The angle a junction chooses

Murray’s law comes out of a minimisation. The cost of a blood vessel is the pumping power needed to drive the flow through it plus the metabolic cost of maintaining the blood inside it; minimise the sum over the radius and the flow goes as the cube of the radius, so at a junction the parent’s cube is the sum of the daughters’. This collection has an essay on the radius that costs least.

Every account of Murray’s law stops there, because the radii are what it is quoted for.

The same minimisation fixes the angles #

Once the three radii are settled, the junction still has a free parameter: where the branch point sits. Moving it lengthens one segment and shortens the others, and since each segment has a cost per unit length, the total cost depends on the position.

Minimising over that position gives

$$\cos\theta_1 = \frac{r_0^4 + r_1^4 - r_2^4}{2r_0^2r_1^2}, \qquad \cos\theta_2 = \frac{r_0^4 - r_1^4 + r_2^4}{2r_0^2r_2^2},$$

which for a symmetric bifurcation obeying Murray’s law is 37.467 degrees each, a total opening of 74.93.

It is worth being clear what has and has not been added. The cube law is a statement about three scalars and it leaves the geometry of the junction entirely open: the same three radii can meet at any angles whatever, and nothing in the radius calculation prefers one arrangement. Adding the branch point’s position adds two more unknowns, and the same cost function — unchanged, with no new assumption — supplies two more conditions.

So the minimisation is complete in a way the usual account does not suggest. A bifurcation has five degrees of freedom once the parent and the two far endpoints are fixed: three radii and two coordinates. Murray’s law fixes one combination of the radii and the flow split fixes the rest of them; the branch point’s position is fixed by the same cost; and there is nothing left over.

The formula is a triangle of forces #

The fourth powers look arbitrary until the reason for them is in view, and the reason is worth having because it makes the result memorable.

At the Murray optimum the cost per unit length of a segment is not free: substituting the optimal radius back into the cost gives exactly $\tfrac{3}{2}b\pi r^2$ — proportional to the square of the radius. So moving the branch point is like moving a knot pulled by three strings whose tensions are those numbers, and the equilibrium of three forces is a triangle of forces.

Writing $K_i \propto r_i^2$, the balance $K_0\hat{u}_0 + K_1\hat{u}_1 + K_2\hat{u}_2 = 0$ gives $K_2^2 = K_0^2 + K_1^2 - 2K_0K_1\cos\theta_1$, which is the law of cosines, and squaring the tensions squares the radii again. The fourth powers are $r^2$ squared, and there is nothing else in it.

Which is worth checking rather than assuming, because it depends on the cost per unit length being proportional to $r^2$ and that in turn depends on the segment being at its own optimal radius. Moving the branch point in the plane by direct minimisation, at four degrees of asymmetry, gives angles agreeing with the closed form to five parts in a billion — which is the minimiser’s tolerance.

The check found an error while it was being set up, and the error is instructive: writing the flow as $Q = r^3$ rather than $Q = \pi r^3\sqrt{b/\mu}/4$ is Murray’s law up to a factor and is not Murray’s law. With the constant wrong, each segment sits off its own optimum, the cost per unit length is not proportional to anything, and the angles are not the measured ones.

What the angles do as the branching becomes asymmetric #

The symmetric case is one point on a curve, and the rest of the curve is the more useful part.

As one daughter shrinks its angle climbs and the other’s falls. At a tenth of the parent’s radius the small branch leaves at 85.9 degrees and the large one turns by 0.57; at a hundredth, 89.62 and 0.006.

A capillary taps its feeding vessel at a right angle, and the feeder does not notice. That is what every microvascular bed looks like, it is what a leaf’s minor veins do, and it comes out of the same minimisation that gave the cube law rather than out of a separate principle.

The limit is easy to see in the force picture. A vanishing daughter exerts a vanishing tension, so the parent and the large daughter must balance each other — which puts them in a straight line — and the small one is then free to leave at whatever angle keeps its own component zero, which is perpendicular.

There is another reading of the same result which is worth having because it makes the angles predictable without any algebra at all.

A junction of three segments whose costs per unit length are $K_0$, $K_1$ and $K_2$ is exactly the Steiner problem with weighted edges — the problem of joining three points by a network of least total weighted length. Its solution has been known since Fermat and Torricelli, and for equal weights the three segments meet at 120 degrees each. Weight them and the angles tilt, in exactly the proportion the force triangle gives.

So Murray’s angles are the weighted Steiner angles with the weights supplied by Murray’s radii, and the two pieces come from different centuries. That also explains why the symmetric answer is 74.93 rather than 120: the parent’s tension is larger than either daughter’s, by a factor of $2^{2/3}$, so it pulls the junction towards itself and squeezes the daughters together.

And it says what would happen with a different cost law. Anything that makes the cost per unit length proportional to $r^n$ gives a triangle of forces with tensions $r^n$ and angles from the law of cosines with $2n$ in the exponent. The fourth powers are not a fluid-mechanical fact; they are $n=2$, and $n=2$ is what Poiseuille flow plus a volume cost produces.

How flat the optimum is #

An optimum located by a vanishing derivative is located to the square root of the precision of its objective, and this one is no exception: five degrees off costs a tenth of a per cent, ten degrees costs half of one, and twenty degrees costs two per cent, rising as the square.

That is the same shape this collection has already met in a wing’s planform and in a bearing’s gap, and it changes how the prediction should be compared with nature.

Measured branching angles in arterial trees, lung airways and leaf venation cluster in the seventies and eighties. Given that twenty degrees costs two per cent, that agreement is what a flat optimum permits rather than a tight confirmation of the principle — and saying so is more useful than claiming a match, because it says what a disagreement would have to look like to mean anything. A tree at fifty degrees would be eleven per cent off the optimum and worth explaining; one at eighty is not.

What the flatness is worth knowing for #

There is a use for the flatness that is more interesting than the caution it implies, and it is worth separating them.

A junction two per cent from optimal is one whose cost could be improved by two per cent. That is not nothing over a whole tree, but it is far smaller than the variability of anything biological, and it means a vascular network has room to satisfy other constraints without paying much for it. It can avoid a bone, follow a fold, route around an obstacle, or grow into a space it did not choose, and the transport cost of doing so is second order.

Read that way the flatness is a design feature rather than a nuisance. A system whose optimum was sharp would have to be built precisely and could not adapt; one whose optimum is flat can be built by a local rule, perturbed by an accident of geometry, and still be within a few per cent of the best arrangement.

The same is true of the engineering version. A pipe network laid out to fit a building rather than to minimise pumping power is paying a penalty that is quadratic in how far it deviates, and quadratic penalties are cheap for small deviations. Which is why the layout of real pipework looks nothing like a Steiner tree and works perfectly well.

What the law does to a tree #

Applied recursively, Murray’s law has a consequence for the whole network that is worth stating because it is the reason capillary flow is slow.

A symmetric bifurcation gives each daughter a radius of $2^{-1/3}$ of the parent, so the total cross-sectional area is multiplied by $2^{1/3}$ at every generation — it grows. Since the volume flow is conserved, the mean velocity falls by the same factor: twenty-one per cent per generation.

Six generations take a metre a second to a quarter of one; twelve take it to a sixteenth. That is the arithmetic behind blood slowing from half a metre a second in the aorta to under a millimetre a second in the capillaries, and it follows from the cube law rather than from any separate design choice about capillary size.

Why the network cannot be optimised branch by branch #

There is a limit to the local result worth stating, because it is the difference between this calculation and designing a network.

The angles here minimise the cost of one junction with its three endpoints held fixed. A real tree’s endpoints are themselves junctions, so moving one branch point moves the cost of its neighbours, and the true optimum is a simultaneous minimisation over every node.

What the local conditions guarantee is that a globally optimal network satisfies them at every junction — a necessary condition, not a construction. That is enough to test a measured tree against, which is what they are used for, and it is not enough to build one with: a network satisfying the local conditions everywhere can still be far from optimal, in the same way that a path satisfying Snell’s law at every interface need not be the shortest.

The practical consequence is that measured agreement with the angle formula is weaker evidence than it looks. It confirms that each junction is locally sensible, which a growth process that responds to local shear could achieve without any global optimisation at all — and there is a substantial literature arguing that vessels do exactly that, adapting their radius to the wall shear they feel. On that account Murray’s law and its angles are the fixed point of a local feedback rather than the solution of a design problem, and the two are indistinguishable from the geometry alone.

Where the principle applies and where it does not #

Two conditions are doing all the work, and both are worth stating plainly.

The flow must be laminar and the resistance must be Poiseuille’s. The cost function’s first term is $8\mu LQ^2/\pi r^4$, which is Hagen–Poiseuille, and it is the fourth power there that produces the cube law. In a turbulent vessel the resistance goes as a different power and the optimal exponent changes — which is why the aorta, where the flow is not laminar, is the vessel Murray’s law fits worst.

And the second cost must be proportional to the volume. For blood that is the metabolic cost of maintaining it, which is the assumption Murray made in 1926 and the one that is hardest to defend for anything that is not blood. A pipe network’s second cost is the capital cost of the pipe, which goes as the surface rather than the volume, and the exponent that comes out is not three.

So the law is narrower than it is usually stated, and the angles inherit that narrowness exactly: they follow from the radii being optimal, so a network whose radii do not obey the cube law has no reason to obey the angle formula either.

The other optimisations in this collection, beside this one #

It is worth putting this result beside the other minimisations here, because the family resemblance is strong and the differences are informative.

The cheapest shape the walls allow minimises dissipation at a fixed flow and gets a shape rather than a number; the answer is a whole velocity field and the condition is a partial differential equation. The depth that costs least minimises a wetted perimeter at a fixed area, which is pure geometry with no fluid in it at all. The radius that costs least is this page’s parent and trades two costs against each other in one variable.

And the optimum that does not matter is the general warning that applies to all of them: a quantity that is stationary at its optimum is insensitive near it, so the optimum’s location is poorly determined by its own objective, and a measured system that sits near one has not thereby demonstrated much.

What distinguishes the junction is that it produces a shape prediction that is not flat. The angles at the symmetric point are within two per cent of optimal over a range of twenty degrees, and that is the soft part. The hard part is the trend: the small daughter’s angle goes to ninety and the large one’s to zero, and there is no way to be near-optimal while getting that qualitatively wrong.

Which is where a flat optimum’s predictions should always be read. Not at the stationary point, where everything nearby is nearly as good, but in how the optimum moves as a parameter changes — because that dependence is a first derivative rather than a second, and it is not flat at all.

What the same cost says about the whole tree #

There is one more consequence of the cost function worth extracting, because it connects the local result to a global one that is measured constantly.

If the cost per unit length of a segment is proportional to the square of its radius, then the total cost of a tree is $\sum r_i^2 L_i$ — which is proportional to the total volume of the tree, since a segment’s volume is $\pi r^2 L$. So minimising the junction cost is minimising the volume of vessel required, at fixed endpoints and fixed flows.

That is a satisfying place for the argument to land, because it is the quantity a body would actually be economising. Blood is expensive to make and to maintain; vessel wall is expensive to build; and the cost function’s two terms — pumping power and metabolic maintenance — conspire, at the optimum, to make the whole problem equivalent to minimising how much of the fluid there has to be.

It also gives a check anybody can run on a measured tree without knowing any of the constants. Murray’s law predicts that the total cross-sectional area grows by $2^{1/3}$ per symmetric generation, so the mean velocity falls by the same factor, and both are measurable without reference to any cost. That prediction is the one that is best confirmed in real vascular beds, and it is the one that follows from the radii alone — which is where the parent essay stops.

What this page adds is that the same cost, with nothing further assumed, also fixes the geometry. The tree is not merely sized by the principle; it is shaped by it, and an optimum that is flat is what allows the shaping to be approximate without being expensive.

What would have to be measured to test this properly #

Since the flat optimum makes the symmetric angle a weak test, it is worth saying what a strong one would look like.

The prediction with real content is the relation between the radii and the angles, junction by junction, across a range of asymmetries. A tree in which the angles track the radii by the formula — small daughters leaving near ninety degrees, large ones nearly straight on, symmetric ones near thirty-seven — is confirming something that no simpler principle predicts. A tree in which the angles cluster near seventy-five whatever the radii do is confirming nothing, since seventy-five is where a flat optimum puts everything.

That is a measurement of a correlation rather than of a mean, and it needs the radii and the angles of the same junctions rather than population statistics of each. Such data exist for retinal vasculature and for lung casts, and the reported correlations are positive and noisy — which is what a flat optimum plus biological variability would give, and is weaker evidence than the tidy agreement usually quoted.

The general lesson is the one from the optimum’s shape. Test the trend, not the value. A stationary point is uninformative about its own location and informative about how it moves, and every prediction from a variational principle should be read that way.

A final note on the cost function’s second term. Murray’s metabolic cost is proportional to the volume of blood, which is what makes the optimal radius depend on the flow to the one-third power. Any other second term gives a different exponent: a cost proportional to the vessel’s surface gives a different law again, and one proportional to nothing at all gives no optimum, since the pumping power alone is minimised by an infinite radius. The cube law is that particular trade, and nothing more general.

What is not claimed #

Murray’s law is not being tested against biology here. Every number on this page is computed from the cost function, and the comparison with measured angles is a comparison with quoted ranges rather than with data collected for this purpose.

The junction is two-dimensional and planar. A real bifurcation is not necessarily planar, and a trifurcation is a different minimisation with a different force polygon — three tensions and a parent, whose equilibrium is not a triangle.

The optimisation is over the branch point alone. The endpoints are held fixed, which is what makes the problem two-dimensional and gives the closed form. A network optimisation moves every node at once and is a much larger problem whose local conditions are these.

And a flat optimum is not evidence of design. The two per cent at twenty degrees says that a range of angles is nearly as good, which means both that biology need not have found the optimum exactly and that finding it exactly would not prove very much. The prediction earns its keep from the shape of the dependence — a right angle for a small branch, a straight line for its parent — rather than from the number at the symmetric point.