Phyllotaxis
the spirals are counted, not admired
Series

Murray — the series

11 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
  8. A crown sized against buckling on clamped bases, coloured by the share of that load each branch keeps on the base its parent really gives. A symmetric crown 9 generations deep, every branch 2^(−1/2) of its parent's length, turned 20° at every fork, with equal loads on the tips, sized against buckling as if every branch were clamped. Each branch is coloured by the share of its clamped buckling load it keeps once its base is the tip of a parent that bends, the parent alone: from 0.296 to 0.795 outside the trunk. A daughter turning back toward vertical off a steep parent keeps least, one turning outward most. Branches pointing level or down carry no compression and are drawn faint.

    A base that gives

    Every branch sized against buckling was a column clamped to a parent that does not move. A parent bends, and a column on a base that turns keeps only a share of its clamped load — the root of φ·tan φ = kL/EI — which no stiffening of the column itself lifts past the load at which a rigid rod on that base would tip. On the crown sized as if clamped, that share is 0.561 at every branch when the loads run along them, falls from the ground to 0.447 when the whole path is counted, and under gravity is a closed form in the turn from a parent's tilt to its daughter's: 0.795 for a daughter turning outward, 0.296 for one turning back upright off a steep parent. Resized for the bases they really have, the crown's junctions alternate about two from generation to generation, 226 branches of a thirteen-generation crown have no radius that holds them, and the cone of directions that divided buckling from bending no longer divides them.

    part 9 · branching
  9. A crown whose parents are sized for the bases their daughters need, coloured by the term that sizes each branch. A symmetric crown 9 generations deep, every branch 2^(−1/2) of its parent's length, turned 20° at every fork, under gravity, sized from the trunk outward by the largest of three terms: the radius its own column needs on the base its parent gives, the base term its daughters demand at a margin of 2.5 over the load at which a rigid rod would tip on that base, and the bending rule, balanced against buckling on the trunk at 50°. The base term sizes 249 branches, the column term 238 and bending 24; no branch is left without a radius.

    A parent sized for its daughters

    A branch whose compression times its length exceeds what its parent's tip could hold as a rigid rod cannot be saved by thickening the branch; the requirement is the parent's. Size every parent for the base its daughters need, at a stated margin over that ceiling, and nothing in a crown is left without a radius. With every load along its branches the junctions stop alternating at one margin, 2.043 at the planar ratio, and conserve exactly two above it. Under gravity a junction sized this way conserves the clamped crown's closed form read one fork further up. Joined with bending, the cone of directions comes back, wider. And the cheapest crown sits below the margin at which the alternation stops, at about 1.6 to 1.7, so the zigzag is not waste.

    part 10 · branching
  10. A crown sized branch by branch for its loads, and the shape it buckles in as one frame. An axial crown seven generations deep, every parent sized for the base its daughters need at a margin of 1.7 over the rigid-rod ceiling and every branch to carry its own load on its parent's spring, assembled as one planar frame of beam-column elements and buckled. The frame buckles at 0.205 of the loads it was sized for. The lowest mode, drawn over the crown, leans the trunk and the first forks and carries the crown above them over: the trunk holds 24 per cent of the mode's strain energy and the first three generations 68 per cent. The branch the per-branch model holds at the smallest margin is in generation 1 and carries 9.4 per cent.

    A crown sized branch by branch buckles as one at a fifth of its load

    Every branch of a crown can be sized to carry its own load on the spring its parent's tip gives it, and every parent for the base its daughters need — and the crown, assembled as one frame, still buckles at a fifth to a third of the loads it was sized for. The spring stood in for the parent; it did not stand in for the crown above a branch, which leans when the branch leans and hangs its loads further out. The mode lives at the bottom, in the trunk and the first forks, not in the branch the spring model marks weakest. And the cheapest margin moves: branch by branch it was 1.7; held as one frame it is 4, on a floor so flat that 1.7 costs under two per cent more.

    part 11 · branching
  11. A crown of 63 branches after 8 steps of moving wood towards its lowest buckling mode. Each branch drawn to its radius, scaled so the whole crown just holds the loads it was sized for. At step 8 it takes 43.76 of wood where the uniform repair takes 44.89 and the best reached takes 43.76.

    Moving the wood saves almost nothing

    A crown sized branch by branch buckles as one frame at a fifth of its load, and the repair so far has been to thicken every branch alike. Its lowest mode lives in the trunk and the first forks, so thickening a twig looks like waste. Moving each branch's wood to match its share of the mode's strain energy tests that, and it finds one crown, the same from every starting margin to four figures. From the crown that margin of 1.7 had made cheapest it saves 2.5 per cent, and from a margin of four, one. A twig holds very little of the mode's energy and also very little of the wood. The best crown does not keep one exponent: from about two and a half at the trunk's fork it falls to about one and a quarter at the twigs.

    part 12 · branching

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