Phyllotaxis
the spirals are counted, not admired
Concept

Systematic error — where it appears

An error that displaces an answer in a known direction and does not average away across specimens. Two here run downward: a single shape exponent fitted to an organ that has four returns their harmonic mean, and symmetric error on a branch radius biases a fitted branching exponent low at every error level.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 One exponent fitted to an organ that has 4 of them. Each dot is one step between consecutive rings, reporting 2 ln φ / ln(s′/s) — the exponent that step would have if the organ had one. They run from 1.980 to 1.697. The line is what a single fit returns, 1.891, which is their harmonic mean of 1.880 and sits below their plain average of 1.887.

What one exponent reports

Fit a single shape exponent to an organ that has four of them and it returns a real quantity — the harmonic mean of what its individual steps report. Harmonic means sit below arithmetic ones, so the fit understates, systematically, in a known direction, and invisibly.

cylinder · Fitted exponent What a 200-organ window allows against what the offset ceiling allows. Two settings decide what the counter can return, and at this window the offset ceiling is the tighter of them. A window of 200 organs holds a family of at most 194, since the counter scores an offset only where it has 6 hops of it inside the band; the largest offset it looks at is 60, a factor of 3.2 between them. The ticks are the rungs of the two ladders this collection is built on, and the largest counted number anywhere in the settling table is 19 — inside both bounds, which is what makes the table inert.

A plateau the instrument should have had

The counting window decides nothing on the settling table, and it has two bounds that decide everything outside it. A window one organ too narrow returns the previous rung of the same ladder, a family past the offset ceiling is reported as a coarser rung at every width there is, and neither failure produces a refusal, noise or a wide error bar.

lattices · Read window The window 16 known pairs need, against the 20 organs the counter refuses under. Cylinders built at a stated divergence and rise, counted at every window from 20 organs to 200, so the pair is known before the counter sees it. The smallest window that reads it is 20 organs for every pair up to 14 parastichies and the larger count plus 6 above that — 13 of 16 measured exactly, on both branches, with nothing fitted. A pair of 76 and 123 needs 129 organs where 2 and 3 needs 20; the open marks are pairs the counter's own offset ceiling refuses at every window, read only once that ceiling is raised.

How many organs a pair needs

A count taken over too few organs does not fail. It returns the rung below, which is a perfectly good pair, and nothing anywhere says so. The window that avoids it is not a constant but the counter's own arithmetic, and 384 settled runs sit exactly where that arithmetic puts them.

lattices · Read window Six handovers relocated, in steps of the sweep that recorded them. The ladder finds a handover by stepping at a ratio of one per cent, which never lands on the grid the rises are named on, so a recorded handover is the nearest rise the sweep visited to a crossing nobody had located. This is the difference, in units of the sweep's own step at that rise: 0.082 to 0.835, mean 0.374. All six are positive and all six are inside a single step of the sweep, and both of those are predictions rather than summaries: the ladder reports the first rise it visits at which the ordering has already changed, so the rise it records must sit on the fine side of a crossing and within one of its own steps. The dashed rule is one step of the sweep, which is the bound the sampling predicts.

The handovers corrected

Six recorded handovers, relocated to the grid against where a one-per-cent sweep put them: all six sit on the fine side of a crossing and all six inside a single sweep step. Nothing about the rung explains the size of the discrepancy, which is what a sampling artefact is supposed to look like.

cylinder · Handover grid

Named alongside it

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

Honest limitsBiasInstrument settingLadderMeasurement errorRiseBranching exponentClaim testingDa Vinci's ruleFittingMeasurementMurray's law

All concepts