Phyllotaxis
the spirals are counted, not admired
Series

Murray — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A branching tree in which every junction obeys the cube law. r₀³ = r₁³ + r₂³ at all 63 junctions, to 2e-16. The widths in the drawing are the radii the law gives, not widths chosen to look right.

    The cube law

    A branching network built to move fluid for the least work obeys one relation at every junction — the cube of the parent radius equals the sum of the cubes of the daughters. It is a minimisation result, it is checkable on a real tree, and it is the rare biological rule with a derivation.

    part 2 · branching
  2. One tree sized for flow and for stress, with each branch 2^(−1/2) the length of its parent. The same symmetric tree, 8 generations deep, each generation's branches 2^(−1/2) the length of the one before and turned 30° at every fork, sized two ways and drawn to one trunk width. On the left each branch's radius cubed is proportional to the tips it feeds — Murray's flow rule — and every junction conserves r³. On the right each branch is sized so that the same load on every tip bends it to the same stress at its base, radius cubed proportional to the sum of its lever arms to its tips; its trunk junction conserves r to the power 1.967, its outermost junctions 1.349, against a deep-tree limit of 2.000. The two trees thin at different rates from the same trunk.

    A cube law with a lever arm

    Murray's exponent of three comes from moving fluid for the least work, and Da Vinci's two has had no derivation here, only the name of the mechanical answer. Size every branch so that the same wind on every tip bends it to the same stress, and a junction conserves r to the power 3/(1 + log₂(1/λ)), where λ is how much shorter each branch is than its parent. A crown that fills a plane gives exactly two; halving lengths gives one and a half; no shortening gives three. Murray's flow rule gives three at every λ, so the lengths of a tree's branches say which mechanism sized it.

    part 3 · branching
  3. One planar crown sized for a load on its tips and for the weight of its own wood. The same symmetric crown, 9 generations deep, each branch 2^(−1/2) the length of its parent and turned 30° at every fork, sized so that every branch is bent to one stress, drawn to one trunk width. On the left the load is on the tips and the trunk junction conserves r to the power 1.980; on the right the load is the weight of the wood, found by iterating the radii until they stop moving, and the trunk junction conserves r to the power 0.969. A crown sized for its own weight thins much faster from the trunk, because a branch's weight grows with the square of its radius.

    A crown that carries its own wood

    Sizing every branch so that equal loads on the tips bend it to one stress gives a crown filling a plane Da Vinci's exponent of two. Move the load onto the wood and the sizing becomes a fixed point, because a branch's load now depends on the radii being solved for. Under the wind on its wood a planar crown still conserves two, but only as a limit its trunk is two tenths short of at fifteen generations. Under its own weight it conserves one — radius rather than area, the stress-similarity law that radius goes as length squared — and a crown carrying leaves and wood reads the leaves' two near its twigs and the wood's one at its trunk, with the handover set by how much of the trunk's load the wood carries.

    part 4 · branching
  4. 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.

    part 5 · branching
  5. 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.

    part 6 · branching
  6. Each junction's exponent against its daughters' lengths, measured exactly and with a two per cent error in every radius. One thirteen-generation crown at a mean length ratio of 2^(−1/2), lengths spread by a half-width of 0.3 in log, its junctions five to ten generations above the tips — every third one drawn. Each junction appears twice, its exponent under equal stress and under equal bending, against the sum of its two daughters' log length ratios; the lines are the least-squares fits. Measured exactly, the stress exponents follow a slope of 0.298 and explain 62% of their scatter, the bending exponents a slope of 0.858 and 93%. With every radius read 2% wrong, the clouds swell — the stress scatter explained falls to 20% — and the slopes read 0.288 and 0.866. The error lands in the residual, and the slope is where the rule is.

    The lengths that name the rule

    A real crown has no single length ratio, and giving every fork a spread of daughter lengths does not blur what a sizing rule conserves: Murray's flow rule still conserves three at every junction, and the two mechanical rules keep their mean exponent, moved only as the square of the spread. What the spread adds is a second number. Each junction's exponent follows its daughters' summed log length with a slope of 0.30 under equal stress and 0.86 under equal bending at the planar crown, where the exponents are both two — and a two per cent error in every radius moves neither slope, while it swamps the scatter that looked like the obvious instrument.

    part 7 · branching
  7. Which tilts buckling sizes and which bending sizes, deep in the crown and near its tips. Crowns at the planar length ratio, forks turned 20°, sized by the larger of buckling and bending, at five balance angles. For each, the upper bar runs from vertical to the widest tilt at which buckling sizes a branch seven or more generations above the tips, and on from the narrowest tilt bending sizes; the lower bar is the same reading within six generations of the tips. Balanced at 30°, buckling sizes the deep crown to 20° and bending from 40°, and near the tips buckling reaches 60°; balanced at 40°, buckling sizes the deep crown to 40° and bending from 60°, and near the tips buckling reaches 60°; balanced at 50°, buckling sizes the deep crown to 40° and bending from 60°, and near the tips buckling reaches 60°; balanced at 60°, buckling sizes the deep crown to 60° and bending from 80°, and near the tips buckling reaches 60°; balanced at 70°, buckling sizes the deep crown to 60° and bending from 80°, and near the tips buckling reaches 60°. In the deep crown the two never overlap at any generation, so the criterion is chosen by direction; near the tips the arms have not converged, the bending term is smaller, and the upright core is wider.

    A crown that would rather not buckle

    A column held below the load at which it buckles and a cantilever held to a fixed deflection need the same radius at every length, because both hold the bending stiffness against a load times a length squared — so the three halves of elastic similarity is also the buckling law, and a crown whose loads all run along its branches conserves the same exponent under either. Gravity does not run along branches. It divides by the cosine of each branch's tilt, so a buckling junction's exponent is set by the direction its parent points, nothing past level is sized at all, and a crown sized by the larger of the two criteria splits by direction into an upright core and a spreading shell whose boundary junctions conserve more than either rule gives.

    part 8 · branching

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