Phyllotaxis
the spirals are counted, not admired
Branching and transport

A swelling at the fork

A branch thickens where it forks, so a parent measured just below a junction and daughters measured just above it carry three different amounts of the same swelling. A swelling that fattens all three alike moves a fitted exponent by exactly nothing. A parent read one per cent fat moves it by as much as 3.6 per cent of random error on every radius, in a sign known in advance, and a tenth of a radius turns a tree built at Murray's three into one that reads Da Vinci's two with no noise at all. Added to the noise, it does not bring the two rules together any sooner: the two errors do not add.
14 min read 8 figures State the claim, state the testCounted, not admiredTwo routes, one number

Worth reading first: The exponent an error moves · Fitting the exponent · The cube law.

The error this thread has priced so far is random. A symmetric error on every radius displaces a fitted branching exponent downward as the square of its size, and two trees built to the two rules become one measurement at twelve per cent of it. That essay closed on a sentence it could not yet back: a swelling at a fork inflates the parent systematically, and that is a different calculation, not done there.

It is done here. A branch does thicken where it forks, and a person who measures a parent just below a junction and its daughters just above it has measured three radii taken at three places, carrying three different amounts of that thickening.

What a swelling does to the exponent read from the informative band, by where the swelling is. 50 junctions of daughter ratio 0.6–1, built at exactly 3 and read with no random error, but with one or more radii measured fat. A parent read fat lowers the reading: 1% gives 2.864, 3% gives 2.631, 10% gives 2.075, and at 11.3% the tree reads Da Vinci's 2. Daughters read fat raise it: 1% gives 3.150, 3% gives 3.499; from 8% some junctions have a daughter measured wider than their parent, which no exponent balances, and the line stops. All three radii read fat by one factor return 3.000 at every swelling — the unswollen reading exactly.
Fig. 1 Fifty junctions of daughter ratio 0.6 to 1 built at exactly three, read with no random error at all, with the parent, the daughters, or all three measured fat. Only the first two move the reading, and they move it in opposite directions.

Three kinds of systematic error

A systematic error is one whose size and sign are the same every time, and three kinds matter at a fork. A common factor multiplies all three radii by one number: callipers that read three per cent high, or a swelling that fattens parent and daughters alike. A differential swelling multiplies the parent by one factor and the daughters by another. A fixed thickness adds the same length to every radius, which is what bark read as wood does.

They are priced on the same fifty-junction combs the random-error measurements used, with the same fitter, so every number here reads directly against the random-error measurements. Nothing random is added until the last comparison.

A common factor moves nothing

The first kind can be settled without running anything. The fitter minimises the discrepancy log(Σrᵢᵏ) − k·log r₀ at each junction, and multiplying every radius by c adds k·log c to both terms. The discrepancy is unchanged — not approximately, identically — so the fitted exponent is the unswollen one at every exponent it could return.

Run anyway, on every band from even forks to twigs and at swellings of one, five and twenty per cent and minus ten, it returns the unswollen reading to the last digit every time. That is the control: anything a differential swelling does below is about the difference, and not about the absolute size of the radii.

A parent read fat

A parent measured fatter than its daughters lowers the reading. On the informative band, one per cent of swelling on every parent takes a tree built at three to 2.864; three per cent to 2.631; ten per cent to 2.075. At 11.3 per cent the tree built at Murray’s exponent reads Da Vinci’s two, with no random error anywhere in the measurement.

Eleven per cent of a radius is not an extreme swelling for a branch point, and it is not an extreme misplacement of callipers along a branch that is thickening. Nothing about the tree has changed; only where on it three radii were read.

Daughters read fat

Daughters measured fatter than their parent raise the reading, by nearly the same amount the other way at first: one per cent gives 3.150 and three per cent 3.499. Then something the parent swelling never does begins.

From eight per cent some of the junctions have a larger daughter measured wider than its parent. No exponent balances such a junction — every power of the daughters exceeds the parent’s — and the fitter cannot use it, so the line in the figure stops there. A swelling in one direction moves the answer; in the other it starts removing the questions.

First order, with a known sign

The central difference from random error is in the arithmetic. Expanding the fit about the true exponent, a parent factor (1 + s) shifts every junction’s discrepancy by the same amount, and the least-squares step is p·Σc₀/Σc₀² per unit of swelling, where c₀ is each junction’s derivative in the exponent. That coefficient is fixed by the daughter ratios before any fit is run: −14.36 on the informative band.

A first-order statement is tested where it is one. At a quarter of a per cent of swelling the measured displacement is within 2.4 per cent of the derived coefficient on the even, informative and middling bands; at one per cent it has fallen 4.8 per cent short on the informative band, which is the curvature of the fit and not a different law. Doubling the swelling to two per cent still roughly doubles the displacement.

Its size against noise

Random error moves the reading as its square, with a coefficient of −113.4σ² on the informative band; a parent swelling moves it as its first power, with −14.36s. So the random error that displaces the reading as far as one per cent of swelling does is √(0.1436/113.4) = 3.6 per cent on every radius.

That comparison has a direction built into it. Below a few per cent, a systematic error is the larger of the two by construction, because a square is smaller than its argument there. And its sign is known before the measurement: a parent read fat always lowers the exponent, where the sign of a random error’s residue is known only after the arithmetic.

A one-per-cent parent swelling on every band, against its first-order prediction and against noise. Each row is 50 junctions built at 3, every parent read one per cent fat, no random error. On even forks the reading moves by −0.125 against a first-order −0.130, as far as 3.9% of random error on every radius would move it; on the informative band the reading moves by −0.136 against a first-order −0.143, as far as 3.6% of random error on every radius would move it; on middling junctions the reading moves by −0.244 against a first-order −0.274, as far as 2.2% of random error on every radius would move it; on lopsided junctions the reading moves by −0.646 against a first-order −1.140, as far as 1% of random error on every radius would move it; on twigs the reading moves by −1.193 against a first-order −6.590, as far as 0.4% of random error on every radius would move it. The first-order coefficient holds on the even bands and fails on the lopsided ones, where one per cent is no longer small beside a junction's margin; the bar stops at −1.3 and the number beside it does not.
Fig. 2 One per cent of parent swelling on every band, from even forks to twigs: the displacement measured, the first-order prediction as a tick, and the random error the same displacement is worth.

Band by band

The same one per cent does very different things on different junctions. On even forks it moves the reading by −0.125 against a first-order −0.130, as far as 3.9 per cent of random error would. On the informative band, −0.136 against −0.143, worth 3.6 per cent. On middling junctions, −0.244 against −0.274, worth 2.2.

On lopsided junctions it is −0.646 against a derived −1.140, and on twigs −1.193 against a derived −6.590 — as far as 0.4 per cent of random error. On the junctions a tree offers in quantity, a one-per-cent swelling costs more than the whole distance between the two rules.

Where the prediction fails

The first-order coefficient holds on the even bands and fails on the lopsided ones, and the reason is the size of what the swelling is compared with. At a daughter ratio of 0.3, a tree built at three has a parent only 0.89 per cent wider than its larger daughter. A one-per-cent swelling more than doubles that excess, and nothing that doubles its own denominator is small.

That is the same boundary the fragile-junction arithmetic located from the other side: the junctions that say least about the exponent are the ones on which any error, random or not, stops behaving like a small perturbation.

The exponent one junction loses to a swollen parent, against its daughter ratio. At a single junction the exponent that balances it moves with a parent swelling at dk/ds = p / Σqᵖ·ln q, exactly. For a junction built at 3 that is 13.0 at a daughter ratio of 1, 13.2 at a daughter ratio of 0.9, 14.1 at a daughter ratio of 0.8, 19.2 at a daughter ratio of 0.6, 25.8 at a daughter ratio of 0.5, 74.0 at a daughter ratio of 0.3, 194.4 at a daughter ratio of 0.2, 399.8 at a daughter ratio of 0.15, 1139.2 at a daughter ratio of 0.1, 7210.1 at a daughter ratio of 0.05; built at 2, 5.8 at 1, 6.9 at 0.6, 14.0 at 0.3. A one-per-cent swelling takes an even fork at 3 to 2.876 and a fork of ratio 0.1 to 1.743.
Fig. 3 At one junction, the exponent lost per unit of parent swelling, in closed form, against the daughter ratio, for a junction built at three and one built at two, on a logarithmic scale.

One junction, exactly

At a single junction nothing needs to be approximated. Differentiating the equation the junction’s exponent solves gives dk/ds = p / Σqᵢᵖ·ln qᵢ, with qᵢ each daughter’s ratio to the parent, and it agrees with a finite difference of the exact bisection to four decimal places.

For a junction built at three that derivative is 13.0 at an even fork, 19.2 at a ratio of 0.6, 74.0 at 0.3, 1,139 at 0.1 and 7,210 at 0.05. One per cent of swelling takes an even fork to 2.876 and a fork of ratio 0.1 to 1.743. For a junction built at two it is 5.8 at an even fork and 14.0 at 0.3.

Why lopsided junctions lose so much

The denominator is where the loss comes from. At a lopsided fork the smaller daughter’s term qᵖ·ln q is vanishingly small, so the sum is carried almost entirely by the larger daughter — whose ratio to the parent is close to one, and whose logarithm is therefore close to zero.

A denominator near zero is a derivative near infinity. That is the closed-form statement of what the band figure measured: the junction whose parent is barely distinguishable from its larger daughter is the junction whose reading any swelling of the parent overturns.

Two trees, one at 3 and one at 2, as their parents are read fatter. Two trees of 50 junctions of daughter ratio 0.6–1, one built at 3 and one at 2, every parent read fat, no random error. At 1% they read 2.864 and 1.941, 0.923 apart; at 5% they read 2.438 and 1.743, 0.695 apart; at 10% they read 2.075 and 1.555, 0.520 apart; at 20% they read 1.630 and 1.295, 0.335 apart; at 40% they read 1.187 and 1.001, 0.186 apart. The gap closes and never crosses; the tree at 3 reads 2 at a swelling of 11.3%, and a tree at 2 reads 3 when its daughters are read 11.4% fat.
Fig. 4 Two trees on the informative band, one built at three and one at two, with every parent read fatter and no random error. Both fall, the tree at three faster, and the rule marks where it reads two.

Two trees, no noise

Put a tree built at two beside the tree at three and swell every parent in both. At one per cent they read 2.864 and 1.941, 0.923 apart; at ten per cent 2.075 and 1.555, 0.520 apart; at forty per cent 1.187 and 1.001, 0.186 apart. The gap closes all the way along and never crosses.

The tree at two falls more slowly, with a coefficient of −6.07 per unit of swelling against −14.36, for the reason the duel found under noise: the lower exponent presents junctions that say more about the exponent, so the same disturbance is divided by more information.

One rule into the other

The two statements worth keeping are symmetrical. A tree built at three reads two once its parents are read 11.3 per cent fat; a tree built at two reads three once its daughters are read 11.4 per cent fat. Neither needs any random error, any small sample or any lopsided junction.

So an exponent measured on the best junctions a tree offers, with perfect callipers, can land on either rule depending only on how far from the fork each radius was read. A published exponent that does not say where along the branch its radii were taken has not said which of those it measured.

The daughter ratio below which a daughter read fat makes a junction impossible. A junction whose larger daughter is measured wider than its parent has no exponent that balances it. A daughter swelling d does that to every junction whose daughter ratio is below ((1 + d)ᵖ − 1)^(1/p): at 0.5%, 0.247 for a tree at 3 and 0.100 for a tree at 2; at 1%, 0.312 for a tree at 3 and 0.142 for a tree at 2; at 2%, 0.394 for a tree at 3 and 0.201 for a tree at 2; at 5%, 0.540 for a tree at 3 and 0.320 for a tree at 2. The shaded strips are twigs, 0.05–0.15, and the informative band, 0.6–1: one per cent of daughter swelling makes every twig junction on a tree at 3 impossible and leaves the informative band untouched. A parent swelling of any size makes no junction impossible.
Fig. 5 The daughter ratio below which a daughter swelling makes a junction impossible, at four swellings, for a tree built at three and one at two, with the twig band and the informative band shaded.

The two directions are not symmetric

A daughter measured fat by d makes a junction impossible whenever its daughter ratio is below ((1 + d)ᵖ − 1)^(1/p). For a tree at three that is 0.247 at half a per cent, 0.312 at one per cent, 0.394 at two and 0.540 at five; for a tree at two, 0.100, 0.142, 0.201 and 0.320.

One per cent of daughter swelling therefore removes every twig junction from a tree built at three, and leaves the informative band untouched until the swelling passes about seven per cent. A tree at two keeps its twigs much longer, because its parents are proportionally wider.

Why the parent swelling is the dangerous one

A parent swelling of any size makes no junction impossible. That sounds like the benign direction and it is the opposite. A daughter swelling announces itself: junctions start failing to fit, a sample shrinks, a careful analyst notices something wrong with the measurements.

A parent swelling never announces anything. Every junction still fits, every sample keeps its size, and the answer simply moves — downward, towards Da Vinci’s rule, by an amount that looks exactly like a biological finding. A sample that is confidently wrong is what random error does to twigs; this is what systematic error does to everything.

Bark read as wood: a fixed thickness on every radius, band by band. A fixed thickness added to all three radii is not a common factor, because it is a larger share of a daughter than of its parent, so it moves the reading where a factor does not — upward. On even forks a thickness of 1% of the larger daughter raises it by 0.027 against a first-order 0.027; on the informative band a thickness of 1% of the larger daughter raises it by 0.030 against a first-order 0.030; on middling junctions a thickness of 1% of the larger daughter raises it by 0.039 against a first-order 0.039; on lopsided junctions a thickness of 1% of the larger daughter raises it by 0.057 against a first-order 0.057; on twigs a thickness of 1% of the larger daughter raises it by 0.089 against a first-order 0.090. The dashed lines are the first-order predictions, with nothing fitted in them.
Fig. 6 Bark read as wood: a fixed thickness added to all three radii of every junction, band by band, with the first-order predictions dashed beside the measured lines.

Bark read as wood

A fixed thickness is not a factor, because the same length is a larger share of a thin daughter than of its thicker parent. So it moves the reading where a common factor did not — and it moves it upward. A thickness of one per cent of the larger daughter’s radius raises the reading by 0.027 on even forks, 0.030 on the informative band, 0.039 on middling junctions and 0.089 on twigs.

The first-order predictions match to the third decimal on every band up to a thickness of two per cent, and within four per cent at five. This is the one systematic error here whose arithmetic behaves as small everywhere, because a thickness is small against every radius at once.

Two errors that can cancel

The parent swelling lowers the reading and the bark thickness raises it. On the informative band, a one-per-cent parent swelling is −0.136 and a thickness of about 4.6 per cent of the larger daughter is +0.136, so a measurement carrying both would read three for reasons that have nothing to do with the tree.

That is not a reassurance. It means a reading that agrees with Murray’s law is not evidence that the radii were measured well; it may be the sum of two protocol errors with opposite signs. Which junctions say anything was always half the question; how their radii were taken is the other half.

The duel at 12% of random error, re-run with every parent read fat. 50 junctions of daughter ratio 0.6–1 from a tree at 3 and one at 2, 300 replicates at 12% of random error on every radius, with the parent swelling added on top. With a 0% swelling the tree at 3 reads 2.035 [1.80, 2.30] and the tree at 2 1.676 [1.53, 1.84], 0.358 apart, and the two first overlap at 12%; with a 1% swelling the tree at 3 reads 2.005 [1.77, 2.25] and the tree at 2 1.644 [1.50, 1.80], 0.361 apart, and the two first overlap at 12%; with a 3% swelling the tree at 3 reads 1.941 [1.73, 2.16] and the tree at 2 1.583 [1.45, 1.73], 0.358 apart, and the two first overlap at 14%. The swelling lowers both trees, the gaps between them differ by 0.002, and the overlap arrives no earlier.
Fig. 7 The duel at twelve per cent of random error on every radius, re-run with every parent read zero, one and three per cent fat. Both trees move down together, and the gap between them does not change.

Swelling and noise together

The natural expectation for a measurement carrying both errors is that they add: a parent swelling lowers the tree at three by more than the tree at two, so under random error the two should meet sooner. The re-run duel was built to confirm that, and it does not.

At twelve per cent of random error with no swelling the two trees read 2.035 and 1.676, 0.358 apart, and first overlap at twelve per cent. With one per cent of parent swelling they read 2.005 and 1.644, 0.361 apart, overlapping at twelve. With three per cent, 1.941 and 1.583, 0.358 apart — and the overlap arrives later, at fourteen.

Why they do not add

The mechanism is the same asymmetry seen twice. Random error’s displacement is larger for a reading that sits higher: −113.4σ² at three against −28.9σ² at two. A parent swelling lowers the tree at three, and a lower reading is pulled down less by the noise, so the swelling has taken away part of the displacement the noise would otherwise have added.

The two effects cancel in the gap to within 0.002 at twelve per cent, and both trees’ intervals narrow as they fall, so the overlap moves to a larger error rather than a smaller one. A swelling does not make the duel’s twelve per cent worse. It moves both answers further from both truths while keeping their order.

What a swelling does to the exponent read from middling junctions, by where the swelling is. 50 junctions of daughter ratio 0.3–0.6, built at exactly 3 and read with no random error, but with one or more radii measured fat. A parent read fat lowers the reading: 1% gives 2.756, 3% gives 2.403, 10% gives 1.765, and at 6.8% the tree reads Da Vinci's 2. Daughters read fat raise it: 1% gives 3.310, 3% gives 4.114; from 1% some junctions have a daughter measured wider than their parent, which no exponent balances, and the line stops. All three radii read fat by one factor return 3.000 at every swelling — the unswollen reading exactly.
Fig. 8 The same response on middling junctions of daughter ratio 0.3 to 0.6. A parent swelling brings the tree at three to two at under seven per cent, and a one-per-cent daughter swelling already makes some junctions impossible.

On middling junctions

Everything is harsher on junctions of ratio 0.3 to 0.6. A parent read one per cent fat takes the tree to 2.756, three per cent to 2.403, and 6.8 per cent is enough for it to read two. Daughters read one per cent fat raise it to 3.310 and already make some junctions impossible, because the bottom of the band sits at the one-per-cent threshold of 0.312.

The band is where the band decides the answer located the practical sample, and it is also where the systematic error is twice as expensive as on the informative junctions.

What a protocol would have to state

The control at the top of this essay is also the remedy. A swelling that multiplies all three radii by one factor moves the exponent by exactly nothing, so a protocol that measures parent and daughters at the same position relative to their own swelling — the same multiple of each branch’s radius away from the fork, say — turns a differential error into a common one and removes it from the answer.

What it cannot do is leave the position unstated. A published exponent carries, invisibly, the difference between how far from the fork its parents and its daughters were read, and one per cent of that difference is worth more than three per cent of calliper error on the informative junctions.

A floor below the callipers

That comparison turns a familiar statement of precision around. A study that reports its radii to two per cent has reported its instrument. If its protocol read parents one per cent further into their swelling than daughters, the exponent it reports is displaced as far as 3.6 per cent of random error would displace it — a floor the callipers cannot see and better callipers cannot lower.

And unlike the random floor, it does not shrink with more junctions. The window that closes found that more junctions narrow an interval without moving its centre; a systematic swelling is all centre.

What this does not establish

No swelling was measured. How much a real branch thickens at a fork, how far from the fork a person measures, and whether that differs between parent and daughters are not known here, and they are the inputs a reader would have to bring. The combs of daughter ratio are designs, fifty junctions to a band, and every junction in a band is swollen by the same factor.

A real tree’s swelling varies from junction to junction, which would add a random part to the systematic one. And the random error beside it is the Gaussian, relative, independent error the rest of this thread uses, which a real instrument is not.

What would withdraw it

A common factor on all three radii that returns any reading but the unswollen one. A quarter-per-cent parent swelling whose displacement is not the first-order prediction to within a twentieth of it. A closed-form derivative that disagrees with a finite difference of the exact exponent. A parent swelling that brings the duel’s overlap to a smaller random error. Each is run every time the measurement runs, and the last is the one the measurement was first written expecting to find.

A correction for the error that can be corrected

Of the two errors, the random one has a standard remedy: add more error on purpose, watch how the fitted exponent moves, and carry the trend back to none. It needs only the sample and the size of the error, and it cannot touch a systematic swelling, since adding random noise to a biased measurement does not remove the bias. The next measurement applies that correction to the duel, and its test is the claim the duel made about itself — that no better estimator removes the displacement — which a correction that keeps the overlap measures.

Where this goes next

Murray's law and Da Vinci's become one measurement at 12% of radius error on the informative band. Two trees measured the same way: 50 junctions of daughter ratio 0.6–1, 300 replicate samples at each of 15 error levels, one tree built at exactly 3 and one at exactly 2. Each shaded band is the central 90% of the recovered exponents. Both run downward, but the tree at 3 runs down faster — -113σ² against -29σ² — so the two close on each other. At 11% they are still apart; at 12% the bands overlap and one study's answer could have come from either tree; at 20.5% the means cross, and above it a tree built at 3 measures lower than a tree built at 2. Where three and two become one Two trees, one built to obey Murray's law and one to obey Da Vinci's, are measured through the same fifty junctions with the same instrument. At twelve per cent of error on each radius the two answers overlap, and above twenty and a half the tree built at three measures lower than the tree built at two. A tree built at 3, measured to 2%, reads 2.957 on the informative band. The exponent recovered from 100 junctions of a tree built at exactly 3, against the relative error placed independently on the parent and on both daughters — half a per cent is a machined section under a microscope, one to two per cent is callipers on a clean branch, five is a branch that is not round, ten is a radius read off a photograph. Each line is a band of daughter ratio and the whiskers are the central 90% of 300 replicate samples. Every band is displaced downward at every error and never upward: at 2% the informative band read 2.957 and the informative band read 2.957. Below, the same rows with the displacement and the spread drawn as separate bars, because only one of them falls when more junctions are measured. The exponent an error moves Every real measurement of a branch radius carries error and no synthetic tree does, so the question is what a symmetric error does to a fitted exponent. It does two things — a bias and a spread — and the bias runs downward at every error level and in every band, by an amount derivable from the daughter ratios alone. Adding error to carry a fitted exponent back to none, at 12%. At 12% of error on every radius each replicate is refitted with more error added at four levels, and the dots are the means over 300 replicates, the tree at 3 above and the tree at 2 below. Uncorrected they read 2.035 and 1.676. Carried back to no error through the mean points, the tree at 3 reads 2.695 by a quadratic curve, 2.380 by a linear curve, 2.914 by a rational curve; the tree at 2 reads 1.961 by a quadratic one, 1.877 by a linear one, 1.982 by a rational one. The curve carried back is a choice the method does not make. A correction that keeps the overlap The duel between a tree built at Murray's exponent and one built at Da Vinci's ended by saying the displacement is the geometry, and that no better estimator removes it. Correcting every replicate by simulation-extrapolation removes 92 per cent of the tree at three's displacement at five per cent of error and 68 per cent at twelve, and the error at which the two means cross leaves the measured range altogether. It pays in spread — the corrected readings are twice as wide at twelve per cent — so the error at which the two trees' intervals overlap does not move. Of the duel's two numbers, the inversion was the estimator's and the overlap is the question's.

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.

Named objects

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

BiasBranching exponentDa Vinci's ruleDaughter ratioEvidenceHonest limitsLeverageMeasurement errorMurray's lawNoiseSystematic errorTruncation