Phyllotaxis
the spirals are counted, not admired
Stems and cones

How long a stem takes to settle

Every result here is grown on a stem that has settled onto a lattice, and settling has always been tested for and never timed. Timed, it takes between nothing and two hundred and ninety organs — against the four hundred every ablation run grows before it cuts anything, and the nine hundred the noise runs carry.
13 min read 27 figures Counted, not admiredState the claim, state the test

Worth reading first: A head is a set of points · Counting the spirals · The rung was not the instrument.

Every stem in this collection is grown, checked for having settled onto a lattice, and then experimented on. Four hundred organs before an ablation run cuts anything; nine hundred before a noise run reads a scatter. Those numbers were chosen once, early, and they have never been justified by anything except that the stems they produce pass the settling test.

Which means settling has been tested for and never timed. This essay times it.

The distinction is not pedantic. A test returns a yes, and a yes carries no information about how much margin there was: a stem that settled at organ eight and one that settled at organ three hundred and ninety both pass a test applied at four hundred, and only one of them says the budget was well chosen.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 1 One row per rise, one mark per starting angle, at the organ from which the divergence stops moving.

The instrument

A stem is settled from the first organ after which every later divergence stays within a degree and a half of the run’s own final value. That is not a new test: it is the recovery test the ablation work uses on a cut stem, pointed at growth from a seed rather than at repair from a hole.

Two runs of the same rule from unrelated starting angles. Both settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.
Fig. 2 What settling looks like on a disc, where this collection first drew it.

Two details in it are load-bearing.

The first is a hold: sixty organs, so that the last handful of a run cannot be called a settling. Without one, the test finds the final few angles of any run and reports it as settled four organs before the horizon, which is what a first version did.

The second is a chirality fold. The rule has no handedness — nothing in it distinguishes a turn one way from the same turn the other — so a run settling on 220.9° has settled on the same arrangement as one settling on 139.1°. Every angle is folded to within half a turn before any mean is taken. Without folding, a mirror-image run reports a wandering mean and a settled stem is called unsettled.

The wrecked stem is the lattice it was cut from, wound the other way. The divergence of each organ placed after two were removed from a settled stem at a rise of 0.032, over the 180 organs following the cut. The upper line is the divergence the undisturbed stem holds, 139.6875°; the lower is 220.3125°, which is 360° minus it and therefore the same lattice with the opposite handedness. The sequence is thrown by the cut, wanders for a few dozen organs, and settles on the second line to four decimal places — 220.3125° against 220.3125° — where it stays. A counter shown the positions afterwards returns 3 and 5, the pair the stem was cut from. Nothing about the pattern has been lost; its chirality has been reversed, which at this rung a single removal cannot do.
Fig. 3 The two representations the fold identifies, which are one arrangement traversed in two directions.

The control that makes it a measurement

Started from the golden angle, a stem at a rise of 0.020 is settled at organ zero.

That is not a fast stem. It is a stem started on its own answer, so it never leaves it, and a measurement of approach that begins at the destination measures nothing at all. The number is real and it is a property of the starting angle rather than of the rise.

What the model settles on, against how fast the meristem grows. A broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 9 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.
Fig. 4 Where the answer comes from, which is the reason starting on it is a control failure rather than a result.

So the table is grown from nine starting angles spanning 40° to 180°, and the golden one is reported beside the others rather than as the result. Two rises out of eight have a zero in them, and both zeros belong to the golden seed.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 5 The same table at a shorter run length, where the zeros sit in the same two rows.

What the nine starting angles are for

Nine angles from 40° to 180°, evenly spread, and the spread rather than the count is the design.

A divergence is defined up to a mirror, so 180° covers the whole range of distinguishable starting lattices; anything above it is a reflection of something below. Nine points across that range put a seed roughly every twenty degrees, which is coarse compared with the width of a basin near an attractor and fine compared with the gaps between attractors.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.
Fig. 6 The range the seeds span, and what a counter makes of arrangements built at angles across it.

That coarseness has a consequence worth stating: the shares reported here — how many of nine settle at each rise — are estimates of a basin’s size with an error bar of about a ninth. Reading a fall from seven of nine to five of nine as a precise measurement would be over-reading. Reading a fall from seven of nine to one of nine as a real change would not.

The angle between the first two peaks, for six starting disorders. The same equations, the same ring, six different starting perturbations: 177°, 47°, 109°, 151°, 47°, 133°. A spiral needs the angle between successive elements to be the same one each time, and a stationary ring does not supply that.
Fig. 7 The same use of several seeds elsewhere in this collection, where a spread across starting conditions is what makes a count a measurement.

The answer

Where a stem settles at all it does so within 290 organs, and most of them inside fifty.

The slowest settling anywhere in the table is 290 organs, at the finest rise where anything settles at all and from a starting angle a hundred and thirty degrees away from where it ends up. The next slowest is 159. The median is under twenty.

How many starting angles reach a lattice at all. The share of nine starting angles, spread from 40 to 180 degrees, whose stem settles onto a lattice at each rise. It falls from 7 of 9 at the coarse rises to 1 at the finest, and the runs that fail do not fail by running out of organs — grown three times as long they fail identically. So what closes at the fine end is the set of arrangements a stem can fall into rather than the time it has to find one, which is a different kind of limit and a more interesting one.
Fig. 8 How many starting angles reach a lattice at each rise, which is the other quantity the table carries.

Against four hundred organs before a cut and nine hundred before a noise reading, that is a factor of one and a half at the very worst and a factor of forty typically. Not one row in the table needs the length it was given.

Take away the organ four places back, and the next one goes into the hole. The last 30 organs of a stem at a rise of 0.013, unrolled. The open circle is the organ removed — four places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 164.1° apart, against a local spacing of 41°, and the vacancy itself is 172.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.
Fig. 9 The kind of run the length was chosen for, which grows four hundred organs before it removes one.

What that settles and what it does not

It settles a worry that has been implicit in this collection for four rounds: that results at the fine end of the ladder might be about stems that had not finished settling. They are not. At every rise where a stem settles, it has settled hundreds of organs before anything is done to it.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 10 The ladder the worry was about, whose fine end is where the run lengths would have bitten if they were going to.

It does not settle the reverse worry, which is subtler and which the table sharpens rather than resolves. A stem that has settled by the site’s test is a stem whose divergence has stopped moving to within a degree and a half. Whether some slower process is still going on underneath that tolerance — a drift of hundredths of a degree over thousands of organs — is not something this test can see, and it is exactly the sort of thing that would matter to a measurement of where a divergence sits inside a rung.

The divergence slides along the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 11 The quantity such a slow drift would move, whose whole variation across a rung is about a degree.

The check for that is a longer run and a tighter tolerance, and it is not the same experiment as this one. What can be said from here is that at the site’s own tolerance, nothing is still moving.

Where the time goes

The distribution is not smooth and it is worth looking at rather than summarising.

Most starting angles settle within twenty organs. A few take fifty to a hundred and twenty. One takes 290. And a large number never settle at all — at the coarse rises two of nine, at the fine ones eight of nine.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 12 The distribution as it is: a cluster near zero, a tail, and a great many runs that never arrive.

That shape is the signature of a system with several attractors and a boundary between them. A starting angle inside a basin goes to its attractor quickly; one near a boundary takes longer; one in no basin at all wanders indefinitely. This collection has drawn those attractors from the other direction — sweeping the rule’s own parameter rather than its starting angle — and the branches there are the destinations these runs are falling into.

What the model settles on, against how fast the meristem grows. A broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 9 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.
Fig. 13 The several attractors, drawn as this collection’s central figure, where the branches are the destinations a run can reach.

The destinations bear that out. Across the whole table the settled values cluster on a handful of angles — near 79°, 99°, 101°, 137°, 139°, 151° — which are the golden and Lucas branches and a few others, rather than a continuum.

six limit divergences, all of them 137.5078 over a whole number. The golden angle is the k = 1 member of a family. Real bijugate plants — teasel, Cephalaria — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.
Fig. 14 The angles those clusters sit at, which are limits of the branch structure rather than arbitrary values.

The number the collection should have had

There is a version of this measurement that would have been worth having from the day the first stem was grown here, and it is not the one above.

The useful quantity is not “how long does a stem take to settle” but “how long does a stem take to settle from a starting angle that is not the answer”, which is what the nine-seed design measures and what a single golden-seeded run cannot. The distinction is the difference between a transient and a control, and it is the same distinction the ablation work draws between a cut run and its own control.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 15 The other place the same distinction is made, where a measurement is only a measurement against a matched control.

Had it been measured early, two things would have been different. The run lengths would have been chosen from a number rather than from caution, which would have made every sweep in this collection cheaper — four hundred organs where fifty would do is a factor of eight on the most expensive tables here.

And the shape of the distribution would have been in view. A quantity that is either small or infinite, with very little in between, is a strong hint about the system producing it, and it is the hint the next essay follows.

Placements that went to a different minimum, per thousand. The 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.
Fig. 16 Basins as this collection has drawn them elsewhere, which is what a bimodal settling time is a measurement of.

The one row that is genuinely slow

Two hundred and ninety organs, at a rise of 0.003, from a starting angle of 150°, settling on 151.3°. It is the only row in the table that comes near the run lengths this collection uses, and it deserves a paragraph because it is also the only settled row at that rise.

Everything about it is unusual. The rise is below where any branch of the ladder reaches, so the arrangement it settles onto is not on the golden or Lucas branch; its divergence of 151.3° is not near a limit angle either. Eight of the nine starting angles at that rise never settle at all.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 17 The row in question, at the bottom of the table, with its single mark far to the right of every other.

So the slowest settling in the collection is a lone survivor in a region where almost nothing survives, on an arrangement nothing else here reaches. That is a curiosity rather than a caution: it does not say that fine stems settle slowly, because at that rise there is no population of fine stems to be slow. It says that one starting angle out of nine found something to settle onto, and took a while.

five limit divergences, all of them 137.5078 over a whole number. The golden angle is the k = 1 member of a family. Real bijugate plants — teasel, Cephalaria — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.
Fig. 18 The angles the branches reach, none of which is where that row ended up.

Whether that arrangement is a lattice in any useful sense — whether a counter returns a contact pair for it, whether it has parastichies a person would draw — is not something this measurement asks. The settling test asks only whether the divergence stopped moving, and a divergence can stop moving on a great many things.

A whorl and a spiral, from one lattice at two divergences. At 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.
Fig. 19 An arrangement whose divergence is perfectly steady and which is not a spiral lattice, as a reminder of what steadiness alone establishes.

What a settling time is not

It is not a plastochron, and the two are easy to confuse. The plastochron is the interval between organs, which here is the rise; the settling time is how many organs pass before the arrangement stops changing. One is a parameter and the other is a response.

The rule, 26 steps in, at a growth of 0.40The next primordium goes where the repulsion is least — the marked minimum at 216°. Nothing in the rule refers to any particular angle.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
Fig. 20 The rule the response belongs to, whose parameter is the spacing rather than the settling.

It is also not a measure of stability. A stem that settles in eight organs is not more stable than one that settles in eighty; both end up on the same lattice and both stay there. What the time measures is distance from the destination in the space the rule moves through, which depends on the starting angle at least as much as on the rise.

Two stems at 0.75° of scatter, one angle at a time. The divergence of each node, with the slow climb of the ladder removed. Both stems scatter by about 52.26° and a botanist would report them identically. The jostled one wanders in runs — its lag-one correlation is 0.70 — and the one with placement noise alternates about the mean at 0.21, because a displacement of one node enters two consecutive divergences with opposite signs.
Fig. 21 Two settled stems, whose subsequent behaviour is the same whatever their transients were.

And it is not, on this evidence, a useful predictor of anything else. The rows that settle slowly are not the rows with unusual scatters, or unusual fronts, or unusual anything — they are rows that started further away. That is worth stating because a transient is exactly the sort of quantity a reader expects to correlate with something, and this collection has been caught before treating a quantity as informative because it was measurable.

The tolerance falls from 4.1° to 0.8° up the ladder. Each dot is one configuration: a stem carried down to the stated rise, swept over the whole amplitude range, with the largest divergence scatter any surviving run showed. It ends on 5/8 at the coarse end and 8/13 at the fine one. The open marks are the width of the band of divergence angles that gives each pair at all — a quantity from a different calculation entirely, moving the same way.
Fig. 22 The quantity that does tighten down the ladder, which is not the settling time.

Reading it against the ablation runs

The reason to care is the run lengths, so it is worth putting the two side by side properly.

An ablation run grows four hundred organs, removes one, and continues for three hundred more. The settling table says the first four hundred are between one and a half and forty times more than the arrangement needs. The three hundred after the cut are a different question — that is how long a wrecked stem takes to reach its new motif, and it is measured separately by the recovery test, which finds a repair within tens of organs where there is one at all.

The next organ moves for the last 8, and for no others. One row per organ removed, counted back from the tip of a stem at a rise of 0.013 whose counted pair is 5 and 8. Removing any of the last 8 moves the next organ by 4.9° to 164.1°; removing an older one moves it by at most 0.70°, which is under the azimuth grid. The boundary is at 8, and 8 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.
Fig. 23 The runs the lengths belong to, whose first four hundred organs exist to guarantee a settled starting arrangement.

So both halves are generous, and generously is the right way to be wrong about a run length. The cost is compute rather than correctness: shorter runs would have made every table here cheaper and no table here different.

The block a wrecked stem settles into is the count it was cut from. A 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.
Fig. 24 One of the tables the length paid for, which would look identical grown from a shorter start.

The one place it might matter is at the fine end, where fewer starting angles settle at all. A run that has not settled is not a stem with a longer transient; it is a stem doing something else, and what that something else is turns out to be the more interesting half of this measurement.

What is left

The tolerance. Everything here is measured at a degree and a half because that is the figure the ablation work uses, and it was chosen for a different question — how close a repaired stem has to be to its control before the repair is believed. Whether a tenth of a degree gives the same table is a re-run rather than a new experiment, and it is the obvious next thing.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 25 The table the tighter tolerance would be run against, whose fast rows are the ones most likely to move.

There is also a version of the measurement that would say something the count of nine cannot. Sweeping the starting angle finely — a degree at a time rather than twenty — would map the basins directly rather than sampling them, and the settling time as a function of the starting angle is a curve with structure in it: flat inside a basin, rising toward a boundary, undefined outside. That is a hundred and eighty runs per rise instead of nine, which is affordable at these lengths, and it would turn a share into a boundary.

Placements that went to a different minimum, per thousand. The 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" — 0.2°, 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.
Fig. 26 The basins such a sweep would map, drawn here at three runs rather than at the resolution the question wants.

And the disc. Everything above is a cylinder, where the rise is a parameter. On a disc the rise falls as the radius grows, so a head is a stem climbing its own ladder as it fills, and what settling means there is not obviously the same question. This collection has drawn that climb and has never asked whether the arrangement keeps up with it.

The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.
Fig. 27 The climb a head makes as it fills, which is the version of this question nobody here has asked.

Where this goes next

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing. A wall and not a budget Below a rise of about 0.005 this collection's stems stop settling onto a lattice, and the limit has been written up four times without anybody asking which kind of limit it is. Grown three times as long, the table is identical row for row: not one stem that failed to settle succeeds. The floor is a wall. The rule, 26 steps in, at a growth of 0.40. The next primordium goes where the repulsion is least — the marked minimum at 216°. Nothing in the rule refers to any particular angle. 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. Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820. A stem too fine to settle Below a rise of about four thousandths the counter stops returning contact families and starts returning pairs like 2/13 and 13/24. Lengthening the stem does not fix it. That is a ceiling on every sweep this collection runs up the ladder, and it has never been written down.

What links here

Computed from the collection, not written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

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

Named objects

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

AttractorBasinCounting blindControlConvergenceDivergence angleHandednessHonest limitsInitial conditionLatticeMeasurementRiseSettling timeTransient