Phyllotaxis
the spirals are counted, not admired
Concept

Optimisation — where it appears

The derivation of a form by minimising a stated cost. It is what separates Murray's law from a rule of thumb, and it is also the move this collection declines for phyllotaxis, where which pattern appears turns out to be a question about basins rather than about optima.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

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.

branching · Murray The uninformative sample does not say so — it says something else. Above: how many junctions of a given asymmetry it takes to distinguish an exponent of 3 from an exponent of 2, at 2% measurement error on each radius. An even fork needs one; a junction whose small daughter is a twentieth of the large one needs 2195. Below: 50 junctions from each end of the range, on a synthetic tree built at exactly 3. The even forks return 3.01; the twigs return 1.69, with an interval no wider — because 47% of them measure as a parent thinner than its own larger daughter, and dropping those keeps only the half where the noise ran the right way.

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 The fork the cost chooses at a daughter ratio of 1: 37.47 and 37.47 degrees. Three ends held fixed, three weights fixed by the radii, and the branch point put where the total cost is least. At the optimal radius the pumping term is exactly half the upkeep term, so a segment's weight is its own cross-section and the three weights here are 1.5874, 1.0000, 1.0000. Minimising directly over the position — a 41×41 grid re-centred and shrunk 220 times, told nothing about any formula — puts the daughters at 37.4673° and 37.4673° from the parent's own forward direction, against the closed form's 37.4673° and 37.4673°.

The angle the cost chooses

Murray's exponent falls out of minimising a cost over the radius of a tube. The same cost minimised over the position of the branch point instead fixes both fork angles, and 281 networks minimised on a grid told nothing about any formula agree with the closed form to under two ten-thousandths of a degree.

branching · Fork angle The three weights at an exponent of 2 and 3: one closes a triangle and one is a straight line. A branch point minimising a weighted sum of three lengths has an interior solution only when the three weights close a triangle, and the weight on a segment here is its own cross-section. At an exponent of 3 the three areas clear that condition by 0.4126, and the triangle they close is what the two fork angles are read off. At an exponent of 2 the parent's area is exactly the daughters' areas summed — that is what area conservation says — so the slack is -4.44e-16, the triangle collapses onto a line, and the fork closes to 0.0000 degrees. Leonardo's rule does not predict a different angle here; it predicts no angle.

A rule that predicts everything

Leonardo's rule says a fork conserves cross-section, and cross-section is exactly the weight the branch point is minimised against. So the three weights land on the boundary of the triangle inequality, the cosine comes out at one to the last bit, and the rule predicts no angle at all — and the free constant its own derivation leaves behind then walks the prediction across every angle a fork could have.

branching · Fork angle What insisting on a fork angle costs: 43.3 degrees for one per cent of the network. The three ends are held where the optimum wants them, the branch point is moved everywhere inside them, and the cheapest network at each total angle is kept. The optimum sits at 74.93°. Everything within one per cent of the least cost runs 55.9° to 99.2° — a span of 43.3°, which read back as exponents covers 2.44 to 5.34 — and within a tenth of a per cent it still runs 13.6°. The prediction is steep in the exponent and the cost is nearly flat in the angle; they are the same curve read along its two axes.

An optimum too flat to reach

One per cent of a branching network's cost buys forty-three degrees of fork angle, covering exponents from 2.44 to 5.34, while the angle the theory predicts moves only fourteen and a half degrees across every daughter ratio there is. The prediction is steep and the cost is flat, and those are the same curve read along its two axes.

branching · Fork angle The two trees this site draws, at 30° and 32° to a side, against the cost's 37.47° and 37.47°. Two trees of 63 segments each, 5 generations deep and 31 junctions apiece, with every junction's radii taken from r₀³ = r₁³ + r₂³ exactly and every junction's angle taken from a constant. Read as an exponent through cos(θ/2) = 2^(2/p − 1), the drawn angles say 2.5237 and 2.6239, in pictures whose widths are built at exactly 3. The cost that fixed those widths wants 37.47° and 37.47° at this daughter ratio, 74.93° in total, and the misses cost 0.573% and 0.292% of the network — which is why a fixed angle can sit in a figure about a minimisation and never look wrong.

The trees drawn at no angle

Two branching figures in these essays set every junction's radii from the cube law exactly and every junction's angle from a constant nobody derived. Read as exponents the drawn angles say 2.52 and 2.62, in pictures whose widths say exactly three — and at a lopsided fork the drawing puts a daughter thirty-four degrees from where the same cost puts it.

branching · Fork angle 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 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.

branching · Murray 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.

branching · Murray 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.

branching · Murray

Named alongside it

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

Branching exponentClaim testingHonest limitsMurray's lawClosed formCriterion dependenceElastic similarityFork angleFree parameterModel scopeBranch pointDa Vinci's rule

All concepts