Phyllotaxis
the spirals are counted, not admired
Concept

Interval estimate — where it appears

A range quoted alongside a fitted number, saying what the data leaves consistent with it. Two things are asked of one here and they move opposite ways with sample size: that it be narrow enough to exclude the rival exponent, and that it still contain the value the tree was built at.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

A round trip on four heads of 400 primordia: the divergence angle recovered from each. The counter is shown the points and nothing else. The worst recovery across the four is 0.035°.

Recovering the angle from the counts

Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates.

lattices · Recovery One junction's bias moves 1.36-fold across the range its leverage moves 308,352-fold, at an exponent of 3. The bias a single junction of daughter ratio γ contributes to a least-squares fit, as a multiple of the squared measurement error, drawn against the leverage that junction carries — both computed from the expansion about a true exponent of 3 rather than fitted to anything. Across the whole range from an even fork to a twentieth the bias moves by a factor of 1.36 and the leverage by a factor of 308,352, and the uninformative junction contributes the larger share: 6.00σ² at γ = 0.05 against 4.50σ² at an even fork.

The fragile junctions are the informative ones

That is the obvious worry once the radii are uncertain, and it is false. Across the whole range of asymmetry a junction's contribution to the bias moves by a factor of 1.36 while its leverage moves by a factor of 308,352, so the junction that says nothing damages the answer as badly as the one that says everything — and a sample is spoiled by counting rather than by weight.

branching · Exponent error Between 3% and 5% of radius error, no sample size answers — 50 junctions among them. One row per error level. The pale bar is the sample sizes whose interval is narrow enough to state a claim from — half-width under ±0.25 and excluding 2 — and it starts where precision arrives. The second bar is the sample sizes whose interval still contains the 3 the tree was built at, and it ends where the displacement overtakes the width. Where the two overlap there is a usable window; at 5%, 7%, 10% they do not overlap at all, so below 50 junctions the answer is too wide to state and above 30 it no longer contains the truth.

The window that closes

The spread of a fitted branching exponent falls as the reciprocal root of the sample and its displacement does not fall at all, so there is a count past which every further junction buys confidence and no accuracy. Between three and five per cent of radius error the count arrives before the answer does, and no sample size both states a claim and contains the truth.

branching · Exponent error 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.

branching · Exponent error 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.

branching · Exponent error Measured radii against the tips each branch carries, at 12% of error on every radius. One replicate of a 50-junction tree, 101 segments, every radius measured with 12% of relative error and drawn against the number of tips that segment carries. The count has no error in it, so the slope is not displaced: the tree built at 3 gives a slope of 0.3404, an exponent of 2.937, and the tree built at 2 a slope of 0.5071, an exponent of 1.972. On the same measured radii the junction-by-junction fit reads 2.218 and 1.736.

A count carries no error

Fitting r₀ᵏ = Σrᵢᵏ junction by junction puts a measured radius on both sides of every equation, and at twelve per cent of error a tree built at Murray's three and one built at Da Vinci's two stop being told apart, however the fit is corrected. Fit the same measured radii against the number of tips each branch carries instead — a count, which nobody measures with error — and the two trees read 2.996 and 1.997 at twelve per cent and 3.015 and 2.001 at thirty, never overlapping. The twelve per cent belonged to the junction fit, not to the tree. The count fails in its own way, and the way is stated.

branching · Exponent error The exponent read from a tree that has lost tips, tips lost one at a time. The 50-junction tree built at 3 and at 2, 12% of error on every radius, 300 replicates of loss and measurement at each level. Shaded: the central 90% of the exponent read against the tips still there. Dotted: the mean read against every tip ever grown. Solid: the junction fit's mean over the junctions that survive. The count's two intervals are still apart with 70% of the tips gone, reading 2.366 and 1.576, and overlap by 80%. The junction fit's intervals overlap by 20%. At the heaviest loss drawn, 90%, the scars still read 2.998 and 1.997.

A count that has lost tips

Radii read against the tips each branch carries keep Murray's three apart from Da Vinci's two where junction fits cannot, because a count has no measurement error in it. A count of the tips a tree has is not a count of the tips it grew. Losing them lowers both trees' readings by one factor that belongs to the losses and not to the rule, so the count stops being right long before it stops telling the trees apart: on fifty junctions at twelve per cent of error, to seventy per cent of the tips lost one at a time, and only to about a quarter lost in whole limbs. Counting scars repairs single losses exactly. Nothing countable repairs a shed limb.

branching · Exponent error A shell whose deposition law changes part way through, beside one that never does. Both panels are the same logarithmic spiral at 3.20 per turn, marked at equal intervals of time. On the left the animal holds a constant angular rate for the first 3.50 whorls and a constant length added after that; on the right it holds a constant length added throughout. The curves are identical to the last bit a double holds, because a curve records no clock at all. The counts per whorl are not: 56, 57, 56, 67, 191, 613, 1960 against the unchanged shell's, and the change is in where the marks crowd rather than in where the shell goes.

A shell that changed its law

An animal that grew as a juvenile under one deposition law and as an adult under another leaves a sequence of whorl ratios rather than one, and the sequence says where the change happened. The ratio across the change is a closed form that is neither law's — 6.72 between a length clock and an area clock at 3.2 per turn, exactly the average of 3.2 and 10.24 — and it is monotone in where inside its whorl the change sits, so it inverts. On a seven-whorl shell of 18,466 lines a change at 3.5 whorls comes back at 3.5001, in a band 0.027 whorls wide that holds the true position. The reading refuses a change in the outer three whorls or the inner three, because a plateau it will trust is two agreeing ratios and two ratios need three untouched whorls.

shells · Spiral What the round trip returns as the organs are displaced: golden angle, 300 organs. Thirty heads at each size of displacement, from none to two and a half spacings, each run through the round trip and sorted by what came back. The undisplaced head counts 21 and 34. At 0.25 spacings, 97% the undisplaced pair, 3% a neighbouring pair; at 0.5 spacings, 50% the undisplaced pair, 50% a neighbouring pair; at 0.75 spacings, 30% the undisplaced pair, 70% a neighbouring pair; at 1 spacing, 13% the undisplaced pair, 80% a neighbouring pair, 7% a straddling pair; at 1.25 spacings, 7% the undisplaced pair, 57% a neighbouring pair, 7% a straddling pair, 30% refused; at 1.5 spacings, 40% a neighbouring pair, 3% a straddling pair, 57% refused; at 2 spacings, 10% a neighbouring pair, 90% refused; at 2.5 spacings, 3% a neighbouring pair, 97% refused. 10 recovered intervals in all exclude the true angle.

A head displaced before it is counted

The round trip from a head's spiral counts back to its divergence angle was tested on heads whose every organ sat exactly where the rule put it. Displaced by a normal error of up to two and a half spacings, heads of 900 organs keep counting a pair from their own sequence and return intervals holding the true angle to a spacing and a half; heads of 300 organs move to the neighbouring pair by half a spacing and then refuse, nine in ten of them by two spacings. Every moved count brings in the family whose chord was third shortest. Of 898 heads recovered, 19 intervals miss the true angle and 17 of those by about a tenth of a degree — displacement makes the reading coarser and then silent, not confidently wrong.

lattices · Recovery A head of 300 organs as the rule places it, and twisted by 2 rad. Two heads at the golden angle, 300 organs each. On the left every organ is where Vogel's rule puts it, and the round trip counts 21 and 34 and recovers 137.515°. On the right the same organs turned about the centre by 2 radians times their radius over the head's, so the rim turns by 2 radians and the centre not at all; the round trip counts 34 and 47, recovers 137.80°, missing, with an interval of 0.230° that excludes the true angle.

A twist is a divergence

Recovering a head's divergence angle from its spiral counts survived independent displacements of whole spacings, moving to a neighbouring pair and then refusing rather than misleading. Displacements with a direction are harder on it in only one case. A head pressed to an aspect ratio of 2.25, spread at the rim by eighty per cent or sheared with a slope of 1.6 is still counted as its own pair or its neighbour, and recovered inside its interval. A head twisted — each organ turned about the centre in proportion to its radius — is not: past a turn of the rim of about a radian and a half the counts leave their sequence and the recovered angle misses, by up to sixty-one degrees, because a twist changes the angle between one organ and the next. The round trip is not fooled. It is reporting the angle the twisted head has.

lattices · Recovery A 900-organ head twisted by 0.5 radians at the rim, with the bands its spirals are counted in. Every organ turned about the centre by 0.5 times its radius over the head's. The rings mark the inner annulus, 0.35 to 0.6 of the radius, and the outer, 0.7 to 0.95. The inner annulus counts 34 and 55; the outer 34 and 89 — skips one; the single band, 0.55 to 0.95, 34 and 89 — skips one. A count that is not two consecutive Fibonacci numbers flags the twist; on this head that happens from half a radian, and the single band is misled only from six.

The count sees the twist first

A twisted head recovers a changed divergence, and the check proposed for it was two annuli: the twist's extra angle falls with radius, so an inner and an outer annulus should disagree. Read on golden heads of 900, 2,400 and 9,000 organs, they never do in time. Their intervals separate at eight radians on 900 organs and never on the larger heads, always after the ordinary reading has been misled — from six radians on 900 organs and from two on 2,400 and 9,000. What catches the twist first, at every size and on every seed, is the count: at half a radian to three quarters some band stops returning two consecutive Fibonacci numbers — 34 and 89, 89 and 233 — which no untwisted golden head, clean or displaced, ever does.

lattices · Recovery The highest and lowest factor a loss of a given share can put on a tip count, by which tips it takes. The 50-junction tree with no error on any radius. For each share of its 51 tips, a search over which tips make up that share for the set that reads the factor highest and the set that reads it lowest, by single swaps from 20 starts: 10%, 0.954 to 1.040; 20%, 0.906 to 1.062; 30%, 0.863 to 1.093; 40%, 0.812 to 1.121; 50%, 0.743 to 1.173; 60%, 0.684 to 1.162; 70%, 0.599 to 1.176; 80%, 0.509 to 1.137; 90%, 0.358 to 0.882. A factor above one reads both rules high. The line is the mean factor when every tip is equally likely to go. The true range is at least this wide, since a search can stop short of the extreme.

A loss that picks its tips can read high

A tree that has lost tips at random reads its branching exponent low, by one factor shared by every rule. Let the loss choose which tips go and the factor is still shared — the ratio of three to two survives any choice — but it is no longer below one. On fifty junctions with half the tips gone, the choice of which half moves Murray's tree from 2.23 to 3.52, and past about half the tips a single tree's reading can no longer say which rule built it. The ways a crown plausibly chooses stay within a few hundredths of a fair loss, all of them low, and counting scars repairs every one.

branching · Exponent error

Named alongside it

The objects these essays reach for when they reach for this one.

Honest limitsMeasurement errorBranching exponentDa Vinci's ruleMurray's lawDiscriminationIdentifiabilityEvidenceRound tripBiasClaim testingDivergence angle

All concepts