Phyllotaxis
the spirals are counted, not admired
Concept

Sampling — where it appears

The choice of which observations to take, which sets how much a study can establish before any of them are made.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

1101001000100000.2000.4000.6000.8001daughter ratio — the smaller branch over the largerjunctions needed to tell Murray's rule from Da Vinci'san even fork — one is enougha twig — 2195the same tree, built at an exponent of 3, measured twicethe truth — 350 even forks3.0150 twigs1.692% error per radius · 47% of twigs impossiblemargin 0.11% against 2.8% noise

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 1101001000100000.2000.4000.6000.8001daughter ratio — the smaller branch over the largerjunctions needed to tell Murray's rule from Da Vinci'san even fork — one is enougha twig — 2195the same tree, built at an exponent of 3, measured twicethe truth — 350 even forks3.0150 twigs1.692% error per radius · 47% of twigs impossiblemargin 0.11% against 2.8% noise

A sample that is confidently wrong

Fifty lopsided junctions from a tree built at an exponent of exactly 3 return 1.7, with an interval that excludes 3 and excludes 2 as well. The sample carrying almost no information does not give a wide answer — it gives a narrow wrong one, and the cause is a selection nobody applies on purpose.

branching · exponent counted from the pointsread from the anglesgolden, rise 0.0323 / 535/5 clear · peak 0.72golden, rise 0.0135 / 855/5 clear · peak 0.59golden, rise 0.0058 / 1385/5 clear · peak 0.78Lucas, rise 0.0323 / 435/5 clear · peak 0.52Lucas, rise 0.024 / 745/5 clear · peak 0.45Lucas, rise 0.0087 / 1175/5 clear · peak 0.59golden, rise 0.052 / 34, 23, 12, 2, 9refused — peak 0.13 under 0.345 runs per rise · the readout sees a list of angles and nothing elsethe refusal is the gate working

A counter that sees no positions

This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.

lattices · counting 00.2000.4000.6000.80000.50011.502reading error on each organ's position, in degreesheight of the peak at the parastichy numberwhat noise alone givesthe threshold a reading must clearpredictedrise 0.008 · 5 runs · pattern scatter 0.75°peak × σ²/(σ² + 2ε²), nothing fitted

What the protractor has to be

The readout that names the parastichy number costs sixty internodes, which is cheap. It also needs every organ's position measured to better than a quarter of a degree, which is not — and the requirement follows from arithmetic rather than from care, so no amount of averaging relaxes it.

wrong · sequence mean sides per cell(forced to six)mean squared departure from six(not forced)whorled, 144°5.9860.023golden, 137.508°5.9900.253rational, 137.5°5.9900.255Lucas, 99.502°6.0360.255137.0°5.9900.291Poisson5.9691.830six433–637 interior cells each, inside 86% of the radiussame cells, same cut, two statistics

What a summary throws away

Four statistics this collection has relied on turn out to be incapable of varying with the thing they describe — one is invariant to shuffling, one is fixed by a theorem, one is a parameter that stopped mattering, one is a fitted number selected into being wrong. In each case the second statistic was free and nobody had taken it.

wrong · second statistic -0.50000.500100200300nodes per rung of the ladder — how slowly the shoot climbscorrelation between one divergence and the nextthe rise held fixed — -0.68no noise · 185, 254, 323, 438, 553, 922, 1474 nodesthe threshold is in the rate, not the rule

A shoot too fast to remember

Sweep the rate at which a stem climbs the ladder and the correlation between one divergence and the next changes sign — negative below about fifty-five nodes per rung, positive above it, with the flip inside one step of the grid. The instrument the previous phase proposed is unavailable on a fast shoot, and nothing said so.

emergence · rising

Named alongside it

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

MeasurementSummary statisticAutocorrelationDivergence angleNoiseSpecimenSurveyBranching exponentCylinderEnsembleEquilibriumParastichy

All concepts