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

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.
12 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: A disturbance with a memory · Counting the spirals · A head is a set of points.

Every sweep this collection runs up the ladder assumes it can keep going. Ask for a finer rise and a finer lattice comes back, with larger contact numbers and a deeper front, and the only cost is a longer stem. That assumption is false, and this essay is about where it fails and what it costs.

The assumption is not idle. Three separate threads reach for finer rises when they run out of room: the ablation census wants larger counted numbers so that more offsets are chain members, the depth thread wants a wider span of contact scales, and every rung sweep wants rungs it has not already swept. All three end up asking the same question of the machinery — how fine can a lattice be and still be countable — and none of them had asked it explicitly.

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. 1 The ladder as this collection draws it, stopping at four thousandths — and this essay is about why it stops there.

Where it stops

Grow a stem at a stated rise, let it settle, and ask a counter what pair it returns. Down to about five thousandths the answer is what the ladder predicts: 3/5, then 5/8, then 8/13, each over its own range of rises.

The two spiral families a counter finds between 0.43 and 0.67 of the radius. 21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.
Fig. 2 What a counter does: it follows chains of near neighbours on the positions and reports how many run in each direction.

The ceiling is not sharp, and that is part of why it went unnoticed. At five thousandths the pair is clean; at four it is marginal; at three it is gone. A sweep stepping in thousandths crosses it in one step, and a sweep stepping in half-decades steps over it entirely.

Below about four thousandths it stops being a Fibonacci pair at all. At three thousandths the counter returns 2/13. At two thousandths, 5/22. At just over one thousandth, 2/17. These are not contact families; they are what a chain-following algorithm reports when the arrangement it is following has not finished sorting itself out.

The obvious explanation is that the stem is too short. A finer rise packs more organs into the same vertical distance, so a fixed node count covers less of the settling. Lengthening the stem should fix it.

It does not. At three thousandths with 1,200 organs the counter returns 2/13; at two thousandths with 1,800 it returns 13/24; at just over one thousandth with 2,600 it returns 13/15. The pairs change and none of them becomes a contact family.

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. 3 Part of why a longer stem does not simply fix it: a count depends on where up the arrangement it is taken, so a stem settled at one end and not the other answers differently in different bands.
How the largest gap behaves as the head fills. The rational angle's gap grows by a factor of 3.8 over this range; the golden angle's stays within 1.30. This is the claim about 137.5° that survives measurement.
Fig. 4 And the general shape of the problem: what an arrangement looks like partway through filling is not a coarser version of what it looks like when full.

The pairs are not random either, which is the part that makes the failure hard to notice. 13/24 and 13/15 contain Fibonacci numbers; 2/13 looks like a jugate pair. A reader shown one of these without the rise attached would have no reason to doubt it, and a sweep that reported it would produce a figure that looks exactly like every other figure on the site.

What that is, and what it is not

It is not a bug in the counter. The same routine returns the right pair at every rise above the ceiling, on stems of the same construction, and it is the routine every count on this site is made with. It is also the routine checked against the positions rather than trusted, on arrangements written down from formulas where the right answer is known in advance. A counter that passes those and fails here is not returning a wrong answer about a lattice; it is returning a right answer about something that is not one.

The two spiral families a counter finds between 0.68 and 0.92 of the radius34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.34 and 55 spiralscounted, not assumed
Fig. 5 The same counter at a coarser rise, where the chains it follows are the contact families and the pair it returns is the lattice’s own.

Nor is it obviously a failure of the rule. The placement rule has no scale in it that would break at four thousandths; it minimises a sum of inverse powers of distance whatever the rise. What changes is how long the arrangement takes to stop rearranging, and that is a property of the settling rather than of the rule.

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. 6 What settling looks like when it works: the divergences converge and stop moving, which is the state every count on this site assumes.
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. 7 And the structure being converged onto, which gets finer as the rise falls and takes correspondingly longer to resolve.

The most likely reading — and it is a reading, not a measurement — is that the number of organs needed to settle grows faster than linearly in the fineness, so that each halving of the rise costs more than a doubling of the stem. If that is right the ceiling is economic rather than physical, and it moves with the compute available rather than sitting at a particular rise.

How nearly each angle is a simple fraction of a turn. A dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.008; the rational angle reaches exactly zero.
Fig. 8 Why finer might be genuinely harder rather than merely longer: the divergences a fine lattice has to separate are closer together in the rationals.
The gaps close faster than the dips narrow. For each Fibonacci fraction, the distance to the nearest other rational with a denominator of 60 or less, and the half-width of its own dip at the smallest head that resolves it. The gaps fall from 0.763° at 3/8 to 0.0735° at 34/89; the dips stay between 0.0077° and 0.0155°. The dips never touch — the closest they come is a factor of 10 — so what stops the measurement is not the dips overlapping but the background between them ceasing to be flat. The clear offsets available fall from 72 to 53.
Fig. 9 The crowding itself, family by family, which is the arithmetic behind that reading.

Testing that reading is a day of compute rather than an idea: grow one rise at four stem lengths and see whether the pair converges to a contact family or to something else. It has not been done, and until it is, the ceiling is a measurement of this collection’s budget as much as of the subject.

What it costs

Three sweeps in this collection are bounded by it, and none of them said so before now.

The corner sweep. Whether the corner is the contact scale or any memory at all is separated by moving the contact numbers, and the plan asked for three to thirty-four. What is reachable is three to thirteen — enough to show the corner is not fixed, not enough to say what it is a function of.

The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not.
Fig. 10 The sweep that ran into it: three rises, three shapes, and no fourth panel because the fourth rise does not settle.
Where the deeper rule changes hands, and where it never does. One row per rise, with the contact numbers on the left and what the comparison does on the right. At the coarsest the deeper rule wins at both ends and loses in the middle; at the middle rise it wins throughout; at the finest it crosses once, from losing to winning. The contact scale runs over a factor of 2.6 across these three rises and the behaviour is not a translation of one curve — it is three different curves. That refutes a corner fixed at a short correlation, and it does not by itself establish one that tracks the contacts.
Fig. 11 The same result reduced, with the reachable span of contact numbers printed on it.

The ablation census. The reading about which family lost a member can only be asked at offsets that are multiples of a contact number, and the cheapest way to get more such rows is a finer rung with larger numbers. The ceiling caps how far that goes.

The offsets where the removed organ belonged to one family. Each row is a stem that never repaired, with the family of the organ that was taken and the family that survived. An organ five places back on a stem counted at 5 and 8 spirals lies on the tip's five-chain, so the question can be asked there; an organ four places back lies on neither chain and it cannot. Of 30 wrecked offsets in the census, 9 remove a member of exactly one family and 21 remove a member of neither. On every one of the 9 the family that lost a member is the family left standing, which is the opposite of what the reading predicted.
Fig. 12 The nine rows that reading currently has, and the finer rungs that would add to them.

And the rung sweeps. Sweeping along a rung is how this collection separates quantities that the counted pair holds constant. Rungs get narrower in rise as they get finer, so a fixed sweep step resolves fewer of them, and past the ceiling there are no settled rungs to sweep at all.

The narrowing is worth a number. The 5/8 rung runs from eighteen thousandths to seven, a span of eleven thousandths that a step of one thousandth divides into twelve. The next rung down is roughly half as wide in rise, so the same step gives six samples, and the one after that three. Two rungs below where the sweeps currently run, a rung sweep stops being a sweep and becomes three points — before the settling ceiling is reached at all. The two limits bite at different depths and the sampling one bites first.

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.
Fig. 13 One rung swept at a thousandth. A rung two rungs finer would need a step several times smaller and stems several times longer.
How many offsets wreck, along the 5/8 rung. The count of offsets that never repair, at each rise on one rung. It runs from 1 at the coarse end to 5 at the fine end, while a counter returns 5 and 8 spirals at every one of them. The front — the run of recent organs at which a removal is felt at all — deepens as the rise falls, so there are simply more places a cut can land and fail to heal. That is the mechanism under the grid: the offsets that appear at the fine end are the ones beyond the smaller contact number, and those are the ones that keep the larger family.
Fig. 14 And the quantity that makes finer rungs worth reaching: the front deepens, which is what puts more offsets in reach of a cut.

Four things it is worth checking before believing

That the settling test is the right one. A stem counts as settled when the wander of its divergences over the last stretch falls below a threshold. A different threshold would move the ceiling, and a much looser one would let through arrangements that are not lattices.

What the finer grid does to the rises already published. The two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the earlier work because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.
Fig. 15 The instrument’s own resolution, which is a separate limit from the stem’s and is easy to confuse with it.

That the failure is not the azimuth grid. The rule is evaluated over a finite set of candidate azimuths, and a finer lattice needs a finer grid to resolve its divergence. If the grid were the binding constraint the symptom would be the same, and the fix would be entirely different — a finer grid rather than a longer stem. This is the one alternative that can be ruled out cheaply, because the grid step is a stated number and the divergences a fine lattice has to separate are computable in advance: at the ceiling the grid is still an order finer than the spacing it needs to resolve. That does not prove the settling reading, but it removes the competitor that would have been easiest to fix.

The 13/21 rung, at two azimuth grids. Five 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.
Fig. 16 The check that the grid is fine enough for the claim being made on it.

That it is not a property of one seed. The pairs reported above come from one run each, and an arrangement that has not settled is exactly the kind of thing that differs between seeds.

And that the counter is being asked in the right band. A count depends on where up the stem it is taken, and a stem that is settled at the tip and not at the base will answer differently depending on the band.

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. 17 The same count taken as a function of the divergence, which is the check that a returned pair belongs to the arrangement rather than to the reading.
A count of m and n pins the divergence to 221°/mn. Each dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 2/3 leaves 38.8° open; 34/55 leaves 0.118°. The line is 221°/mn, taken from the three highest pairs and drawn back through the rest.
Fig. 18 And what a count is worth once it is trusted, which is what all of this is protecting.

Two of those four are checks this collection has already built and one is not. Recording which is which is the point of the section.

The one that is not is the seed check, and it is the cheapest of the four. A single run at a rise below the ceiling could be unlucky; four runs at four seeds would say whether the returned pair is stable nonsense or unstable nonsense, and those are different diagnoses. Stable nonsense across seeds would point at the counter or the grid — something deterministic reading the arrangement wrongly. Unstable nonsense would point at the settling, which is the reading assumed here. That the assumed reading has not been separated from its alternative is exactly the kind of thing this section exists to record.

There is a fourth cost, and it is larger than the other three, because it is not about a sweep that had to stop early but about what all of these stems stand in for.

The arrangements this subject is known for are past the ceiling. A sunflower head is commonly counted at 34 and 55 spirals, and a large one at 55 and 89; a pine cone at 8 and 13; a pineapple at 8, 13 and 21. What settles reliably here is 3/5, then 5/8, then 8/13, and 8/13 is the finest of them. The lattices these runs can be trusted on are the coarse end of the range a hand lens meets, and every claim made here about the ladder is a claim verified on its first three rungs.

That does not make the claims wrong and it is not an argument for stopping. The mechanism under test is a placement rule, the rule is identical at every rise, and a result holding across three rungs on two branches is a result about the rule rather than about a rung. It does change what can be said without a hedge attached. This rule produces the observed families is supported. This rule produces the families observed on a sunflower is not, because no stem in this collection has ever been settled at a sunflower’s rise.

The distinction has teeth wherever a claim depends on the numbers being large. The share of offsets that are chain members falls as the two counted numbers grow apart; the front deepens; rungs narrow in rise. Each of those is a trend read across three points and extrapolated to arrangements four rungs finer. An extrapolation is a respectable thing to publish and an indefensible thing to publish silently — and a stated ceiling is what makes the difference visible, because it says exactly where the measurements stop and the extrapolation starts.

Why write down a ceiling

Because a bound that nobody has stated gets crossed silently. A sweep that asks for a rise below the ceiling does not fail loudly: it returns a pair, the pair looks like a number, and every downstream figure and claim inherits an arrangement that is not a lattice.

This collection has a habit that would have caught it and did not apply here. Every generator asserts what is true of its own arguments, so a figure drawn at a rise below the ceiling would fail if it asserted that its pair was a contact pair — and most of them do not, because most of them were written for rises where the question does not arise. The fix is one line in the right places: assert the pair against the contact geometry rather than accepting whatever the counter returns. That is now done in the sweep this essay came out of, which refuses to draw a panel whose rises do not each carry a genuine contact pair, and it is not done anywhere else.

This is the same shape as the collection’s other instrument results — the rung that was not the instrument, the window that was not the neighbourhood — and it belongs with them. An instrument has a range, the range is a fact about the instrument rather than about the subject, and the honest thing is to publish it before somebody quotes a number from outside it.

Where each kind's lattice gives way. The largest amplitude at which every run still has a lattice, and the scatter it produces there. The amplitudes are incomparable — field 0.015 (fraction of the barrier), jostle 1 (degrees of azimuth), placement 0.8 (degrees of azimuth) — and the scatters agree to 19%. The boundary belongs to the pattern rather than to the disturbance: a lattice fails at about a degree and a half of scatter, and which of three mechanisms produced it does not move where.
Fig. 19 The tolerances the rest of the collection works inside, which are the same kind of fact and are stated in the same way.
The order of the angles carries the count. Three stems, each held at a fixed rise so the pattern sits on one rung of the ladder. At a rise of 0.032 the positions count 3 and 5 spirals and the angles peak at 3; At a rise of 0.013 the positions count 5 and 8 spirals and the angles peak at 5; At a rise of 0.005 the positions count 8 and 13 spirals and the angles peak at 8. Each panel marks the peak and its multiples; the pale strip is what an uncorrelated sequence of this length gives.
Fig. 20 And one of the readings that depends on a settled arrangement, which is what a silent failure would corrupt.
Every open question here needs under 28 specimens. The sample size at which each comparison reaches 80 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.
Fig. 21 And the form the same caution takes when a claim is to be made about plants rather than runs.
The band that never heals is what two fixed edges leave over. Each row is one rung. A stem is grown to 400 organs, one organ is removed, and the stem is followed for 300 more. A filled mark is an offset the stem recovers from, with the number of organs it took printed above it; an open ring is an offset it never recovers from. At the 3/5 rung, a front of 5, every offset heals. At the 5/8 rung, a front of 8, the offsets 4, 5 never heal. At the 8/13 rung, a front of 13, the offsets 4, 6, 7, 8, 9 never heal. What is the same at every rung is the two edges: the first three offsets heal, and so do the last few of the front. What is left in the middle is the band, and it widens because the front does.
Fig. 22 The coarsest version of the point: where a stem sits on the ladder decides what can be measured on it at all.

Where this leaves it

The ladder this collection draws runs from a third of a turn down to four thousandths, and every result on it is a result about that range. Whether the same things happen at 13/21 and 21/34 — the pairs real plants most often show — is not answered here and is not currently answerable with this machinery.

Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.
Fig. 23 The pairs a real head most often shows, which sit above this ceiling rather than below it — the ladder’s fine end is where the subject is, and where the machinery is not.
A count of m and n pins the divergence to 223°/mn. Each dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 3/4 leaves 20.7° open; 18/29 leaves 0.426°. The line is 223°/mn, taken from the three highest pairs and drawn back through the rest.
Fig. 24 And the other branch, which has the same problem at the same rises.

That is an uncomfortable place for a collection about phyllotaxis to be, and saying so is better than the alternative, which is a set of sweeps that stop where they stop for reasons nobody wrote down.

It also suggests where the next piece of machinery should go. Every result here is bounded by how long an arrangement takes to settle, and nothing in the collection measures that directly — settling is tested for, not timed. A measurement of organs to settle against rise would convert this ceiling from a place where sweeps stop into a number that can be planned around, and it would say whether the barrier is a budget or a wall.

Where this goes next

The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not. The corner moves with the rise The corner was either the contact scale or simply any memory at all, and nothing in the thread had ever varied the rise — the one knob that moves the contact numbers while leaving the rule, the amplitude and the run length alone. Swept over it, the comparison does not keep its shape. Every rung has a window it can be read in. For each rung of the ladder, the range of lag windows in which the pair can be read: at least three times the smaller parastichy number, so the main comb has three teeth inside the window, and less than three times the larger, so the larger number cannot itself be a candidate spacing. The band is [15, 24) at 5/8, [24, 39) at 8/13, [39, 63) at 13/21. It is non-empty at every rung, because it is empty only when the larger number is no larger than the smaller. The line at 30 is the window the earlier work used everywhere: it sits inside the 8/13 band and outside the 13/21 one, which is what was mistaken for a ceiling. The rung was not the instrument The earlier work said the pair readout has a ceiling one rung above where it works, that this is arithmetic rather than statistics, and that no amount of stem fixes it. The arithmetic is right and gives a band of lag windows that is never empty; what was actually stopping the reading was an eight-node seed and a grid of 384 azimuths. The two spiral families a counter finds between 0.68 and 0.92 of the radius. 21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle. 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.

What links here

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

Shares its objects with

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

  • What a count cannot decide — both name counting blind, claim testing, contact network, control, honest limits, lattice, measurement, negative result, parastichy, parastichy pair, rise, rung, underdetermination
  • One offset, two answers — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung, underdetermination
  • One rise per rung is a sample — both name counting blind, claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, rise, rung, underdetermination
  • The band was not the sampling — both name counting blind, claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • The shortest hop was a coin flip — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung, underdetermination
  • Two accounts of one number — both name counting blind, claim testing, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung, underdetermination

Named objects

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

Counting blindClaim testingContact networkControlHonest limitsInstrument ceilingLatticeMeasurementNegative resultParastichyParastichy pairThe placement ruleRiseRungUnderdetermination