Phyllotaxis
the spirals are counted, not admired
Concept

Elastic similarity — where it appears

Sizing a branch so that its own weight deflects it by a fixed share of its own length, rather than so that it reaches a fixed stress. For a single beam it gives radius as length to the three halves, and inside a crown it gives an exponent that depends on how fast the branches shorten, agreeing with the stress rule only at the crown that fills a plane.

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

Sizing for equal bending and sizing for equal stress cross at one length ratio. Equal stress holds r³ against the sum of a load's arms and gives 3/(1 + ℓ); equal deflection holds r⁴ against the sum of the arms squared and gives 4/(1 + 2ℓ), with ℓ = log₂(1/λ). Setting them equal gives 3(1 + 2ℓ) = 4(1 + ℓ), whose only root is ℓ = 1/2 — the crown that fills a plane, λ = 0.707107 — and there both are exactly two. Below that ratio the stiffness rule reads the lower exponent of the two and above it the higher, so the two criteria size the same crown at one length ratio in the whole family and it is the one Da Vinci's rule names.

A crown sized for how far it bends

Stress is one criterion for sizing a branch and stiffness is another. Holding every branch to the same deflection as a share of its own length sizes r to the fourth against the sum of each load's arm squared, where equal stress sized r cubed against the arm, and the junctions of a deep crown then conserve 4/(1 + 2·log2(1/λ)). A single cantilever under its own weight comes out at radius as length to the three halves — McMahon's elastic similarity, fitted here rather than assumed — against the square that equal stress asks for. And the two criteria agree at exactly one length ratio out of the whole family: λ = 2 to the minus a half, the crown that fills a plane, where both give exactly two.

branching · Murray Three ways of sizing a crown, and the exponent each one conserves. Murray's flow rule sizes r³ against the tips a branch feeds and conserves three at every length ratio, reading nothing of the lengths at all. Equal bending stress conserves 3/(1 + ℓ) and equal deflection 4/(1 + 2ℓ), where ℓ = log₂(1/λ). So an exponent measured on a tree names a rule only with a length ratio beside it, and even then not everywhere: the stress and stiffness curves meet at λ = 0.7071, the stiffness curve passes three at λ = 0.8909, and the stress curve reaches three only as the branches stop shortening.

Three rules, one exponent

A measured branching exponent is quoted as evidence for a sizing rule, and it cannot be. Murray's flow rule conserves three at every length ratio and reads nothing of the lengths at all; equal stress conserves 3/(1 + l) and equal deflection 4/(1 + 2l), where l is log2(1/lambda). So an exponent names a rule only with a length ratio beside it, and even then not everywhere: of ninety-six length ratios between 0.3 and 0.99, thirteen have two rules within five hundredths of each other at a precision of 0.05, in three bands with three different reasons — stress against stiffness where they cross at the planar crown, stiffness against flow where the stiffness curve passes three at 0.8909, and stress against flow only as the branches stop shortening.

branching · Murray

Named alongside it

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

Branching exponentClaim testingCriterion dependenceDa Vinci's ruleHonest limitsModel scopeClosed formConvergenceDegeneracyIdentifiabilityMeasurement errorMurray's law

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