plants are Fibonacci because they begin before there is anything to be” — and a plant that somehow began part-way down, already committed, would keep whatever it was committed to.

Which is a sharper statement than the one it replaces, and it has a consequence: any organ whose pattern is established at a fine rise from the start — a lateral bud on a mature stem, say, or an inflorescence whose first primordia form on an apex already narrow — is exactly the kind of thing that could be non-Fibonacci, and should be more often than a shoot that starts from a seedling.

The branch is kept below 85 nodes per rung and lost above 92Each row is one rate. The Lucas seed keeps its ladder at 46, 58, 65, 75, 85 nodes per rung and abandons it at 92, 108, 131. The golden seed ends on 8/13 at every one of them.nodes per rungseeded Lucasseeded golden467/11 — kept8/13 — Fibonacci587/11 — kept8/13 — Fibonacci657/11 — kept8/13 — Fibonacci757/11 — kept8/13 — Fibonacci857/11 — kept8/13 — Fibonacci928/13 — gone to Fibonacci8/13 — Fibonacci1088/13 — gone to Fibonacci8/13 — Fibonacci1318/13 — gone to Fibonacci8/13 — Fibonacciseeded with 40 nodes at a rise of 0.12threshold between 85 and 92
Fig. 2 The other half of the mechanism. A branch entered with room to spare survives a hurried descent and is lost in a leisurely one; a start above all the forks has no branch to survive.
Two runs of the same rule from unrelated starting anglesBoth settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.1201301401500255075100stepdivergence angle produced at that step (°)golden anglegrowth 0.40settled spread 0.00°
Fig. 3 The convergence the sixteen runs inherit. Two runs of the disc rule from unrelated starting angles reach the same divergence within a few dozen steps, and a coarse cylinder converges faster still.

What the runs look like individually

The aggregate hides the interesting part, which is that the sixteen histories are not the same history with different labels.

Each starts at its own divergence — 67.4, 67.4, 100.8, 113.9, 130.9, 146.9, 164.3, 174.4, 191.7, 202.7, 205.0, 235.1, 241.8, 274.1, 288.0 and 298.7 degrees, two of which happen to land within a tenth of a degree of one another — and the arrangement at the coarse end genuinely reflects it: a run started at 192° places its first nodes nearly opposite one another, and one started at 101° does not. For the first few dozen placements the runs are visibly different objects.

What happens next is that the rule reorganises them. At a rise of 0.4 there are only two or three nodes within a circumference of the tip, so the arrangement is almost entirely determined by the last few placements, and the last few placements are where the rule’s preference lives. By the time the rise has fallen a rung the runs have converged, and after that they are walking the same ladder at the same rate and stay together.

That is a convergence rather than a coincidence, and it is the same convergence the settling figure shows on a disc: two runs from unrelated starting angles reaching the same divergence within a few dozen steps. The cylinder version is faster, because a coarse cylinder holds fewer nodes in play than a growing disc does.

The consequence worth stating is that the starting divergence is not preserved for long enough to matter, and that is exactly what makes the census unanimous. If the rule kept its start, sixteen starts would give sixteen answers and the grown census would look like the static one.

What sixteen runs can and cannot support

The number is 16 and not 1,600, and that is a real limitation with a boring cause: each run is a few hundred placements against a growing neighbourhood, and the whole census takes about a second. A thousand runs would take a minute, which is affordable, and would not change the answer — the interesting question is not the precision of a share that is already unanimous.

What sixteen runs cannot do is find a rare exception. If one history in fifty ends somewhere else, this census will usually miss it, and reporting “100%” would then be reporting the sample rather than the population. So the honest statement is sixteen of sixteen, written that way, and not “always”.

The starting angles are drawn from a small deterministic generator rather than from a random source, which is a choice this site makes everywhere for the same reason: a census should be a measurement and not a mood, and a figure that redraws differently on every build is a figure nobody can check.

What this does not explain

Three things, and the first two are the ones a reader is most likely to take away wrongly.

It does not explain why the golden angle. The rule settles there, and why a repulsion rule has that attractor is a separate question this site has answered as far as it can — the angle is where the rule goes, over a range of one parameter, and it is not singled out by any packing criterion. Continuity explains why a plant ends up on the branch it started near; it does not explain where the branch is.

It does not explain the counts. 8 and 13 at that final rise is the rung, and the rung is set by the rise, which is set by the organ’s geometry — which is what the cone essays are about. Continuity explains why the pair is Fibonacci; the rise explains which Fibonacci.

And it does not describe a plant. A repulsion rule on a cylinder with a declining rise is a model of form. The same patterns come out of magnetised droplets with no biology in them, and what a mechanism would have to show is a much longer list than any of this touches. What has been shown is that a history produces a different distribution of outcomes from a sweep, which is a statement about the two calculations and about which of them the popular claim was implicitly about.

What a growing stem counts, against what a static lattice wouldThe steps are the blind counter's answer as the stem grows at 65 nodes per rung; the dashed verticals are the rises at which the static ladder changes. 55 of 56 counting windows agree, and the mean gap between where a transition happened and where the ladder puts it is 0.011 of a rung.11.502falling rise, as −log₁₀which rung the pattern is on1/22/33/55/88/1365 nodes per rung · 259 nodes55 of 56 windows agree
Fig. 4 The middle step of the funnel, at the rate these runs use. Once a pattern is on the Fibonacci branch it walks the rungs where a static lattice says it should, so the final pair is decided by the rise it stops at rather than by anything that happened on the way.
The Fibonacci share, read two waysAt a rise of 0.0048 only 10.8 per cent of divergences give a pair of consecutive Fibonacci numbers. Allow a common factor to be divided out first and it is 49.3 per cent. The whole of the increase is pairs of the form k and 2k — k copies of the bottom rung — which is a generous reading that describes no plant.rise 0.12, strictly67.5%rise 0.12, up to jugacy72.4%rise 0.03, strictly31.1%rise 0.03, up to jugacy55.7%rise 0.0048, strictly10.8%rise 0.0048, up to jugacy49.3%share of divergences3 rises · 3600 divergences each10.8% → 49.3% at rise 0.0048
Fig. 5 The static census this is measured against, at three rises and read two ways. Every share here is over divergences at a fixed rise, which is the quantity the grown census replaces rather than corrects.

The shape of the claim, and why it is not circular

There is a way of reading this result that makes it trivial and it is worth heading off, because the arrangement invites it.

The reading is: the rule converges on the golden angle, so of course everything grown by the rule ends up Fibonacci. Nothing was learned; the answer was in the rule.

Two things are wrong with that.

The first is that the rule’s attractor was measured, not assumed, and it was measured on a different geometry. The bifurcation diagram sweeps the growth parameter of the disc model and finds a golden branch, a transition, and a two-whorl regime — the golden angle is one branch of a diagram rather than the model’s output, and outside the golden range the same rule settles elsewhere. The cylinder version here has a different parameter and could have behaved differently.

The second, and the important one, is that the comparison is not between the rule and nothing. It is between a history under the rule and a sweep over divergences, and both are things one might mean by “how often is phyllotaxis Fibonacci”. The popular claim — that Fibonacci counts are what nature prefers — is ambiguous between them, and the two give 100% and 10.8%. Making the ambiguity explicit and putting a number on each side is the work.

Which is the general form of what this site does. The claim was not false and it was not true; it was a sentence with two readings and no measurement attached to either. Attaching one to each is what turns it into a statement that could have come out differently.

What the phase’s three results say together

This is the last essay of the group and the three results are worth putting in one place, because separately each is smaller than the set.

A growing pattern is at equilibrium. Its transitions land where a static ladder puts them, to better than a tenth of a rung, over a fifteenfold range of rate. So three phases of static calculation on this site are statements about growing organs and not only about periodic lattices.

But which ladder it is on is a matter of history and rate. A seeded branch survives a hurried descent and is abandoned in a leisurely one, with a threshold at about ninety nodes per rung. The Lucas branch is metastable rather than stable, and a plant is on it only if it entered early and moved fast.

And a coarse start funnels everything onto the default. Sixteen histories from divergences spread over most of the circle all end on the same pair, where a sweep over divergences at the same rise gives that pair one time in seven.

The composition of those three is an answer to the question the site’s wrong field has been circling since the foundation phase. Fibonacci counts are not what the geometry hands over — that was measured, and it is 14.7%. They are what a history hands over, provided the history begins before there is a pattern and the rule it follows has the golden angle as its attractor.

Both of those conditions are substantive, both are stated, and neither is a property of the number 137.5°.

Why the static share moves with the rise

A detail worth pausing on, because it affects how the two numbers should be read against each other.

The static Fibonacci share is not a constant. It is 67.5% at a rise of 0.12, 31.1% at 0.03, 14.7% at 0.008 and 10.8% at 0.0048. It falls monotonically, and it keeps falling: there is no floor visible in the range the counter can reach.

The reason is that the interval of divergences giving any particular pair narrows as the pair gets deeper. At a coarse rise the pair 1/2 occupies most of the circle; at a fine one, 21 and 34 occupy a sliver, and every other sliver belongs to some other pair that nobody has a name for. The named sequences do not get more common as the pattern refines — they get rarer, and there are more unnamed alternatives to be rare among.

So the comparison in this essay is between a history’s outcome and a sweep’s outcome at one particular rise, and choosing a different final rise moves the sweep’s side and not the history’s. A stem grown from a coarse start ends Fibonacci whatever rise it stops at, because the branch is chosen early and the rungs are walked without lag. The sweep’s share, meanwhile, keeps shrinking.

That widens the gap rather than narrowing it, and it is the sharpest form of the whole argument: the finer the pattern, the less the geometry accounts for and the more the history does.

Where this implementation stops convergingBelow about G = 0.18 the settled angle wanders over 113° however long the run. That is the model's limit, not a fact about plants.0501000.2500.5000.7501growth parameter Gspread of the last 30 steps (°) — 0 means settledfilled dark: convergedthe usable range is stated, not implied
Fig. 6 The limit every result in this group is bounded by. A discrete rule on a sampled circle stops converging below a stated parameter, and what is claimed here is claimed about the range it can be run at.

What is still owed

The one thing this phase most wanted and did not get is unchanged: a real dataset.

Every frequency here is about the geometry or about a model. The grown census is sixteen runs of a rule, not sixteen plants. The threshold at ninety nodes per rung is a property of inverse-cube repulsion against a truncated neighbourhood. And the survey fields that would let any of this be compared with material — counting radius, element count, symmetry order — are exactly the ones the literature does not record.

The claim this essay closes was an interpretation for two phases and is now a measurement of a model. That is one step, and the next one is not a computation.

Where this goes next

Fibonacci59.6%Lucas20.8%whorled10.7%other8.9%5 distinct pairs over 1200 divergencesrise 0.100Fibonacci 59.6% 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. 10011012013014011.502falling rise, as −log₁₀divergence the stem is producing (°)137.51°, Fibonacci99.50°, Lucasseeded at 99.50°, rise 0.127/11 against 8/13 The rate decides the branch Forty nodes of Lucas lattice, carried down to the same fine rise twice. Hurried, the pattern holds the Lucas ladder through three more forks at 99.5°. Given room, it abandons it at the first fork it reaches and walks to 8/13 at 137.5°. Same seed, same rise, two ladders — with a threshold between them at about ninety nodes per rung. 5/107.9%5 × 1/23/67.9%3 × 1/22/46.1%2 × 1/24/85.5%4 × 1/27/145.2%7 × 1/26/122.8%6 × 1/28/160.1%8 × 1/2share of divergences at this riserise 0.008 · 3600 divergences35.4% share a factor What "whorled" was hiding The expansion phase's census put 35% of divergences in a bucket labelled whorled and moved on. Opened, every pair in it is k and 2k — the coarsest rung of the ladder, repeated k times — and reading the census up to jugacy takes the Fibonacci share from 14.7% to 50.1% without describing a single extra plant.