Phyllotaxis
the spirals are counted, not admired
Ladder

Exponent error — the ladder

4 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · branching
  2. 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.

    rung 2 · branching
  3. 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.

    rung 3 · branching
  4. 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.

    rung 4 · branching

All ladders