Phyllotaxis
the spirals are counted, not admired
What a plant might be doing

Two-ranked, by two different routes

The rule produces a two-ranked stem at a coarse rise, where 180° is the only thing available, and that has been in the bifurcation diagram since the beginning. It also produces one at a fine rise, at a rise whose own answer is the golden angle, if a single organ is removed. The diagram cannot show the second, and the reason it cannot is how it is drawn.
19 min read 13 figures Form is not mechanismThe angle is an outputState the claim, state the test

Worth reading first: The angle is an output · The bifurcation diagram · The organ that was taken away.

The central picture of this whole subject is a bifurcation diagram: run the placement rule at a parameter, let it settle, plot the divergence it settles on, and step the parameter. Out of it comes the golden angle over a range, other angles outside that range, and the branches whose existence is the reason the Fibonacci numbers turn up in plants at all.

It is drawn by running to convergence from an arbitrary start at each parameter. That sentence is the subject of this essay.

The stem version of the sweep is the same construction with the rise in place of the growth parameter: fix a rise, grow a stem, record the divergence it settles on, step the rise. It is how every rung on this site’s ladder was established.

Two-ranked, the way a sweep shows it

At a coarse rise the rule has nothing else to do. With the organs far apart in height there is no useful room around the circumference, and the arrangement that minimises the repulsion is alternate: one organ, then one opposite it, then one above the first.

Measured on this site’s own rule at a rise of 0.20, the settled divergence is 180.00° — not near it, at it, to the resolution of the azimuth grid.

What the model settles on, against how fast the meristem growsA broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 16 of 40 converged settings land within 4° of the golden angle; 17 land more than 20° away.1001251501750.2000.4000.6000.80011.20growth parameter Gangle the model settles on (°)goldentwo-whorlfilled: converged · hollow: still wanderingone rule, one knob
Fig. 1 The diagram, over the parameter the disc version of the rule is swept in: how fast the elements move apart between one arrival and the next. There is a stretch where the answer is the golden angle and settings outside it where the answer is something else. Every point is one run, started from an arbitrary condition and left to settle.

That is two-ranked phyllotaxis as a parameter regime: it is what the rule does when the parameter is coarse, it is reached from any start, and it is a property of the rise rather than of the history.

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.120130140150050100stepdivergence angle produced at that step (°)golden anglegrowth 0.42settled spread 0.00°
Fig. 2 One of the runs behind one of those points. The pattern is started arbitrarily and the divergence walks to its attractor over a few dozen organs, which is what “run to convergence” means and why one point on the diagram is one number.

What “run to convergence” leaves out

The construction deserves a paragraph of its own, because the objection to it is not that it is careless. It is the standard way to draw such a picture and it is the right way to answer the question it asks.

At each value of the parameter the rule is started from some condition — a seed lattice, or a scattering, or the end state of the previous parameter — and iterated until the divergence stops changing. The number recorded is where it stopped. Sweep the parameter and the recorded numbers trace out branches, and the branches are the object: where they meet, where they split, and where the golden angle appears without being asked for.

What the picture answers is what does a run at this parameter settle on. What it does not answer is what can a run at this parameter settle on, and the two differ exactly when a parameter has more than one attractor.

There is a second, subtler omission in the same construction. Continuation — the usual economy of starting each parameter from the previous one’s answer — makes the diagram a statement about a path through parameter space. A stem whose rise falls slowly is following such a path, so continuation is the right choice for modelling a growing shoot; it is the wrong choice for asking what a fixed parameter permits, because it visits one basin all the way along.

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. 3 The two readings side by side: a pattern grown with a declining rise against the static ladder computed at each rise on its own. They agree closely, which is a result about this rule — and it is a comparison of two paths rather than a survey of what each rise allows.

Two-ranked, the way an ablation shows it

At a rise of 0.005 the same rule settles on 137.84° and holds it, with a pair of 8/13 and a scatter of a fifth of a degree. Nothing about that rise is two-ranked and nothing on the diagram suggests it could be.

Remove one organ from the middle of the front and the stem never returns to 137.84°. What it holds instead is a block of eight angles repeating exactly, an average divergence of 182.8°, and a precession of 22.7° per block: a two-ranked arrangement with a slow twist, at a parameter whose diagram entry says 137.8°.

The block a wrecked stem settles into is the count it was cut fromA stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 8/13 rung the sequence is eight angles long and advances 22.5° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 16 rows.8/13 rungblock of 8+22.5° a blockand again182.8° on average300 organs after the cut · 1536 azimuthsgenerated from a stated rule, not drawn to look right
Fig. 4 The second arrangement, at the fine rise. Eight angles repeating, five of them near the settled divergence and three of them large, and the effect of the whole block is an organ every 182.8°. The parameter has not changed; the history has.

One rung coarser, at 0.013, the same experiment gives a block of five and an effective divergence of 208.8°, precessing the other way. Both are two-ranked arrangements with a twist, and both sit at parameters whose diagram entries are spiral.

The block a wrecked stem settles into is the count it was cut fromA stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block; At the 8/13 rung the sequence is eight angles long and advances 22.5° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 and 16 rows.5/8 rungblock of 5-36.1° a blockand again208.8° on average8/13 rungblock of 8+22.5° a blockand again182.8° on average300 organs after the cut · 1536 azimuthsgenerated from a stated rule, not drawn to look right
Fig. 5 Both of them. The block is the smaller parastichy number of the lattice that was cut and the precession is one part in twice the block, so each is a two-jugate arrangement — an object the diagram has no way to plot, because the diagram’s vertical axis is one divergence per parameter.

So the rule is bistable at fixed parameter, and one of its two attractors is absent from every picture ever drawn of it.

Why the diagram cannot show it

The omission is not an oversight in this site’s version of the figure. It is what the construction does.

A bifurcation diagram of this kind samples the parameter, and at each parameter it runs one trajectory to convergence and records where it ended. That reports one attractor per parameter by construction — whichever one the chosen start falls into. A parameter with two attractors and two basins produces exactly the same picture as a parameter with one, because only one number is plotted.

The heads three settings of the one knob produceG=0.3 → 139.2° · G=0.62 → 143.0° · G=1.1 → 180.0°. The model was not told any of these angles.G = 0.30139.2° — goldenG = 0.62143.0° — otherG = 1.10180.0° — whorled (half)one rule, three growth rates, repulsion as 1/d^3the angle is an output
Fig. 6 The regimes the diagram is usually read as showing. Each is a statement about what a run settles on, and each is measured the same way: start somewhere, wait, record. Nothing in the method asks what else the rule could have settled on at the same parameter.

The repair is standard and expensive: run many initial conditions per parameter and plot the set of attractors, with the share of starts that reach each. That turns a curve into a picture of basins, and it is a different figure from the one everybody draws.

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.0501000.2000.4000.6000.800growth parameter Gspread of the last 30 steps (°) — 0 means settledfilled dark: convergedthe usable range is stated, not implied
Fig. 7 The model at parameters where its answer is already known to be delicate. The honest reading of any of these is a statement about a run rather than about the rule — which is the same distinction this essay is making, at a place where the site had already noticed it.

There is a cheaper diagnostic, and this site stumbled into it. The second attractor is reachable by discretisation. On a 384-azimuth grid — the grid most flat runs here use — the noiseless rule at a rise of 0.005 has no golden lattice to perturb: it falls into the same periodic orbit an ablation drives it to, at a divergence of 185.8°, before anything is removed. The runs that do not fall into it are the noisy ones, and the noise is what keeps them off it.

The 13/21 rung, at two azimuth gridsFive stems at each of three disturbances, all at a rise of 0.0019, read through a lag window of 45 from a seed of 40 nodes. A filled mark is a stem that returned 13/21, which is what the position counter finds in it; an open mark is a refusal. The only difference between the two rows is how many azimuths the placement rule samples when it takes its minimum: 384, which every run on this site has used since the earlier work, against 1152. At the coarse grid the rule locks onto its own sample points and the readout refuses 14 of 15 stems; at the fine one it reads all 15. The ceiling was a parameter of the program.disturbance0.080.130.18384 azimuthsstep 0.94°0 of 15 read 13/21scatter 46.8°1152 azimuthsstep 0.31°15 of 15 read 13/21scatter 0.4°rise 0.0019 · seed 40 nodesgenerated from a stated rule, not drawn to look right
Fig. 8 The grid finding, which was made while chasing something else. At the coarse grid the rule with no disturbance at all lands in the wrecked arrangement; with a disturbance it does not. A quantised azimuth is a disturbance of its own, and here it is one large enough to decide which attractor a run reaches.

That is a strange sentence — noise keeping a pattern on its lattice rather than off it — and it is exactly what a second attractor with a nearby basin looks like from inside a run that never asked the question.

Are the two two-ranked things the same thing?

They are not, and the differences are measurable rather than interpretive.

The regime is exact and the attractor is not. At a coarse rise the settled divergence is 180.00°. The wrecked stems average 182.8° and 208.8°, and neither is a fixed divergence at all — each is a block of angles, most of them nowhere near 180°.

The regime has no precession and the attractor does. A two-ranked stem at a coarse rise stays put: organ i and organ i+2 are in the same row for ever. The wrecked stems turn by 22.7° or −36.1° per block, so their rows are helical.

The regime is reached from anywhere and the attractor is reached from one place. Every start at a coarse rise gives 180°. At a fine rise, the wrecked arrangement is reached only by removing an organ from a particular part of the front — three offsets of thirteen at one rung, two of eight at another — or by running on a grid coarse enough to act as a deletion.

A whorl and a spiral, from one lattice at two divergencesAt 120° the nodes fall on 3 rows and the two counts share a factor: the elements of each turn are level with one another. At 137.51° they never are.120° — a third of a turn3 and 6 — whorled137.51°3 and 5 — Fibonaccirise 0.030 in both panelsthe counts decide, not the eye
Fig. 9 The distinction the counts alone cannot make. Two-ranked, two-jugate and spiral arrangements overlap in what they report to a counter, and telling them apart needs the symmetry. Both objects in this essay are two-ranked by symmetry; only one of them is a fixed divergence.

What it does to a claim this site has made

The bifurcation diagram has been used here for a specific argument: that the golden angle is an output of a rule that contains no reference to it, over a range of one parameter. That argument is unaffected. The rule does settle on 137.5° from ordinary starts over that range, and the branches it produces are where the Fibonacci ladder comes from.

What is affected is a weaker sentence that has been standing beside it — that the diagram shows what the rule does at each parameter. It shows what the rule does from one start. At a fine rise it also does something else, and the something else is a recognisable phyllotactic arrangement rather than a numerical artefact.

What the divergence does while the pattern climbsThe stem produces a sequence rather than a constant. Over the second half of the run it stays within 2.8° of 137.51°, and the vertical marks are where the counted pair changed — the wander is largest around them.136137138100200300nodedivergence from the node before (°)137.51°353 nodes at 87 per rungspread 2.81° over the second half
Fig. 10 A run that crosses the whole range with a declining rise, which is how a real shoot would meet these parameters. It starts two-ranked because the rise is coarse, and finds the spiral branch as the rise falls — the ordinary route, and the one the diagram is a summary of.

The stronger version of the claim survives and is worth stating in its stronger form: the golden angle is an attractor of this rule, and it is not the only one. A model whose answer depends on its history is a better model of a plant, not a worse one — a meristem has a history, and the one thing every real shoot has that a bifurcation diagram does not is a beginning.

The Fibonacci share, read two waysAt a rise of 0.005 only 11.0 per cent of divergences give a pair of consecutive Fibonacci numbers. Allow a common factor to be divided out first and it is 49.2 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 0.005, strictly11.0%rise 0.005, up to jugacy49.2%share of divergences3 rises · 3600 divergences each11.0% → 49.2% at rise 0.005
Fig. 11 How much of the divergence space belongs to arrangements of this family at the two rises. The wrecked stems land somewhere ordinary rather than somewhere exotic, which is another way of saying the second attractor was always available and nothing had looked.

What a survey of the basins would cost

The measurement this essay asks for is worth pricing, because it is the obvious next thing and it is not free.

At one rise, the rule has to be run from many starting conditions — not many seeds of the same disturbance, but genuinely different arrangements: a seed lattice at each of a range of divergences, a scattering, an arrangement with a hole in it. Each run has to be grown far enough to settle, which at a fine rise is a few hundred organs, and its end state has to be classified: the same settled divergence, or a block of angles, and if a block, which one.

The classification is the part with judgement in it. Two blocks of eight angles whose motifs are rotations of one another are the same orbit seen from different places; two whose precessions differ by a degree are probably the same orbit measured on different stems, and two whose block lengths differ are not the same object at all. The machinery for that exists here — an exact period search with no tolerance to tune, and a precession that comes out of the motif’s own sum — so the decision is stated rather than eyeballed.

The cost is then a few hundred runs per rise, against the handful a bifurcation diagram needs. For a picture of one rung that is an afternoon; for a sweep across the ladder it is a different kind of undertaking, and it is why the diagram everybody draws is the one everybody draws.

What becomes of a seeded branch at 65 nodes per rungEach bar is 3 runs at one amplitude, divided by what the blind counter found at the top of the stem. With no noise this rate ends on the Lucas branch. Placement noise displaces the node after the rule has chosen; field noise perturbs the energy the rule chooses over. Across 48 runs, 1 reached the Fibonacci branch with the lattice intact.placement noisedegrees off the minimum00.250.50.7511.251.52field noisefraction of the barrier00.00250.0050.00750.010.01250.0150.02kept its branchchanged branchanother pairno latticeseeded 40 nodes of Lucas lattice · 3 runs per amplitude1 escape in 48 runs
Fig. 12 The nearest thing this site has to such a survey: many runs at many disturbances, each classified by what it ended up on. The method is the same one a basin survey would need, applied to a different question — and it is what makes the cost above an estimate rather than a guess.

What this does not say

It does not say the diagram is wrong. Every point on it is a correct statement about a run. The complaint is about what a summary omits, and the omission is structural rather than an error.

It does not say plants sit in the second basin. Nothing here measures a plant. Two-ranked phyllotaxis is common in real plants and this rule reaches it in two different ways, and the essay’s whole content is that those two ways are different from each other — not that either one is what a grass does.

Placements that went to a different minimum, per thousandThe rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.field 0.0052.10.70° of scatterfield 0.00750.01.00° of scatterfield 0.010.01.12° of scatterjostle 0.41.10.87° of scatterjostle 0.81.11.12° of scatterplacement 0.40.00.94° of scatterplacement 0.80.01.42° of scatter3 runs each · a basin change is half a local spacingplacement noise: zero by construction
Fig. 13 The basins measured the only way this site can measure them: by disturbing a settled pattern and asking whether it changes arrangement. Noise of a size that leaves the lattice intact does not move a stem between attractors, in either direction, which is what makes the two of them attractors rather than transients.

And it does not say how large the basins are. That is the measurement this essay is asking for and not making. It would need many starts per parameter, a classification of what each one settled into, and a rule for when two blocks are the same arrangement — all of which exist here in pieces and none of which has been run as a sweep.

The claim in its careful form

It is worth writing the surviving statement out once, because two of its words are doing work that the loose version leaves out.

At a fixed rise on the 8/13 rung, the placement rule has at least two attractors: the settled golden-angle lattice, and a periodic orbit of eight angles whose average is 182.8°. A single deletion from the middle of the front moves a stem from the first to the second, and no disturbance tested moves it back.

At least two, because nothing here has surveyed the space of arrangements; a third attractor would not contradict a word of it. Tested, because the disturbances tried are a jostle at 0.1° and 0.4° and a deletion at every offset — which is a small sample of what could be done to a stem.

The same sentence at the 5/8 rung has five angles and 208.8° in it, and that is what makes the pair of them evidence about the rule rather than about one rise. A property found at one parameter is a curiosity; the same property at two parameters, with the numbers changing in the way the geometry says they should, is a property of the rule.

The check

The claims that can fail are the numerical ones, and they are asserted where the stems are grown.

The two-ranked regime must be at 180° to within the azimuth grid at a coarse rise — a claim about a number the rule produces without being told about it. The wrecked stems must have a spread of tens of degrees over their last stretch, so that “a block of angles” cannot be quietly satisfied by a stem that has drifted a little. And the block must precess, at both rungs, by one part in twice its own length, which is what makes the arrangement two-jugate rather than merely periodic.

The grid claim is checked separately and is the most fragile of them, because it is a claim about a coarse grid producing something a fine one does not. It is stated as it was found: at 1,536 and 4,608 azimuths the front, the displacements and the recoveries agree, and at 384 the undisturbed rule is already in the other attractor.

Where this goes next

5/8 rungblock of 5-36.1° a blockand again208.8° on average8/13 rungblock of 8+22.5° a blockand again182.8° on average300 organs after the cut · 1536 azimuthsgenerated from a stated rule, not drawn to look right The block is the count it was cut from A stem that never recovers from an ablation settles into a repeating block of eight angles precessing by 22.7°. Eight was also the smaller parastichy number of the lattice that was cut, which left two possibilities and no way to choose between them. Cut a stem one rung coarser and the block is five. 05e+51e+61.5e+60100200300angle around the boundary (°)repulsion from what is theregrowth 0.40 · 14 elements in playthe minimum is where the next one goes 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. 100200300050100organs placed after the removaldivergence, in degreescycle of 8rise 0.005 · cut 4 backgenerated from a stated rule, not drawn to look right The pattern the cut leaves behind A stem that never recovers from a removal is not disordered. Its divergences settle into a cycle of eight angles and repeat it exactly for the rest of the run, and a counter reading the positions calls the result 8/16 — a two-jugate lattice. The rule has a second attractor at the same growth parameter, and an ablation is how you get to it.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • A front with no middle — both name ablation, artefact, discretisation, divergence angle, ensemble, equilibrium, honest limits, measurement, the placement rule, rise
  • A rule that cannot heal a hole — both name ablation, artefact, divergence angle, ensemble, equilibrium, honest limits, measurement, the placement rule, rise
  • The organ that guards the second slot — both name ablation, artefact, discretisation, divergence angle, equilibrium, honest limits, measurement, the placement rule, rise
  • What a sample grid decides — both name artefact, discretisation, divergence angle, ensemble, equilibrium, honest limits, measurement, the placement rule, rise
  • The grid was in the number — both name artefact, discretisation, divergence angle, ensemble, honest limits, measurement, the placement rule, rise
  • The response with a hole in it — both name ablation, artefact, discretisation, divergence angle, honest limits, measurement, the placement rule, rise

Named objects

A flat tag is an object no other essay names yet.

AblationArtefactAttractorBifurcationDiscretisationDivergence angleEnsembleEquilibriumHonest limitsInitial conditionMeasurementMechanismThe placement ruleRiseWhorl