The angle the cost chooses

A tube carrying fluid costs work to pump through and costs upkeep to maintain, and the sum of the two has a minimum. Minimising it over the tube’s radius is the step that gives Murray’s exponent: flow goes as the cube of the radius, so the cubes of the daughters sum to the cube of the parent.

That minimisation is done over one number. The same cost has another variable in it that nobody usually turns, and it is a position rather than a size. Hold three destinations fixed — where the parent comes from and where each daughter must reach — and ask where the branch point should sit. The answer is a point, and a point fixes two angles.

So the exponent and the fork angles come out of one optimisation rather than two, and the second half of it has a closed form. What follows is that closed form earned rather than quoted: 281 networks minimised directly, on a grid that was told nothing about any formula, agreeing with it to a worst of 1.83 × 10⁻⁴ degrees.

What the radius optimum leaves behind #

Before the branch point can be placed, each segment needs a weight, and the weight is not a modelling choice here. It is what the first minimisation returns.

Minimising the cost over the radius at six flows spanning a thirty-two-fold range gives the same ratio of flow to the cube of the radius every time — 0.70710679, to under a part in a million across the whole range. That constant is the exponent. It is also, read a different way, a statement about the two terms.

The two terms settle at one half #

At the optimal radius the pumping term is exactly half the upkeep term, at every flow tested, and the total cost per unit length comes to exactly one and a half times the upkeep term.

The one-half is the part that matters and it is easy to walk past. It means the cost per unit length of an optimally sized tube is proportional to its own cross-section and to nothing else. There is no free multiplier left over, no constant to be fitted, no scale to be chosen. Whatever the flow, whatever the pumping coefficient, whatever the metabolic price of the tissue — a segment’s weight in the network’s total is its own area.

Why that closes the question rather than opening it #

A weighted-shortest-network problem needs three weights, and a model that leaves them free predicts nothing, because the three weights are exactly what the fork angle is a function of. Handing the weights to a second optimisation would be the ordinary move, and it would also be the move that makes the answer unfalsifiable.

Here the first minimisation supplies them. That is the whole reason this argument reaches a number, and it is precisely what the rule with an exponent of two cannot do — where the same reasoning leaves a constant in the weight and the prediction walks across every angle a fork could have.

The network the cost is minimised over #

Three ends, three weights, one free point. The parent’s end sits below, the two daughters’ destinations sit above at stated bearings and distances, and the branch point is moved around inside the triangle they make. The cost of a configuration is the sum of each segment’s weight times its own length.

The two fork angles are read off afterwards, each measured from the direction the parent was already travelling in. Nothing about them is chosen; they are what the winning point happens to make.

The formula under test #

Written down before any of this was run: the cosine of the first daughter’s angle is the fourth power of the parent’s radius plus the fourth power of that daughter’s, less the fourth power of the other daughter’s, divided by twice the product of the two squared radii. Exchange the daughters for the second angle.

Every quantity in it is a radius. Nothing in it refers to where anything ends, how long any segment is, or how the three destinations are arranged. If that is right, the fork angle is a property of three numbers and of nothing else in the picture.

What was actually run #

Fifteen daughter ratios, from an even fork down to a twig one hundredth the radius of its sister. Twenty placements of the three ends at each — daughter destinations from fifteen to fifty-five degrees off the axis, at half a unit to four units away. Three hundred networks.

Each was minimised over separately, on a grid of forty-one by forty-one points re-centred on its own best cell and shrunk two hundred and twenty times. The minimiser was given the three ends and the three weights. It was not given the formula, and it has no term in it that could have been derived from one.

Nineteen have no angle at all #

Of the three hundred, 281 were compared and nineteen were not, because their cheapest configuration puts the branch point on one of the network’s own ends rather than inside it.

That is not a failure of the search. A weighted network with three destinations sometimes has no interior solution, and when it does not, the right answer is a junction with no fork rather than a fork with a small angle. Those nineteen are reported as having no angle to check, which is the honest thing to do with them and is also the first sign of what an exponent of two does to every placement.

The agreement #

Across all 281, the worst disagreement between the direct minimisation and the closed form is 1.83 × 10⁻⁴ degrees. Restricted to daughter ratios of a twentieth or more it is 4.0 × 10⁻⁵.

Two ten-thousandths of a degree is not a proof and is not meant to be one. It is the statement that a search which knows nothing about the formula lands where the formula says, everywhere the formula claims to apply, at every ratio and every arrangement of ends tried.

Where the disagreement concentrates #

At the lopsided end, and for a reason that is about the geometry rather than about the code.

The three weights have to close a triangle for an interior branch point to exist at all, and the margin by which they do it — the two daughters’ areas summed, less the parent’s — collapses as the square of the daughter ratio. At an even fork the margin is 0.4126; at a hundredth it is 9.93 × 10⁻⁵.

So at a twig the weight triangle is nearly flat, the cost surface around the optimum is nearly featureless, and a search has almost nothing to descend. The angle is still predicted sharply. The position of the branch point that makes it is barely determined, and at ratios of a thirtieth and below all twenty placements sit within a thousandth of having no interior branch point whatever.

The answer does not depend on where the branches end #

This is the claim the twenty placements exist to test, and it is worth stating as a claim rather than as a property of the formula, because the formula is what is being checked.

Across the twenty arrangements at a given ratio, the answers agree with each other to between 1.7 × 10⁻⁶ and 1.4 × 10⁻⁴ degrees. Move the destinations, lengthen the segments, swing them round — the branch point moves, and the two angles it makes do not.

An agreement that could not have failed is not a test #

The habit this collection runs on is that a claim gets given something it could fail on, and an assertion that has never rejected anything proves nothing. So the comparison was handed two deliberate mistakes.

The first: minimise the same network with each segment weighted by its radius rather than by its cross-section. The branch point moves and the fork opens to 101.906 degrees, which is 26.97 degrees from where the correct weight puts it.

The second: read the formula at an exponent of four on a junction whose radii were built at three. That quotes 90.000 degrees against the minimisation’s 74.935, and the two are 15.07 degrees apart.

A refutation that had to be checked for being one #

The first version of that check was worse than useless and passed anyway, which is the part worth recording.

It handed the minimisation a weight of the radius cubed against a junction built at an exponent of three. With a weight that goes as the radius to some power and a junction exponent, the three weights stop closing a triangle exactly when the two match — so that pairing is not a wrong weight, it is a degenerate one. It returned an angle of 35.000 degrees, which was the bearing of the ends themselves, and it returned it confidently.

A refutation that is really a degeneracy looks identical to a refutation that works. The difference is that the second one reports a number about the cost and the first reports a number about the picture, and only reading the value gave it away.

The angles themselves #

At an even fork each daughter leaves at 37.4673 degrees, and the total is 74.9346 degrees, which is twice the arccosine of two to the minus one third. That number is not fitted to anything; it is what the cube law does to a symmetric junction.

As the fork goes lopsided the large daughter straightens and the small one swings out. At a daughter ratio of a hundredth the large daughter is at 0.006 degrees — very nearly continuing the parent — and the small one is at 89.615.

What the sum hides #

One daughter’s angle runs across nearly ninety degrees over that range. Their sum runs from 74.935 to 89.621 — fourteen and a half degrees, monotone, with no interior turning point.

That matters because the sum is the one quantity readable from a photograph of a fork without callipers on three branches. It is the robust measurement and it is the uninformative one, which is a shape this collection keeps finding in its own instruments.

The cost around the branch point is nearly flat #

Everything so far says the optimum is sharp. The cost that locates it is not, and the second half of this essay is about what that does to the search.

Hold the three ends where the optimum wants them and move the branch point everywhere inside them. Positions costing one per cent more than the least cover a 43-degree span of total fork angle. A tenth of a per cent still covers nearly fourteen.

Which makes the search a measurement of its own settings #

A grid search that re-centres and shrinks converges because each round is smaller than the last. How much smaller is a number somebody chose, and on a surface this flat that number decides the answer.

Run the whole sweep at four shrink factors. The worst disagreement with the closed form anywhere comes out at 14.199 degrees at a shrink of 0.4, 5.880 at 0.5, 3.383 at 0.6, and 0.00018 at 0.75. The first three all land at the most lopsided ratio in the sweep, which is exactly where the weight triangle is nearly flat.

Fourteen degrees is not a rounding error. It is most of the range the total fork angle has.

And the cost cannot tell which one is wrong #

Here is the whole of the problem. At that junction the four searches report network costs agreeing with each other to about three parts per million, and fork angles fourteen degrees apart.

A search that has stalled does not return an error. It returns a point, at a cost indistinguishable from the right one, with an angle that is wrong by more than the quantity being measured. Every internal signal the search has says it has converged, because on a surface this flat convergence in the cost is nearly free and convergence in the position is nearly impossible.

What that would have been read as #

If the closed form had been in hand and the search had been left at a shrink of 0.4, the comparison would have come back disagreeing by fourteen degrees at the lopsided end and agreeing everywhere else.

The natural reading of that is that the formula is right in general and fails for very uneven forks — a plausible, publishable, entirely wrong conclusion, with a mechanism ready to hand in the collapsing weight triangle. The instrument’s failure and the hypothesis’s failure have the same shape, and this collection has mistaken one for the other before, where three consecutive samples of a fixed grid were reported as a repeating block that a standing prediction had asked for.

Which is why the convergence study is part of the result #

Reporting an angle from a search of this kind without reporting what the search’s own settings do to it is reporting one number from a family and calling it the answer. The family here spans fourteen degrees.

That is the same lesson a sample grid taught this site on its central discriminator: the quantisation was larger than the effect, and nobody had varied it because the routine was working perfectly.

A number withdrawn #

A first reading of the shrink sweep put the worst disagreement at forty-two degrees at a shrink of 0.4. It does not reproduce, and the figures above are the ones the sweep’s own worst-case column returns.

The point survives the correction, and is worth less than it would have been at forty-two, which is the correct amount to be worth. A number that changes when it is measured again is exactly the kind that should be re-measured before an argument is built on it, and the argument here needs the effect to be degrees rather than tens of degrees.

Two more places the instrument decides the answer #

The scan of the cost surface has to be kept away from the corners of the triangle of ends. The fork angles are undefined at a corner and unbounded beside one, so a scan allowed to run into a corner reported a one-per-cent band running past 300 degrees — a number about the corner and about nothing else.

And where the exponent is run down towards two, the placement compared against the formula is chosen by how far it sits from a network that does not fork at all, rather than by whatever the loop reaches first. Taking whatever came first put one such case half a degree out, because a nearly degenerate weight triangle admits only marginal placements, and those are precisely the ones a direct search resolves worst.

What the agreement establishes #

That the closed form is the minimiser’s answer, everywhere the minimiser has one, to a precision that leaves no room for a competing formula. That the fork angle depends on the three radii and on nothing else about the network. And that both facts come from the same cost that fixes the exponent, with nothing added between them.

That is a tighter chain than most of what this collection reports, because there is no fitting anywhere in it. Fitting an exponent to junctions returns whatever the junctions say; this returns what the cost says, and the only question is whether the cost is the right one.

What it does not establish #

Nothing about a plant, and the gap is wide enough to be worth spelling out.

This is arithmetic about a model, in a plane, at one junction, with the radii already set to their Murray values and the three destinations held fixed. A real branch has to hold itself up, has to have got where it is by growing rather than by being placed, and reaches ends that move while it is reaching them.

It also says nothing about whether an observed fork angle is evidence for or against the cost, because the cost is too flat in the angle for a departure to mean much — a separate argument with its own numbers, and the one that most limits what any of this is good for.

What would refute it #

A junction whose radii satisfy the cube law, whose branch point minimises this cost, and whose angles are not the ones the formula gives. That is a well-posed thing to look for and the sweep above is 281 attempts at finding one.

What would not refute it is a measured fork on a real tree at some other angle. That is a statement about trees, and it would need the argument to be about a mechanism rather than about a form before it could bear on the arithmetic at all.

Where the angle is worth using #

At the point where it becomes an instrument. The predicted total angle climbs 23.03 degrees per unit of exponent at Murray’s value, so one degree of fork angle is 0.043 of an exponent — and that is a sharper instrument than callipers on radii, at the precision this collection has assumed for its junction measurements.

The obstacle is not sensitivity. It is that the same flatness which makes a search stall also means nothing pushed a real tree to the minimum in the first place, and that an exponent of two predicts no angle at all rather than a different one.