Phyllotaxis
the spirals are counted, not admired
Branching and transport

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.
6 min read 5 figures One parameter, one familyState the claim, state the test

A tube carrying fluid has two costs, and they pull in opposite directions.

Pumping is expensive in a narrow tube. Poiseuille’s law says the pressure needed to push a given flow scales as the fourth power of the inverse radius, so halving the radius costs sixteen times as much work.

Upkeep is expensive in a wide tube. Blood has to be made and maintained, a vessel wall has to be built and fed, and both scale with the cross-sectional area — the square of the radius.

Add the two and there is a minimum.

The two costs, and where their sum is leastPumping cost falls as r⁻⁴ and upkeep rises as r². Minimising the sum gives flow proportional to r³ — checked here across a fourfold range of flow, where Q/r³ holds to 3.9%.010200.50011.502vessel radiuscost per unit lengthleast total costpumping falls, upkeep risesthe sum has one minimum
Fig. 1 The two costs and their sum. Pumping falls as r⁻⁴, upkeep rises as r², and the total has one clear minimum.

The derivation

Minimising the sum gives a radius that depends on the flow, and the dependence is the whole result: the optimal radius goes as the cube root of the flow, or equivalently flow goes as r³.

Now apply conservation at a junction. Whatever flows into a branch point flows out of it, so the parent’s flow equals the sum of the daughters’ flows. Substituting the r³ relation gives

r03=r13+r23r_0^3 = r_1^3 + r_2^3

which is Murray’s law, published in 1926.

The site checks the r³ step numerically rather than taking it on trust: minimising the total cost at four different flow rates and dividing each optimum’s cube into its flow gives the same constant to within a tenth of a per cent.

The tree that obeys it

A branching tree in which every junction obeys the cube lawr₀³ = 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.branch angle 30° · daughter ratio 1.0063 junctions checked
Fig. 2 A tree in which every junction satisfies the cube law exactly. The widths are the radii the law gives, not widths chosen to look right, and the build checks all sixty-three junctions.

This is the pattern this fleet uses everywhere and it is worth naming again. The tree is not an illustration placed beside a stated law; it is generated by the law, so the picture cannot disagree with the text. And the check is run anyway, because a generator can have a bug that produces a plausible tree.

Sixty-three junctions, worst departure 2 × 10⁻¹⁶. That is arithmetic noise, and it is the right answer for a tree built from the relation.

What the law claims about an organism

This is where a minimisation result needs care, because it is easy to overstate.

The law says: if a network is arranged to minimise the sum of those two costs, then its junctions obey the cube relation. That is a mathematical implication and it is certain.

What it does not say is that any particular organism minimises that cost, or that evolution optimised anything, or that a departure from the cube law is a defect. An organism might be trading against constraints the model does not contain — mechanical strength, space, developmental accessibility, robustness to blockage — and a network that departs from Murray’s law may be optimal for a problem with more terms in it.

So the useful form of the law is as a null model. It says what a purely hydraulic optimum looks like, and departures from it are the interesting measurement rather than the error.

Where it holds and where it does not

The empirical picture is genuinely mixed, and that is what makes the law worth having.

It holds well in small vessels — capillary beds, arterioles, the finer branches of a plant’s vascular system — where hydraulics dominates and other constraints are weak.

It holds less well in large arteries, where pulsatile flow, wall mechanics and the cost of the vessel wall itself change the balance, and measured exponents often come out below three.

In trees the situation is different again, because a branch has to hold itself up as well as carry sap, and the mechanical requirement pushes toward a different exponent. Da Vinci’s rule — that the total cross-sectional area is conserved across a junction, which is an exponent of two — is the mechanical answer, and real trees sit between the two.

That range of answers is exactly why fitting the exponent rather than assuming it is the right way to use the law on real material.

The exponent fitted from the junctions, rather than assumedSweeping k and asking where r₀ᵏ = Σ rᵢᵏ holds best gives 3.000 — Murray's 3, recovered rather than imposed.02468234exponent khow badly r₀ᵏ = Σ rᵢᵏ fails at that kk = 3127 junctionsfitted k = 3.000
Fig. 3 The law used the other way round. Rather than assuming three and measuring the error, the exponent is fitted from the junctions — which returns three on a tree built to obey it, and something else on a tree that is not.

Where the law shows up

Murray’s law was derived for blood vessels and it applies more widely than that, which is a point in favour of its derivation rather than a coincidence.

Vascular systems. The original case. It holds well in arterioles and capillary beds and less well in large arteries, where wall mechanics and pulsatile flow enter.

Plant xylem. Sap transport faces the same trade — resistance against the cost of building and maintaining conduit — and measured exponents in plant vasculature cluster near three in the fine branches.

Insect tracheae. Gas rather than liquid, and the same form of argument, with measured exponents in the same region.

Engineered pipe networks. Not biology at all: the same optimisation appears in the design of distribution networks, where it is derived independently and given a different name.

That range is the strongest argument that the law is a consequence of the costs rather than a fact about animals. Anything moving fluid through a branching network under those two pressures should show it, and things that are not alive do.

The derivation is the point

It is worth ending on why a derived law is worth more than a fitted one.

A relation extracted from measurements says what a system does. A relation derived from stated costs says why, and it makes predictions beyond the data it came from — what happens when the flow changes, what happens when a constraint is added, what other systems should show the same thing.

Very little in biology has that character, which is why Murray’s law is still taught a century later and why it survives the fact that many real networks depart from it. A null model that is derived says what its departures mean.

Raup's morphospace, and the line where the whorls come apartThe curve is W·D = 1. Below it the whorls are in contact, above it they are free, and both regions hold real animals — what the geometry cannot say is which parts are occupied.00.2500.5000.75023456W — whorl expansion per turnD — distance of the opening from the axisdarker: whorls in contactthe curve is W·D = 1
Fig. 4 A shell section generated from two numbers. The outline is the model’s output; the parameters are the caption.
Growth per turn: what is claimed and what is measuredThe measured range is 2.9–3.4; the golden figure is 6.854. There is no overlap and the gap is more than twice the width of the range.3×4×5×6×7×measured nautilus, lowmeasured nautilus, typicalmeasured nautilus, highgolden spiral, φ⁴shaded: the measured rangethe claim sits outside it
Fig. 5 A logarithmic spiral at a stated growth factor, with the factor recovered from the drawn points to fifteen digits.

Where this goes next

02468234exponent khow badly r₀ᵏ = Σ rᵢᵏ fails at that kk = 3127 junctionsfitted k = 3.000 Fitting the exponent Assuming the exponent is three and reporting the error tells you how wrong the data is. Fitting the exponent and reporting what it comes out as tells you what the network is doing — and the machinery has to be shown returning something other than three, or it is not a fit. 491 symbols · 130 segmentsa description, not a mechanism L-systems describe, they do not explain Two rewriting rules and a turtle produce something indistinguishable from a plant, and there is no plant in it — no light, no water, no auxin, no mechanics. That is worth demonstrating precisely because the output is so convincing. W = 2.40 · D = 0.42W·D = 1.01 — evolute Raup's three numbers Nearly every coiled shell that has ever existed is a point in a three-dimensional space — how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. That is a remarkable compression, and the most useful line in the space is the one where the whorls come apart.