Phyllotaxis
the spirals are counted, not admired
Concept

Least-squares fit — where it appears

Choosing a line's parameters to minimise the summed squared departure of the points from it. It recovers both a shell's growth factor and a branching exponent here, and it weights information by leverage and bias by counting, so a junction that says nothing about the exponent still contributes its full share of bias.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A logarithmic spiral growing by 3.20× per turn. Fitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r.

Growth as a rule

A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit.

shells · Spiral 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 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 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

Named alongside it

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

Branching exponentDa Vinci's ruleHonest limitsMeasurement errorMurray's lawBiasDaughter ratioInterval estimateNoiseSample sizeSystematic errorControl

All concepts