The cube law

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 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

$$r_0^3 = r_1^3 + r_2^3$$

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 #

This is the pattern this collection 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.

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.

The derivation, in full #

The two costs are worth writing down together, because the result falls out in three lines and the three lines are the reason to believe it.

For a tube of radius r carrying volumetric flow Q, Poiseuille’s law gives a power dissipation proportional to Q²/r⁴. The metabolic cost of maintaining the tube and its contents is proportional to its volume, hence to r². The total cost per unit length is

$$C = \frac{aQ^2}{r^4} + br^2$$

Setting dC/dr = 0 gives −4aQ²/r⁵ + 2br = 0, so r⁶ = 2aQ²/b, and therefore

$$Q \propto r^3$$

The flow a tube should carry, if it is sized for least total cost, is proportional to the cube of its radius.

Now impose conservation at a junction: the flow into it equals the flow out. If every tube is optimally sized, substituting gives

$$r_0^3 = r_1^3 + r_2^3 + \dotsb$$

which is Murray’s law. The exponent 3 is not fitted or assumed; it is 4 and 2 — the two exponents in the two costs — combined by a minimisation.

Where the exponent would be different #

Because the derivation is explicit, changing an assumption changes the answer in a predictable way, and that turns the exponent into a diagnostic.

If the maintenance cost scaled with surface rather than volume — the wall, not the contents — the second term would go as r rather than r², and the optimum would give $Q \propto r^{5/2}$.

If the flow were turbulent rather than laminar, dissipation would scale differently and the exponent would fall toward 7/3.

If the constraint were mechanical rather than hydraulic — a branch that must support its own daughters — the relevant balance is cross-sectional area, and the exponent is 2. That is Leonardo da Vinci’s rule for trees, and it is not a competing theory of the same thing; it is the right answer to a different question.

So an exponent measured on a real network is not a test of whether Murray was right. It is a measurement of which cost is binding, and fitting it rather than assuming it is the whole value of the exercise.

What a minimisation result claims #

This is the part that gets over-read, and it is worth being careful.

The derivation says: if a network is sized to minimise the sum of these two costs, then the cube relation holds at every junction. It does not say that organisms minimise anything, that evolution optimises, or that a vessel that deviates is defective.

What makes the result useful is that it is checkable independently of its motivation. A vascular tree either satisfies the relation at its junctions or it does not, and the measurement stands whatever one thinks about optimisation as an explanation.

The empirical answer is that it holds well in small vessels and less well in large ones, which is informative in the direction the derivation predicts: pulsatile flow and wall mechanics matter more where the vessel is big, so the assumptions behind the exponent 3 fail exactly where the exponent is measured to drift.

A theory that fails in a specified place for a stated reason is in much better shape than one that fits everywhere.

Why it is on this site #

Murray’s law is not phyllotaxis, and the connection is the method rather than the subject.

A seed head is one angle. A coiled shell is three numbers. A branching network is one exponent. In each case a family of biological forms is compressed to a small parameterisation, and in each case the parameterisation is what lets a claim be turned into a measurement.

The cube law is the cleanest of the three, because it is the only one with a derivation from stated premises rather than a description fitted to observations. The L-system essay makes the contrast explicit: a grammar can produce a convincing tree while claiming nothing, and Murray’s law claims something narrow and checkable while drawing nothing in particular.

What the tree in the figure is, and is not #

The branching tree drawn here is generated to satisfy the relation, which makes it a demonstration of the geometry rather than evidence for the law.

That distinction is worth being blunt about, because a picture of a plausible tree next to a statement of a law reads as support for it. It is not. The tree obeys the cube law because it was built to; the only thing the picture shows is what a network obeying it looks like, which is a modest and legitimate thing for a figure to do.

The evidence, such as it is, lives in the fitted exponent — where a network built to a different law is measured and the fit returns that different law. That figure can be wrong. This one cannot, which is exactly why it proves nothing.

Keeping the two apart is a habit worth naming: a figure that illustrates and a figure that tests are different objects, and a site that does not label which is which invites the reader to take credit from the wrong one.

Reading the relation off a junction #

For anyone with a picture of a vascular tree, the arithmetic is quick and the intuition is worth having.

Take a parent of radius 1 splitting into two equal daughters. The relation gives 2r³ = 1, so $r = 2^{-1/3} \approx 0.794$. Each daughter is about four-fifths the parent’s radius, and the combined cross-sectional area is 2 × 0.794² = 1.26 — a quarter larger than the parent’s.

That growth in total area is the physically meaningful part. It means the flow slows as it moves into the periphery, by a factor of 1.26 at each generation, which is what gets blood velocity down from a metre per second in the aorta to under a millimetre per second in the capillaries where exchange happens.

An unequal split works the same way: a parent of radius 1 dividing into daughters of radius 0.9 and 0.68 also satisfies the relation, and the asymmetry is how a network serves regions of different size without any junction violating the rule.

Why the exponent and not the ratio #

A final framing point, because there is a competing way to state the same relation and it is worse.

Some accounts quote the daughter-to-parent radius ratio for a symmetric bifurcation, 0.794, as the content of Murray’s law. That number is easy to check against a picture and it is a much weaker statement: it applies only to symmetric junctions, it says nothing about the three-way ones, and it cannot be fitted because there is nothing to vary.

The exponent form applies to any junction with any number of daughters of any sizes, and it has a free parameter, which is what makes it measurable rather than merely checkable.

The general preference is worth stating: given two equivalent forms of a law, the one with a parameter in it is more useful, because it can be fitted, and a fit can return an answer the person doing it did not expect.

The relation as a diagnostic #

One last practical use, which is the one clinicians and plant physiologists actually make of it.

Because the relation is a local statement — it holds at each junction independently — a network can be scanned for junctions that violate it, and a violation localises something. In vasculature, a junction well off the relation marks a vessel that has remodelled, narrowed, or is serving a territory whose demand has changed.

That is a use the exponent form supports and the derivation does not require: one can apply the relation as a null expectation without believing anything about optimality, in exactly the way a fitted exponent can be reported without believing the fit means the organism is minimising anything.

A rule that is useful whether or not its motivating story is true is a good rule to have, and it is rarer in biology than the number of stated rules would suggest.

What the derivation costs to state #

A closing observation about the shape of this result, since it is unusual in its neighbourhood.

The derivation is four lines, uses two assumptions that are stated, and produces an exponent that can be measured. Everything about it is exposed: change the maintenance cost from volume to surface and the exponent changes in a predictable way; change the flow from laminar to turbulent and it changes again.

Compare that with the claims this site spends most of its length on. “The nautilus is a golden spiral” has no derivation and no assumptions to inspect. “The golden angle packs best” has no criterion. “Sunflowers are Fibonacci” has no scope.

The difference is not that Murray was cleverer. It is that his result was stated in a form where being wrong was possible, and the two centuries of pattern-in-nature claims that were not so stated are the reason a site like this has anything to do.

Where the exponent sits on the checklist #

Expansion produced a list of what an account of a biological pattern would have to establish, and Murray’s law does better on it than most of this site’s material, which is worth recording because the site’s usual conclusion is the opposite.

It is derived from an optimisation with a stated cost — pumping power plus the metabolic cost of maintaining blood — rather than fitted to observed junctions. So the exponent is a prediction, and its value follows from the form of the cost rather than from the data.

Its parameters correspond to measurable quantities: vessel radii, which anyone can measure on a cast or a scan.

And it is falsifiable in an ordinary way. A network whose junctions gave an exponent of 2.4 would refute it, and the fit used here can return values other than 3 — shown junctions built to a square law it returns 2.

What it does not establish is that any particular organism optimises anything, which is the same gap every model on this site has. The difference is that here the gap is narrow: the prediction is quantitative, the measurement is direct, and a disagreement would be informative rather than absorbable.

Narrow is not the same as safe, and the thing that closes it is not precision. The measurement is direct and it is selected by what a person can reach: almost all the information about the exponent sits in the junctions whose two daughters are comparable, one of those outweighs a hundred twigs, and a sample taken as it comes returns a confident 1.7 from a network built at 3. So the gap this model has to worry about is not between prediction and measurement but between the junctions that carry the answer and the junctions that are easy to get callipers onto.