Phyllotaxis
the spirals are counted, not admired
Branching and transport

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.
5 min read 4 figures The round tripState the claim, state the test

There are two ways to use a law like Murray’s on real measurements, and they answer different questions.

Assume the exponent. Take k = 3, compute how badly r₀³ = Σrᵢ³ fails on the junctions, and report the error. This asks: how close is this network to the hydraulic optimum?

Fit the exponent. Sweep k, find where r₀ᵏ = Σrᵢᵏ holds best, and report the k. This asks: what relation does this network actually obey?

The second is more informative and it is much less often done.

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. 1 Sweeping k and asking where the relation holds best. On junctions built to Murray’s law it returns 3.000, recovered rather than imposed.

Why the fitted version says more

Because the answer is a number with meaning rather than a distance from an assumption.

An exponent of 3 is the hydraulic optimum — Murray’s law, minimising pumping plus upkeep.

An exponent of 2 conserves total cross-sectional area across a junction. That is Leonardo’s rule for trees, and it is what a purely mechanical requirement gives: a branch that must support itself.

An exponent between them says the network is trading the two, and where it sits says how the trade is struck. Reporting “12% departure from k = 3” throws that away; reporting “k = 2.4” keeps it.

The same reasoning applies whenever a law is used as a null model. The departure is the measurement, and expressing it as a fitted parameter rather than as a residual keeps its meaning.

Showing the fit can find something else

A fitting routine that always returns three would confirm Murray’s law on any data at all, which is the failure mode this site cares about most.

So the gate builds a set of junctions that obey a square law by construction — daughters sized so that r₀² = r₁² + r₂² — and requires the fit to return 2, not 3. It does, to within a twentieth.

That check costs four lines and it is what makes the 3.000 above worth quoting. Without it, the recovery of Murray’s exponent from a Murray tree is consistent with a routine that returns 3 unconditionally, and there would be no way to tell from the output.

The shape of the fit

The curve matters as much as its minimum.

A sharp minimum means the data pins the exponent down: only a narrow range of k comes close to satisfying the relation. A shallow one means the junctions are consistent with a wide range, and quoting a single value would overstate the case.

Real branching data is noisier than a generated tree, and the width of the minimum is the honest expression of that noise. A published exponent without one is a point estimate with no error attached, and on this subject the error is often wide enough to include both 2 and 3 — which would mean the measurement does not distinguish the hydraulic story from the mechanical one at all.

What the fit still cannot do

Two limits worth stating.

It cannot say why. A network with k = 2.6 is doing something between the two idealisations, and which constraints produce that particular value is a question the fit does not answer. It narrows the field and does not close it.

And it assumes the relation has that form. Fitting an exponent to r₀ᵏ = Σrᵢᵏ presumes the junction obeys a power law of that shape at all, and a network organised on some other principle would be badly described by the best-fitting k. The residual at the minimum is what says whether the form is right, and it should be reported alongside — which is the same discipline the spiral fit uses, where a residual that is not arithmetic noise means the model does not apply.

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 The tree the fit is run on. Every junction obeys the cube law by construction, so a fit that returned anything other than three would be a fault in the fit.
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. 3 And where the three comes from. The exponent is not a convention; it is what minimising the sum of two competing costs produces.

Reading a fitted exponent

Some values worth having in mind when one comes out of real data.

k ≈ 3 — the hydraulic optimum. Pumping cost plus upkeep cost, minimised. Expect it in small vessels and fine vasculature where mechanics is not the binding constraint.

k ≈ 2 — cross-sectional area conserved. The mechanical answer, and Leonardo’s rule for trees. Expect it where a branch has to hold itself up.

Between — a trade, and the position says how the trade is struck.

Below 2 — the daughters are unusually thin relative to the parent, which happens where the network is dividing many times in a short distance.

Above 3 — rare, and worth checking the measurement before believing it.

Why this generalises beyond branching

The move here — fit the parameter rather than assume it and report the residual — is the same one the spiral fit makes and the same one the divergence-angle recovery makes.

In each case there is a value the theory predicts and a value the data implies, and the informative thing is the second. Reporting a departure from the predicted value collapses a measurement into a verdict; reporting the fitted value keeps it a measurement.

And in each case the routine has to be shown returning something else on data built to a different rule, or its agreement with the theory is uninformative. That check is four lines and it is the difference between a fit and a formality.

Two runs of the same rule from unrelated starting anglesBoth settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.1201301401500255075100stepdivergence angle produced at that step (°)golden anglegrowth 0.40settled spread 0.00°
Fig. 4 The rule itself, part way through a run. The next element goes to the minimum of the repulsion curve, and nothing in the rule names an angle.

Where this goes next

branch angle 30° · daughter ratio 1.0063 junctions checked 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. 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. 00.2500.5000.7501120130140150divergence angle (°)closest pair, as a fraction of the mean spacing (higher is better)137.508°400 points per anglethree criteria, three winners Packing, measured four ways The claim is that the golden angle packs best. It is measurable, and the measurement gives three different winners on three criteria — all near 137.5° and none of them it. That does not make the claim wrong; it makes the usual statement of it wrong.