Phyllotaxis
the spirals are counted, not admired
Stems and cones

The ratio was the floor of a curve

The last phase left one number standing between a placement rule and a transported disturbance, measured at one rise, with the explanation that the geometry there happens to favour the larger parastichy number. Swept across two rungs the number is a U — a floor of about 0.79 two thirds of the way up a rung, climbing past 2.8 as a transition approaches — and the geometry is flat exactly where the curve is steepest.
21 min read 13 figures State the claim, state the testTwo routes, one numberCounted, not admired

Worth reading first: Errors that pass between organs · The sequence has a memory · A pattern with a rate.

When the previous phase finished retracting the comb, one number was left. Both a placement rule and a lattice with transported errors produce two combs in the divergence sequence, and both return the counted pair; what separated them was how the correlation divides between the two combs. The rule gave 0.65 and a distance-weighted transport gave 1.30.

That was measured at a rise of 0.005, and the phase plan said so:

The ratio 0.65 belongs to a rise of 0.005, where the 13-hop is shorter than the 8-hop by eight per cent. At a coarser rise the asymmetry has a different size and possibly a different sign, so a survey comparing a plant against “the rule’s ratio” needs a curve rather than a number, and nobody has swept it.

This essay sweeps it. There is a curve, it is the same curve on both rungs measured, and the reason offered above for expecting one is wrong.

Holding the disturbance fixed, which is not obvious

A sweep over the rise is a sweep over two things unless the disturbance is held at something that means the same at every rise.

Every earlier comparison used a three-entry lookup — a quarter of a degree at 0.005, three tenths at 0.013 — and a constant at every rise in between. A constant amplitude in degrees is a larger disturbance at a fine rise, because the organs are closer together: at 0.005 the neighbours are 25° apart and at 0.032 they are 65° apart, so the same quarter of a degree is two and a half times as large a fraction of the spacing.

So the disturbance here is a fixed share of the local spacing — a hundredth of √h, in degrees — which comes to 0.25° at 0.005 and reproduces the site’s own figure there to two decimal places.

A lattice or a wreck, with nothing in betweenEvery run in the sweep, at both kinds of noise. The line at 6° is where a run stops being counted as a lattice, and nothing lands near it: the intact runs reach 1.97° and the destroyed ones start at 8.06°, a factor of 4.1 away.00.50011.5001234567amplitude, by stepscatter, log₁₀ degrees6°intactno latticeplacementfield48 runs · both kindsan empty factor of 4.1 at the cut
Fig. 1 Why the amplitude has to be defined against something: the scatter a stem records is not the amplitude it was given, and the relation between them is a property of the rise as well as of the disturbance.

The curve

Fifteen rises across two rungs, five stems each, with the pair taken from the positions rather than from the angle readout — because near a transition the readout starts returning the next rung’s pair while the arrangement is still on this one, and a sweep that took its pair from the readout would compare two different quantities and call the difference a trend.

The ratio is a U across every rung, and its floor is the number that was reportedThe ratio of the second comb to the main comb, on five stems at each of 15 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.29, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.1200.2500.5000.7501where the rise sits in its rung — 0 at the next transition down, 1 at the last onesecond comb ÷ main comb5/8 rung8/13 runga transported disturbance, no rulethe floor, 0.795 stems a point · 1152 azimuthsgenerated from a stated rule, not drawn to look right
Fig. 2 The ratio against the ladder’s own coordinate for where a rise sits inside its rung: 0 at the transition the rise is falling into, 1 at the one it has come through. Two rungs, one shape.

Both rungs give a U. The floor is 0.71 on the 5/8 rung and 0.79 on the 8/13 rung, and on both it sits about two thirds of the way up the rung. Away from the floor the ratio climbs, and at the fine end of the coarser rung it reaches 2.8 — four times its own floor.

The previous phase’s rise of 0.005 sits at 0.66 of the way up the 8/13 rung. It measured the bottom of a curve nobody had swept, which is the third time this collection has found an extremum reported as a constant: the second moment against the divergence angle turned out to be a staircase whose 0.253 was a plateau, and the tolerance of a lattice to noise turned out to be a scale rather than a number.

The numbers, since the shape is the argument:

On the 5/8 rung, reading from its coarse end down towards the transition, the ratio runs 1.10, 1.02, 0.78, 0.72, 0.71, 1.26, 2.13, 2.80 at rises of 0.0175, 0.0165, 0.0155, 0.0145, 0.0125, 0.0105, 0.0090 and 0.0080.

On the 8/13 rung it runs 1.26, 1.06, 0.79, 0.82, 0.90, 0.98, 1.05 at rises of 0.0066, 0.0058, 0.0050, 0.0046, 0.0042, 0.0038 and 0.0034.

The two are not identical. The floor is a little lower on the coarser rung and the climb towards the fine end is far steeper there — a factor of four against a factor of 1.3. What they share is the shape and the place the floor sits, and whether the depth of the climb is a property of the rung or of how close to the boundary the sweep was able to get is not decided by two rungs.

The main comb is a hump, and the ratio is a quotient of two moving things

A ratio hides its numerator and denominator, and both of these move.

The main comb — the correlation at multiples of the smaller parastichy number — is weakest at both ends of a rung and strongest in the middle: 0.13, 0.14, 0.20, 0.24, 0.34, 0.28, 0.20, 0.15 across the 5/8 rung. That is the pattern being most itself in the middle of its rung and least itself near a transition, which is the same thing the mixture thread found when it discovered that a window straddling a boundary reads worse than a window inside one.

The second comb does the opposite at the fine end: 0.14, 0.15, 0.15, 0.17, 0.24, 0.35, 0.42, 0.41. It rises as the main comb falls, which is what the two-family account predicts and is stronger evidence for that account than the ratio alone.

So the U is not one quantity varying; it is two quantities crossing. That matters for anyone who wants to use the ratio on a plant, because a stem whose main comb is 0.15 is a stem whose readout is close to refusing altogether, and a ratio computed just above the refusal threshold is a ratio of two small numbers.

Why it climbs, which is arithmetic about the comb

The second comb is not a fixed set of lags. On a stem whose pair is m and n the main comb collects the multiples of m and the second collects the residue class containing n — so at the 5/8 rung it collects lags 3, 8, 13, 18, 23 and 28.

Thirteen is in that list, and thirteen is the larger member of the pair the stem is about to transition to.

As the rise falls towards the transition, the next rung’s family is already forming in the arrangement. The second comb picks it up, the main comb does not, and the ratio climbs. Near a boundary the ratio is not measuring the transmission of errors at all: it is measuring the transition.

The order of the angles carries the countThree 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.rise 0.032counted 3/5angles say 336912150.5rise 0.013counted 5/8angles say 55101520250.5rise 0.005counted 8/13angles say 8816240.5151015202530lag, in internodescorrelation between a divergence and the one that many internodes later3 runs per rise · 320 internodes eachthe counter is never shown a position
Fig. 3 The readout that the ratio is built out of, at three rises. What the second comb collects is a residue class, and which offsets fall in that class is decided by the smaller parastichy number rather than by anything about the disturbance.
What is visible in the outer part of a 4000-element organBoth surfaces have the same ladder in element number — the rise is 1/(2πi·flare) on a cone and 1/(4πi) on a disc, and c and the internode step both cancel. What differs is where the elements are. Counting outside 50 per cent of the extent, a cone shows 1 change and a disc 2, because half a cone's length holds half its elements and half a disc's radius holds three quarters of them.024680.2000.4000.6000.8001counting only outside this fraction of the organ's length or radiustransitions inside the counted partdisc: 2 beyond 50%cone: 1 beyond 50%flare 0.12 · 4000 elements1 against 2 in the outer 50%
Fig. 4 Where the transitions are. The ratio’s climb begins about a third of a rung from one and is steep by a sixth of a rung, which on a real shoot is a stretch of tens of internodes.
What a growing stem counts, against what a static lattice wouldThe steps are the blind counter's answer as the stem grows at 92 nodes per rung; the dashed verticals are the rises at which the static ladder changes. 82 of 83 counting windows agree, and the mean gap between where a transition happened and where the ladder puts it is 0.012 of a rung.11.502falling rise, as −log₁₀which rung the pattern is on1/22/33/55/88/1392 nodes per rung · 365 nodes82 of 83 windows agree
Fig. 5 A growing stem crossing the boundaries. The arrangement does not change pair at a point; it carries both families for a stretch, which is exactly the stretch in which the ratio is inflated.

What the blind readout says over the same sweep

The ratio here is computed at the pair the positions give, which is why the sweep can be taken at all. It is worth recording separately what the angle readout — the instrument that is shown a list of numbers and told nothing — makes of the same fifteen sets of stems, because the two disagree in a way that is itself informative.

On the 8/13 rung the readout returns 8/13 on all five stems at all seven rises, including the two closest to a transition. On the 5/8 rung it returns 5/8 on all five stems at exactly two of the eight rises — the two nearest the floor of the U — and refuses at the other six.

The refusals are not random. Four of them are at the coarse end, where the main comb is 0.13 to 0.24 and barely clears its threshold, and two are at the fine end, where the second comb has grown until the search’s ordering breaks down and the readout can no longer decide which spacing is the main one. So the instrument reports exactly where the ratio is near its floor and refuses where the ratio is inflated — which is a piece of good fortune rather than a design, and it is the strongest argument in this essay for using the readout’s refusal as a guard.

A plant whose stem gives a clean two-comb reading is, on this evidence, a plant whose rise is near the middle of its rung. That is a testable side-claim and it is the sort of thing this collection has been wrong about before, so it is offered as a pattern in fifteen measurements rather than as a rule.

The geometric explanation, refuted on its own ground

The plan’s reason for expecting a curve was the hop asymmetry. At a rise of 0.005 the 13-hop is 0.069 circumferences and the 8-hop 0.075, so the larger index offset is the geometrically nearer neighbour — and a d⁻³ interaction would weight them by the cube of that ratio, which is 0.80. Change the rise, change the asymmetry, change the ratio.

The asymmetry does change: across the two rungs it runs from 0.77 to 1.23, and it passes through one, so the two hops change places. But it does not change where the ratio changes.

Inside the 5/8 rung there are four rises whose hop asymmetry agrees to within a twentieth — 0.792, 0.793, 0.806 and 0.806 — and whose ratios are 0.71, 1.26, 2.13 and 2.80, a spread of a factor of 3.9. Inside the 8/13 rung there are five rises whose asymmetry spans 0.769 to 0.795 and whose ratios span a factor of 1.33.

Which offsets give short hops, at a rise of 0.005The two lowest points are at 8 and 13, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.0123102030index offsetmedian hop between node i and node i+m23300 nodes, 34 offsets triedshortest at 2 and 3
Fig. 6 The quantity that was supposed to explain the curve, at one rise. It is a fact about the lattice’s geometry, it varies smoothly and slowly with the rise, and the ratio does not follow it.

Several rises with the same asymmetry and different ratios is a falsification rather than a weak correlation. Whatever moves the ratio inside a rung, it is not how far apart the two contact neighbours are.

A stem gathers neighbours linearly; a growing disc barely gathers them at allOn a cylinder of circumference 1 with a rise of 0.005, the nodes within distance d number 2d/0.005 once d exceeds one turn — a fitted exponent of 1.011 and 400 per unit against the 400 the geometry fixes. In the disc model an element of age k sits at radius e^(0.4k), so each doubling of the distance adds the same two or three neighbours rather than twice as many.01234-0.50000.50011.50distance from the node, log₁₀neighbours, log₁₀a stemslope 1a disca logarithmrise 0.005 · 24000 nodes · meristem growth 0.4slope 1.011 against slope 1
Fig. 7 And the weighting that would follow from the geometry if it did explain the ratio. The transported disturbance is built with exactly this weighting, and it is why the rival’s number is where it is — but the rule’s is not a weighting at all.

What it costs the discriminator

The rival’s curve across a rung is flat, and it is flat for a reason that is worth stating rather than measuring: a kinematic lattice’s angle sequence does not contain the rise. The arrangement is organ i at i times the divergence with an inherited error, the divergence cancels when consecutive angles are differenced, and the errors know only the two offsets they are inherited at. Two stems on the same rung with different rises give the same sequence statistics.

So the comparison is between a flat line at 1.29 and a U with a floor at 0.79 — and the U crosses the line.

At the coarse end of the 8/13 rung, a twentieth of a rung from the transition, the rule gives 1.26 against the rival’s 1.29. At the fine end it gives 1.05. Three of the fifteen rises measured put the rule within a fifth of the rival’s value, and at those rises the quantity distinguishes nothing at all.

The two combs, in the proportions the rule gives themThe ratio of the second comb to the main one, for a kinematic lattice whose errors are inherited from its two contact neighbours, against how unevenly that inheritance is split. The horizontal line is where the placement rule's own stems sit, at 0.65. Weighted by distance — the coupling a d⁻³ interaction would give, which at this rise favours the 13-neighbour by 1.26 to one because the 13-hop is the shorter — the forgery sits at 1.46, well above the rule. It reaches the rule's value only at about 3 to one the other way, which is a factor of 4 against what distance supplies and in the opposite direction.0.4000.6000.80011.201.40-0.30100.1760.3010.4770.699how much more strongly the error is inherited from the 8-neighbour than from the 13-neighbourthe second comb's strength as a fraction of the main comb'sthe placement rule: 0.65equal combs1:21:11.5:12:13:15:1at 3:1 the ratio is 0.75kinematic lattice · 3 seeds a pointgenerated from a stated rule, not drawn to look right
Fig. 8 The discriminator as the previous phase left it: one number for the rule and another for the transport, a factor of two apart. The factor of two is real in the middle of a rung and gone at the ends of one.
Which arrangements carry a comb, and what each one reportsThe largest comb mean in five arrangements at a rise of 0.005, all read by the same instrument at the same length, with the sampling band of 0.073 marked. Only the first is a placement rule; the other four are kinematic lattices with no rule in them, differing from one another only in how their azimuth errors are structured. Independent errors and errors with a memory leave nothing to read. A repeating error puts up a comb and names a partner that is not the lattice's. Errors inherited from the contact neighbours reproduce both the comb and the pair.three sampling bandsthe placement rule0.6428/13independent errors0.031refusedan error with a memory0.014refusedan error that repeats0.4338/10, 8/12errors passed between neighbours0.5538/13one rule, four kinematic latticesgenerated from a stated rule, not drawn to look right
Fig. 9 The five arrangements the ratio was asked to separate. What this essay adds is that the separation depends on where the plant is between two transitions, which is not a property of the arrangement at all.

There is a compensation, and it is a real one. At the coarser rung the rival does not merely give a different ratio: it gives no main comb at all. A disturbance transported at offsets 5 and 8 puts almost nothing on the comb through the origin at spacing 5 — a mean of 0.033 against a sampling band of 0.10 — while the rule puts between 0.13 and 0.39 there at every rise on that rung. At the 5/8 rung the discriminator is not the ratio but the presence of the main comb, and that is a much cruder and much more robust test.

Two combs, at a rise of 0.013The autocorrelation of 760 divergence angles from one stem held at a rise of 0.013. The filled teeth are the lags at multiples of 5; the open teeth are the second comb, at the same spacing offset by 3. Reading the spacing off the first and the offset off the second gives the pair 5 and 8, which is what the position counter reports for the same stem — from angles alone, with no coordinate anywhere in the calculation.-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later510152025303813182328spacing 5 · offset 3pair 5/8 — counter says 5/8the shaded strip is the sampling bandone stem · 760 divergences · disturbance 0.3generated from a stated rule, not drawn to look right
Fig. 10 The two combs on the coarser rung, where the rule has both and a transported disturbance has only one. A ratio between a number and nothing is not a ratio, and the readout’s own refusal is what reports it.

What this does not do

It does not restore the comb as evidence of a rule. The previous phase’s retraction stands: a comb is evidence that something was transmitted between contact neighbours, and both accounts transmit. This essay is about the size of the one quantity that was left.

It does not explain why the floor is two thirds of the way up rather than halfway. The coordinate here is logarithmic in the rise, because the ladder is geometric, and on that coordinate the floor sits at 0.61 and 0.66 on the two rungs. A symmetric contamination from the two neighbouring rungs would put it at a half. The asymmetry is in the same direction on both rungs and is worth about a sixth of a rung, and nothing here says where it comes from.

It does not establish that the curve is universal. Two rungs is two, and both are in the middle of the ladder. The 3/5 rung is coarse enough that the readout’s own thresholds start to bite — the phase before last measured it refusing three runs in five there — and the 13/21 rung needs stems this sweep does not build. Whether the floor is 0.79 at every rung or drifts with the pair is unmeasured.

And the floor is not 0.65. That is the next essay’s subject and it is not a detail: the number changes because the azimuth grid the previous phase measured on was coarse enough to be a disturbance in its own right.

At 384 azimuths the ratio is 0.62; converged it is 0.82The comb ratio and the recorded divergence scatter at a rise of 0.005, against how finely the rule samples the circle when it takes its minimum. At 384 azimuths — the grid every flat run on this site uses, and the grid the previous phase's 0.65 was measured on — the step is 0.94°, which is larger than the 0.25° disturbance the stems carry. The quantisation is white noise, it dilutes both combs, and it does not dilute them equally. The ratio settles at 0.82 from 1152 azimuths up, and the scatter loses 0.19° that belonged to the grid rather than to the stem.0.4000.6000.8001azimuths the rule samples the circle at (logarithmic)ratio, and the scatter a protractor would record, in degrees384768115215362304ratioscatterthis phase works hererise 0.005 · 5 stems a pointgenerated from a stated rule, not drawn to look right
Fig. 11 Which is where this thread goes next. The ratio at the bottom of the U is 0.79 on a converged grid and 0.62 on the grid the original measurement used, and the difference is not noise.

And it leaves the survey specification worse than it found it. Before this sweep, a botanist wanting to use the ratio needed a long stem, a good protractor and a plant that was not too quiet or too disturbed. Now they also need to know where on the ladder the plant sits, which means measuring the rise — the height gained per organ, in units of the stem’s circumference — on the same stretch of stem the angles came from. That is not a hard measurement, and it is one more thing that has to be right.

Where the disc's counts change, predicted from a cylinderThe dashed lines are the transition radii the cylinder's ladder gives through h = c²/4πr², with nothing fitted. The dots are what the blind counter returns from the disc: 15 of 16 bands agree, and the ones that do not sit on a transition.501001501020304050radius in the disclarger parastichy number, measured on the disc21/3434/5555/8989/144prediction from the cylinder, counts from the disc15 of 16 bands agree
Fig. 12 The rise as a measurable quantity rather than a model parameter. It is what the ratio now has to be indexed by, and on a real stem it is a length divided by a circumference.
A window that fits inside a rungStems that climb the ladder at four rates, read over a window at the fine end. The condition is a ratio: the window has to be shorter than a rung. 250 internodes at 130 per rung is 1.92 rungs and agrees on 0 of 3; 400 internodes at 130 per rung is 3.08 rungs and agrees on 0 of 3; 250 internodes at 260 per rung is 0.96 rungs and agrees on 3 of 3; 400 internodes at 260 per rung is 1.54 rungs and agrees on 1 of 3; 250 internodes at 520 per rung is 0.48 rungs and agrees on 2 of 3; 400 internodes at 520 per rung is 0.77 rungs and agrees on 3 of 3; 250 internodes at 1040 per rung is 0.24 rungs and agrees on 3 of 3; 400 internodes at 1040 per rung is 0.38 rungs and agrees on 3 of 3. Read over the whole stem instead, every rate returns nothing — 0 of 3, 0 of 3, 0 of 3, 0 of 3 — because the quantity the comb is periodic in changes as the pattern climbs.nodes per rung250-node window400-node windowwhole stem1301.92 rungs0/3 · 1 wrong3.08 rungs0/30/32600.96 rungs3/31.54 rungs1/3 · 1 wrong0/35200.48 rungs2/30.77 rungs3/30/310400.24 rungs3/30.38 rungs3/30/33 stems per cell · rise falls from 0.4 to 0.004 on every onefilled where the angles and the positions agree
Fig. 13 And the standing difficulty this joins. A reading taken inside one rung means something; a reading taken across a boundary is a mixture, and the ratio is now a third quantity that has to know where in its rung it was measured.

Where this goes next

-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later816245132129reads 8/13 · no rule in itmain 0.514 · band 0.073the shaded strip is the sampling bandkinematic lattice · 759 anglesgenerated from a stated rule, not drawn to look right Errors that pass between organs An organ's neighbours are the ones eight and thirteen places back — that is what a parastichy pair is. So a disturbance transmitted by contact is correlated at exactly the two lags the readout examines, and it does not have to be told them. Driven into a lattice with no rule in it, it returns the counted pair on eight stems out of eight. -0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later816245132129spacing 8 · offset 5pair 8/13 — counter says 8/13the shaded strip is the sampling bandone stem · 760 divergences · disturbance 0.25generated from a stated rule, not drawn to look right The second comb The autocorrelation of a divergence sequence has peaks at the smaller parastichy number and at every multiple of it. It also has a second set of peaks, at the same spacing, offset by the difference of the pair — so a list of angles with no coordinate in it returns both numbers rather than one. nodes per rung250-node window400-node windowwhole stem1301.92 rungs0/3 · 1 wrong3.08 rungs0/30/32600.96 rungs3/31.54 rungs1/3 · 1 wrong0/35200.48 rungs2/30.77 rungs3/30/310400.24 rungs3/30.38 rungs3/30/33 stems per cell · rise falls from 0.4 to 0.004 on every onefilled where the angles and the positions agree A window inside a rung A stem that climbs the ladder has no comb in it at any rate, because the quantity the comb is periodic in changes as it goes. Read a window instead and it comes back, on one condition: the window has to be shorter than a rung — which makes the shoot's rate the thing that decides whether a plant can be asked.

Shares its objects with

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

  • The rung was not the instrument — both name artefact, autocorrelation, divergence angle, ensemble, honest limits, ladder, measurement, parastichy pair, rise, rung
  • Two windows on one stem — both name autocorrelation, discrimination, divergence angle, ensemble, honest limits, ladder, measurement, parastichy pair, rung, transitions
  • A disturbance with a memory — both name artefact, autocorrelation, discrimination, divergence angle, ensemble, honest limits, measurement, null model, parastichy pair
  • The disturbance that travels — both name autocorrelation, discrimination, divergence angle, ensemble, honest limits, measurement, nearest neighbour, parastichy pair, repulsion
  • The organ that was taken away — both name discrimination, divergence angle, honest limits, measurement, nearest neighbour, null model, parastichy pair, repulsion, rise
  • The survey loses its second outcome — both name artefact, autocorrelation, discrimination, honest limits, measurement, null model, rise, rung, transitions

Named objects

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

ArtefactAutocorrelationDiscriminationDivergence angleEnsembleHonest limitsLadderMeasurementNearest neighbourNull modelParastichy pairRepulsionRiseRungTransitions