Phyllotaxis
the spirals are counted, not admired
Concept

Exponent fitting — where it appears

Recovering a power from measurements rather than assuming it, and reporting what it comes out as. A fitted exponent on a shape whose exponent varies turns out to be the harmonic mean of what its individual steps report, which is a number about the fit.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The exponent fitted from the junctions, rather than assumed. Sweeping k and asking where r₀ᵏ = Σ rᵢᵏ holds best gives 3.000 — Murray's 3, recovered rather than imposed.

Fitting the exponent

Assuming the exponent is three and reporting the error says how far the data is from that assumption. Fitting the exponent and reporting what it comes out as says what the network is doing — and an estimator has to be shown returning something other than three, or it is not a fit.

branching · Exponent The uninformative sample does not say so — it says something else. Above: how many junctions of a given asymmetry it takes to distinguish an exponent of 3 from an exponent of 2, at 2% measurement error on each radius. An even fork needs one; a junction whose small daughter is a twentieth of the large one needs 2195. Below: 50 junctions from each end of the range, on a synthetic tree built at exactly 3. The even forks return 3.01; the twigs return 1.69, with an interval no wider — because 47% of them measure as a parent thinner than its own larger daughter, and dropping those keeps only the half where the noise ran the right way.

Which junctions say anything

Da Vinci's rule and Murray's law differ by 12% at an even fork and by a tenth of a per cent at a twig. So one even fork settles which is right, and two thousand twigs do not — a factor of two thousand across a tree, decided entirely by the shape of the junction and not by how carefully it is measured.

branching · Exponent 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.

branching · Exponent error Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by four different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry ninety-nine hundredths gives 284, 263, 149, 54, 9. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.

Four ways to count a neighbourhood

How deep does the placement rule look? Counting the organs that carry nine tenths of its profile gives 182 down to 3 as the falloff steepens. Counting the ones that carry half gives 33 down to 1. The weighted mean lag gives 68 down to 11. The four disagree by an order of magnitude about the size and agree exactly about the order.

mechanism · Noise colour Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by three different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.

The nearest organ is not the nearest neighbour

Rank the terms of the sum the rule minimises and read off which organs the biggest ones belong to. At every falloff exponent from 1.5 to 6 the answer is the same five: lags 13, 8, 5, 21 and 26. The organ placed immediately before is not among them, and counting the neighbourhood in organs was the wrong unit.

mechanism · Noise colour

Named alongside it

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

Da Vinci's ruleHonest limitsMeasurementBranching exponentThe range of the interactionMeasureMurray's lawNeighbourhoodNeighbourhood depthNoiseThe placement ruleSummary statistic

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