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

Two windows on one stem

A pair read off a climbing shoot can only be read through a window, and a window can straddle a transition. Read a second window half a length lower and the outcomes fall into four kinds — and agreement between them never happens on a shoot whose rung is shorter than the window, which turns the most awkward of the four refusal causes into something a reading can certify.
20 min read 12 figures State the claim, state the testCounted, not admiredThe round trip

Worth reading first: A pattern with a rate · The sequence has a memory · The survey this site cannot do.

A real shoot does not sit at one rise. Its internodes shorten as it grows, the pattern climbs the ladder, and the parastichy pair changes underneath any measurement made on it. The pair can still be read from the angles — the previous phase established that — but only through a window, and only if the window fits inside one rung of the ladder.

That left a problem the previous phase named and could not solve. A stem that returns no reading is reporting that one of four things is true: too quiet, too disturbed, too fast, or a window in the wrong place. The sequence distinguishes none of them, and a refusal that could mean four things is close to a refusal that means nothing.

The proposal was a second overlapping window. This essay implements it, and the first surprise is that the decision about what agreement means is harder than the measurement that follows.

Two windows on a shoot at 400 nodes per rungA stem grown at 400 nodes to the rung with a disturbance of 0.25, its rise falling from 0.4 to 0.004 over 1914 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads 8/13; the lower reads 8/13. The verdict is agree, and the window holds 0.63 of a rung.-2.50-2-1.50-1-0.50005001e+31.5e+3node, counted from the base of the shoot — the rise falls as it climbsthe rise, logarithmicupper: 8/13lower: 8/13verdict: agree0.63 of a rung in the windowone stem · 400 nodes a runggenerated from a stated rule, not drawn to look right
Fig. 1 The arrangement. A shoot climbing at 400 nodes to the rung, its rise falling from 0.4 to 0.004; the upper window is the last 250 internodes, where a count would be made on a real plant, and the lower is the same length shifted down 125. Both windows read, and both return the same pair.

Four outcomes, and each is given a name

Two windows, each of which either reports a pair or refuses. Four cases:

Agree — both report and report the same pair. Disagree — both report and report different pairs. One-sided — exactly one reports. Silent — neither reports.

There is nothing clever about that enumeration, and the temptation is to skip past it to a rule like take the reading if either window gives one. The enumeration is the point: each of the four is a different piece of evidence, and collapsing them into report-or-not throws away most of what the second window bought.

Agreement between two windows happens only on a slow enough shootFive stems at each of four rates and five disturbances, each read through two overlapping windows of 250 internodes. A filled mark is agreement — both windows reported the same pair; a half mark is a disagreement; a small mark is one window reporting and one refusing; an open mark is silence. Agreement appears 0 times in 25, 1 times in 25, 14 times in 25, 14 times in 25 at 130, 250, 400, 700 nodes per rung, and the two rates it is almost absent from are the two at which a rung is no longer than the window.nodes per rung0.050.150.250.50.9disturbance1301.92 rungs0 of 25 agree2501.00 rungs1 of 25 agree4000.63 rungs14 of 25 agree7000.36 rungs14 of 25 agreeagreedisagreeone-sidedsilentfive stems a cellgenerated from a stated rule, not drawn to look right
Fig. 2 Five stems at each of four rates and five disturbances, one mark per stem. The filled marks are agreements and they are absent from the top two rows and common in the bottom two — and the top two rows are the rates at which a rung is no longer than the window. That is the result of the essay, and it is a property of the rate rather than of the disturbance: agreement appears at 0.05 and at 0.5 alike, and only below a certain speed.

Agreement certifies the rate

The numbers are stark enough not to need statistics. Over twenty-five stems at each rate: at 130 nodes per rung, 0 agreements, and 1.92 windows to a rung; at 250, 1 agreement and 1.00; at 400, 14 agreements and 0.63; at 700, 14 agreements and 0.36.

The transition is not gradual and it sits exactly where the arithmetic puts it: when a rung is shorter than the window, at least one of the two windows contains a transition, and a window containing a transition has no single pair in it to find.

That makes agreement a certificate. A botanist who reads two overlapping windows and gets the same pair from both has established — from the angles alone, without measuring internode lengths or counting transitions or knowing anything about how fast the shoot grew — that the shoot is slow enough for the reading to be about one rung.

Two windows on a shoot at 130 nodes per rungA stem grown at 130 nodes to the rung with a disturbance of 0.25, its rise falling from 0.4 to 0.004 over 623 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads nothing; the lower reads nothing. The verdict is silent, and the window holds 1.92 of a rung.-2.50-2-1.50-1-0.5000200400600node, counted from the base of the shoot — the rise falls as it climbsthe rise, logarithmicupper: refused, undecidedlower: refused, undecidedverdict: silent1.92 of a rung in the windowone stem · 130 nodes a runggenerated from a stated rule, not drawn to look right
Fig. 3 The same measurement on a shoot climbing three times as fast. The window now holds nearly two rungs, so the pattern changes inside it; the upper window still returns something and the lower does not, and the verdict is one-sided. Twenty-five stems at this rate produced no agreement at any disturbance.

That is the cause the previous phase most wanted removed, and it is the one a botanist can least easily check by other means. The others have external evidence available: a protractor gives the scatter, and a ruler down the stem gives some idea of the rate. But the rate that matters is not the rate in centimetres per week — it is nodes per rung, which needs the ladder, which needs the pair, which is the thing being measured. The certificate breaks that circle.

Two windows on a shoot at 700 nodes per rungA stem grown at 700 nodes to the rung with a disturbance of 0.25, its rise falling from 0.4 to 0.004 over 3350 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads 8/13; the lower reads 8/13. The verdict is agree, and the window holds 0.36 of a rung.-2.50-2-1.50-1-0.50001e+32e+33e+3node, counted from the base of the shoot — the rise falls as it climbsthe rise, logarithmicupper: 8/13lower: 8/13verdict: agree0.36 of a rung in the windowone stem · 700 nodes a runggenerated from a stated rule, not drawn to look right
Fig. 4 And a slow shoot, at 700 nodes to the rung — a third of a rung inside the window. Both windows read and agree. Twenty-five stems at this rate produced fourteen agreements, four disagreements, four one-sided reads and three silences, and the silences are all at the extreme disturbances rather than at the workable ones.

Why the enumeration is worth keeping

A reader who has followed the survey thread will notice that this essay produces no new statistic. It runs the same readout twice on the same stem and looks at the pattern of outcomes, and that is all.

That is the point, and it is worth defending because the alternative was tried. The obvious way to use two windows is to combine them into one better estimate — average the two correlograms, or concatenate the internodes and read four hundred instead of two hundred and fifty. Both make the reading better and both destroy the thing the second window is for. A combined reading is a single number again, and a single number cannot say whether the pattern changed underneath it; that information lives in the disagreement between two readings, which averaging is designed to remove.

So the second window is not more data. It is a second measurement of the same thing under a slightly different condition — a lower stretch of stem — and the value is entirely in whether the two agree. This is the standing pattern of this site’s strongest results: the round trip, the two instruments that share no code path, the angle readout against the position counter. Each is two ways of getting one number, and each is worth more than either way alone by exactly the amount that a disagreement would have cost.

What is different here is that the two readings are not independent, which is what the next section is about.

What the overlap costs

The lower window is shifted by half a window, so the two share half their internodes. That is a choice and it is the one parameter here worth arguing about.

Disjoint windows would be better evidence. Two readings from the same internodes are not independent: a wrong reading driven by a peculiarity of the data can appear in both, and agreement between them is then agreement of a measurement with itself.

And disjoint windows would be unusable. Two 250-internode windows laid end to end span 500 internodes. At every rate this instrument works at, 500 internodes is longer than a rung — 500 against 400, or against 700 with the pair sitting near a boundary — so the lower window would be almost guaranteed to straddle a transition, and the verdict would be one-sided whatever the shoot was doing.

So the overlap is a trade between independence and placement, and half is neither end of it. What that costs is measured in the next essay: agreement is not correctness, and at the quiet end of the disturbance range there are stems where both windows agree on a pair the position counter does not report.

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. 5 The single-window measurement this replaces, from the previous phase: the window has to fit inside a rung, and the fraction of stems that read tracks the ratio directly. The second window turns that from something you have to know into something the reading tells you.

Reading the grid the other way

The rows of the grid are rates and they carry the result. The columns are disturbances and they carry a second one that is worth extracting, because it says which of the four verdicts a survey should expect to spend its time on.

Over the twenty stems at each disturbance, pooled across all four rates, the tallies of agree, disagree, one-sided and silent run: 8, 1, 9, 2 at a disturbance of 0.05; 6, 5, 7, 2 at 0.15; 9, 1, 6, 4 at 0.25; 6, 0, 12, 2 at 0.5; and 0, 0, 2, 18 at 0.9.

Three things fall out of that.

One-sided is the commonest verdict over most of the range, and it is the least informative of the four. It happens when one window is on a rung and the other is not, which is what a stem at an awkward rate looks like from below, and it happens at every disturbance from the quietest to the strongest. A survey should expect to discard roughly a third of its specimens for this reason alone, and discarding them is correct: a one-sided read is a reading with no certificate attached.

Silence is concentrated at one end. Eighteen of the twenty-two silences in the whole grid are at the largest disturbance, where the pattern has come apart. That is the reason the ruler is in the specification, and it is why silence is much less ambiguous in practice than the four-cause enumeration makes it sound: most silences are accompanied by a scatter that says immediately which end of the range the stem is at.

And disagreements cluster in the middle. Five of the eight are at a disturbance of 0.15, which is not where the pattern is fragile and not where it is quietest. That is the disturbance at which the second comb is weakest relative to the first — strong enough to clear, not strong enough to be unambiguous — so it is where a window is most likely to name the wrong partner. A disagreement that is not between adjacent rungs is therefore a signature of a particular disturbance range rather than of anything about the shoot, which is useful to know and would have been read as noise without the grid.

A disagreement can name a transition

The rarest of the four verdicts is the most informative when it happens.

Across the grid there are eight disagreements. Six of them are the two windows agreeing on the smaller parastichy number and differing on the larger — 8/13 against 8/11, or 8/13 against 8/10 — which is one window misreading the second comb and is not a fact about the plant.

Two of them are different. At 250 nodes per rung, one stem returns 8/13 above and 5/8 below: two rungs that the ladder puts next to each other, in the right order, on a shoot whose rung length is about the window length. That is a transition caught inside the read.

The distinction is mechanical rather than a matter of judgement: check whether the two readings are consecutive pairs on the Fibonacci ladder. If they are, the windows are on opposite sides of a transition and the shoot’s rate can be estimated from where the boundary must lie. If they are not, one of the two readings is wrong and the stem should be discarded.

Two of eight is not many, and the reason there are so few is instructive rather than disappointing. A genuine transition inside the read requires the rate to be in a narrow band: fast enough that a boundary falls between the two windows, slow enough that each window is mostly on one side of it. Faster than that and both windows straddle boundaries and neither reads; slower and no boundary falls in the pair of them at all. So the disagreement-names-a-transition case is a diagnosis that is available exactly at the rate where it is needed, and is absent — correctly — everywhere else.

It also means the count of adjacent-rung disagreements across a survey is itself a measurement: it estimates how many of the sampled shoots are climbing at around one rung per window, which is the population’s rate distribution, which is a quantity no survey of phyllotaxis has ever reported.

What is visible in the outer part of a 900-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 60 per cent of the extent, a cone shows 0 changes and a disc 1, because half a cone's length holds half its elements and half a disc's radius holds three quarters of them.02460.2000.4000.6000.8001counting only outside this fraction of the organ's length or radiustransitions inside the counted partdisc: 1 beyond 60%cone: 0 beyond 60%flare 0.35 · 900 elements0 against 1 in the outer 40%
Fig. 6 Why an adjacent-rung disagreement is the expected form of a genuine transition: consecutive rungs of the ladder share a member, so a pattern crossing a boundary goes from m/n to n/(m+n) or back. A disagreement between two pairs that are not adjacent is not a transition, because a pattern does not skip a rung.

The certificate, stated as a claim a plant could refute

It is worth writing the result in the form the site uses for anything it expects to be held to, which is a statement with a stated way of failing.

Claim. If two windows of 250 internodes, overlapping by half, both return a pair and return the same one, then the shoot’s rung is longer than the window.

How it could fail. A shoot fast enough that both windows straddle transitions and both nevertheless return the same wrong pair. That is not impossible in principle — a pattern spends part of a rung near each boundary, and two windows sharing half their internodes share whatever the overlap contains.

What the measurement says about that. Over fifty stems at the two fast rates it happened once, and the once was at 250 nodes per rung with the window exactly one rung long, which is the boundary case. At 130 nodes per rung — two windows per rung — it did not happen at all in twenty-five.

So the claim is not a theorem and is not offered as one. It is a rate of false certificates measured at one in fifty at the worst rate tried, which is the form a claim about an instrument should take.

What this does not fix

The second window removes one cause of refusal and identifies a second. It does not remove the other two, and the reason is that they are properties of the whole stem rather than of where the window sits: a plant that is too quiet is too quiet in both windows, and one that is too disturbed is too disturbed in both.

Both ends of the window are silent, and a ruler tells them apartThe scatter recorded on stems at 400 nodes per rung, against the disturbance that produced it, with the stems that returned no reading at all marked as open. Silence at the quiet end comes with a scatter of 0.38 and 0.44°, which any botanist would call an orderly plant; silence at the disturbed end comes with 56°, which nobody would call a pattern. The two refusals look identical in the instrument's output and are three orders of magnitude apart in a quantity measured with a protractor.-0.50000.50011.502-2.30-2-1.50-1-0.5000disturbance, in degrees of azimuth (logarithmic)the scatter a protractor would record, in degrees (logarithmic)no lattice left above here3 of 5 silent3 of 5 silent400 nodes a rung · five stems a pointgenerated from a stated rule, not drawn to look right
Fig. 7 Both ends of the disturbance range are silent, and the quantity that separates them is one a protractor reads directly: 0.38° of scatter at the quiet end against 56° at the disturbed end. The second window contributes nothing to that separation and does not need to — the ruler was already in the specification, for this reason, since the measurement phase.

The full accounting, and what is left after it, is the next essay. The short version is that the four causes go to three separated and one remaining, and the one remaining is the one this instrument was built to remove.

What a growing stem counts, against what a static lattice wouldThe steps are the blind counter's answer as the stem grows at 400 nodes per rung; the dashed verticals are the rises at which the static ladder changes. 464 of 470 counting windows agree, and the mean gap between where a transition happened and where the ladder puts it is 0.014 of a rung.0.50011.502falling rise, as −log₁₀which rung the pattern is on1/22/33/55/88/1387 nodes per rung · 415 nodes95 of 95 windows agree
Fig. 8 The object all of this is a measurement of: a shoot walking down the ladder, its counted pair changing rung by rung as the rise falls. The whole difficulty of reading a real plant is that the thing being measured is moving while it is measured, and a window is the only way to hold it still.
The lag that is not thereEach dot is one rate: the mean gap between where the grown pattern changed its count and where the static ladder puts that transition, in rungs. Over rates from 9 to 135 nodes per rung the worst is 0.068 of a rung. A lag of one rung would put a dot on the top line.-0.500-0.25000.2500.50011.251.501.752nodes the stem spends per rung, log₁₀transition late by, in rungsone rung late8 rates · rise 0.4 → 0.004worst mean lag 0.068 rungs
Fig. 9 And the reason the rate matters twice over. A pattern does not merely pass through rungs as it climbs; it lags behind where the static ladder says it should be, and how far it lags depends on how fast it climbs. A reading taken across a transition is not an average of the two rungs — it is a reading of something that was never in either.
The branch is kept below 85 nodes per rung and lost above 92Each row is one rate. The Lucas seed keeps its ladder at 46, 58, 65, 75, 85 nodes per rung and abandons it at 92, 108, 131. The golden seed ends on 8/13 at every one of them.nodes per rungseeded Lucasseeded golden467/11 — kept8/13 — Fibonacci587/11 — kept8/13 — Fibonacci657/11 — kept8/13 — Fibonacci757/11 — kept8/13 — Fibonacci857/11 — kept8/13 — Fibonacci928/13 — gone to Fibonacci8/13 — Fibonacci1088/13 — gone to Fibonacci8/13 — Fibonacci1318/13 — gone to Fibonacci8/13 — Fibonacciseeded with 40 nodes at a rise of 0.12threshold between 85 and 92
Fig. 10 The rate at which this site’s other rising results were measured, for comparison. The shoots that can be read by two windows are four times slower than the ones this collection has grown throughout, which is a constraint on which plants a survey can use and was arrived at from a different direction one phase ago.
Both statistics, on the same stems, at a rise of 0.005Five seeded stems at each disturbance, held at a fixed rise. Bars are how many returned the pair the position counter finds; open portions are refusals. The pair comes out from 0.1 to 0.25, and across that whole range the lag-one correlation of the *same* sequences is -0.33, -0.58, -0.59 — decisive, negative and flat. There is no trade between the two: one stem supplies both. Below the window the sequence has locked onto the sampling grid and is a cycle rather than a sample; above it there is no lattice left, at 116° of scatter.012345-1.70-1.30-1-0.824-0.602-0.398-0.2220disturbance amplitude, degrees of azimuth per nodestems out of five returning the counted pair0.020.050.10.150.250.40.61lag-one correlation = 0lag one, on the same sequencesrise 0.005 · 5 stems per point · bars are the pair, line is lag onefilled where the pair agrees with the position counter
Fig. 11 The fixed-rise version of the same question, which is where the disturbance window was measured. Everything in this essay is about carrying that window onto a stem whose rise is falling, and the answer is that it survives if the shoot is slow and the reading is checked twice.
Two windows on a shoot at 250 nodes per rungA stem grown at 250 nodes to the rung with a disturbance of 0.15, its rise falling from 0.4 to 0.004 over 1197 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads 8/13; the lower reads nothing. The verdict is one-sided, and the window holds 1.00 of a rung.-2.50-2-1.50-1-0.50002505007501e+3node, counted from the base of the shoot — the rise falls as it climbsthe rise, logarithmicupper: 8/13lower: refused, undecidedverdict: one-sided1.00 of a rung in the windowone stem · 250 nodes a runggenerated from a stated rule, not drawn to look right
Fig. 12 The boundary case, at exactly one window per rung. This is the stem whose two windows report one rung above and the rung below it beneath — two rungs the ladder puts next to each other, which is a transition caught inside the read rather than an instrument failing.

Where this goes next

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. -0.50000.50011.502-2.30-2-1.50-1-0.5000disturbance, in degrees of azimuth (logarithmic)the scatter a protractor would record, in degrees (logarithmic)no lattice left above here3 of 5 silent3 of 5 silent400 nodes a rung · five stems a pointgenerated from a stated rule, not drawn to look right A refusal with a reason Three phase plans running have recorded that a refusal has four causes and the sequence separates none of them. With a second window and a protractor, three are separated: silence at 0.38° of scatter is a quiet plant, silence at 56° is a disorderly one, and agreement certifies the rate. The fourth survives, and so does a worse discovery — agreement is not correctness. 012345-1.70-1.30-1-0.824-0.602-0.398-0.222disturbance amplitude, degrees of azimuth per nodestems out of five returning the counted pair0.020.050.10.150.250.40.6lag-one correlation = 0lag one, on the same sequencesrise 0.005 · 5 stems per point · bars are the pair, line is lag onefilled where the pair agrees with the position counter What a refusal does not say The readout can decline for four different reasons — too quiet, too disturbed, too fast, or a window in the wrong place — and a stem that returns nothing does not say which. That is the third time this thread has failed to close the mixture problem, and the first time the failure has a shape.

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.

  • The rung was not the instrument — both name autocorrelation, divergence angle, ensemble, honest limits, identifiability, ladder, measurement, parastichy pair, rung
  • What the pair costs — both name autocorrelation, discrimination, divergence angle, ensemble, honest limits, identifiability, measurement, parastichy pair, specimen
  • A periodicity is not a lattice — both name autocorrelation, discrimination, divergence angle, ensemble, identifiability, measurement, parastichy pair, specimen
  • A disturbance with a memory — both name autocorrelation, discrimination, divergence angle, ensemble, honest limits, measurement, parastichy pair
  • The angles name the branch — both name autocorrelation, discrimination, divergence angle, ladder, measurement, parastichy pair, rung
  • The control a survey would need — both name autocorrelation, discrimination, honest limits, identifiability, measurement, parastichy pair, specimen

Named objects

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

AutocorrelationDiscriminationDivergence angleEnsembleHonest limitsIdentifiabilityLadderMeasurementParastichy pairRateRungSamplingSpecimenTrackingTransitions