A trend that stops at Murray's angle

On a crown sized for equal bending stress, the transport cost predicts forks that open as they get smaller: 105 degrees a decade of radius on a planar crown, shrinking as the branches shorten more slowly. The prediction had a constant in it that nothing fixed. One constant for every fork showed it is one number for a whole tree and enters each fork scaled by its size, and to draw anything it was set so that a middle-sized fork opened at Murray’s angle — a calibration, not a measurement.

That essay ended on the tree a real one is more likely to be. A trunk carries sap and bends in wind at once, so each branch has to be thick enough for both, and which requirement binds can change from trunk to twig. The question it left was whether the fork-angle trend changes sign at the generation where the sizing hands over.

Whichever asks for more #

The crown here takes, at every branch, the larger of two radii: the one Murray’s flow rule gives, whose cube goes as the tips the branch feeds, and the one equal bending stress gives, whose cube goes as the sum of those tips’ lever arms. Both are measured on the same symmetric crown, every branch a fixed fraction of its parent’s length.

The flow radius falls by the cube root of two a generation. The stress radius falls faster, because a branch’s lever arm shrinks with its length as well as with its tips. So two lines that cross once, and the crossing is a generation: scaled to meet at generation six of a thirteen-generation planar crown, the stress radius is 2.03 times the flow radius at the trunk, and the flow radius is 2.11 times the stress radius at generation eleven.

Which end is which #

That ordering is not a choice. Deep in a crown a branch’s moment is its tip count times its length times a constant, so the ratio of the stress requirement to the flow requirement goes as the branch’s length. Long branches are near the trunk. The trunk end is sized by stress and the twigs by flow, whatever the constants, and the only free number is where the handover falls.

That fits what a cube law with a lever arm said a stress crown conserves and what three rules, one exponent said a flow crown conserves, stacked: the stress rule’s exponent near the trunk, and exactly three at the twigs.

The constant stops being free #

The fork angle comes from the cost each branch carries per unit length: pumping, which goes as the flow squared over the radius to the fourth, and upkeep, which goes as the radius squared. Written as the upkeep times one plus a share, each branch’s weight in the fork’s minimisation is its radius squared times $(1 + \rho)$, where $\rho$ is the fraction of its upkeep the branch spends again on pumping.

A flow-sized branch has one value of $\rho$ everywhere, because its radius to the sixth goes as its tips squared, and that single value is what the free constant was. The cube law fixes it: minimising the cost over the radius leaves pumping at exactly half the upkeep. So a crown whose twigs are the radii the transport cost itself prefers has $\rho = 1/2$ at every flow-sized branch, and there is no constant left to set.

Every branch’s pumping share #

A stress-sized branch is thicker than flow alone asks for, so it spends less on pumping, and the share falls towards the trunk by roughly the square of the length ratio a generation. On the planar crown it is 0.242 one generation inside the handover and 0.007 at the trunk; on a crown shortening by 0.87 a generation it is 0.338 and 0.064.

That share is the only thing separating a fork’s three weights from its radii squared, and radii squared are exactly the weights that predict a fork at every angle and none on an area-conserving tree. A trunk that spends almost nothing on pumping has almost Da Vinci’s weights.

The trend, measured #

Handing over at generation six, the forks of a planar crown open from the trunk outwards at 13°, 18°, 26°, 36°, 50° and 66°, and every fork from the handover outward at 74.93°. Handing over at three, the trunk opens at 38° and the angle reaches Murray’s by generation three; handing over at eight, the trunk opens at 4°.

It stops, and it does not turn #

No fork on any of those six crowns opens wider than Murray’s angle, and none turns back. The trend Da Vinci’s rule predicts is confined to the stress-sized generations; it rises through them and meets Murray’s 74.93° at the handover, and the flow-sized generations beyond carry that angle exactly. So the answer to the question as asked is no. The sign does not change; the slope goes to zero.

Across four length ratios and seven handovers, every crown whose handover sits five or more generations above the tips rises at every generation. With the handover four generations above the tips, the two crowns that shorten slowest let their trunk forks narrow by under half a degree before the rise, which is the arms near the tips not having converged rather than any turning at the handover.

Why nothing passes Murray’s angle #

At a symmetric fork the cosine of half the angle is the parent’s weight over twice a daughter’s, and the weight is the radius squared times one plus the pumping share. Two things move that ratio, and they pull in opposite directions. Each radius is the larger of two requirements and both at least double from daughter to parent, so the radii squared stand in at least the ratio flow gives them, which on its own closes the fork towards the trunk. But a stress-sized parent spends a smaller share on pumping than its daughters, which lowers the parent’s weight against theirs and opens the fork.

Which one wins is a matter of size, not of sign. With twigs at the cost’s own radius the radii win at every generation inside the handover, the fork stays narrower than Murray’s, and the only fork that reaches Murray’s angle is one whose branches are all flow-sized. The measured crowns agree to the last decimal: the widest fork on any of them is Murray’s. That the pumping share can win instead is the whole of the section on thin twigs below.

The stress region has a closed form #

Deep in the stress region the ratio of a parent’s radius cubed to a daughter’s is $2/\lambda$, and the pumping share falls by $\lambda^2$ a generation towards the trunk. A fork whose parent sits $k$ generations inside the handover then has

$$\cos\frac{\theta_k}{2} = \Bigl(\frac{2}{\lambda}\Bigr)^{2/3}\frac{1 + \rho\,\lambda^{2k}}{2\,\bigl(1 + \rho\,\lambda^{2k-2}\bigr)}$$

with $\rho$ the twigs’ share, which is Murray’s angle at $k = 0$.

With the handover eleven generations above the tips, the planar crown’s forks read 27.6°, 38.1°, 51.4° and 67.0° against 27.9°, 38.4°, 51.7° and 67.1° from the form, a largest gap of 0.32°. The gap is 0.41° at 0.74, 0.67° at a volume-filling crown’s 0.794, and 1.26° at 0.87, where the arms converge slowest.

Its limit is the stress tree’s #

Far inside the handover the pumping share goes to nothing and the form tends to $(2/\lambda)^{2/3}/2$, which is exactly the small-fork limit the stress-sized crown found, $2\arccos\bigl(2^{2/p-1}\bigr)$ with $p$ the stress rule’s exponent: 0° for a planar crown, 28.07° at 0.74, 44.40° for a crown filling a volume and 58.87° at 0.87.

So the joined crown is the stress crown’s trend with the calibration replaced by a boundary. Its trunk end runs down towards the stress tree’s narrowest possible fork, its handover end runs up to Murray’s, and the number of generations in between sets how far the trunk gets.

The gap near the tips #

The form is a deep-crown statement and it degrades where the arms have not converged. On the planar crown its largest gap is 0.32° with the handover eleven generations above the tips, 0.83° at nine, 2.2° at seven, 6.1° at five and 10.5° at four, rising at every step. A handover close to the tips puts the whole stress region in the part of the crown where the stress radius itself is still approaching its limit.

That matters for reading a real crown, which has fewer generations than any of these: the form is the shape of the trend, not a table to look angles up in.

The exponent jumps at the same generation #

Above the handover the junctions conserve a little under the stress rule’s deep limit; from the handover outward, exactly three. On the planar crown the trunk conserves 1.997 and the junction just inside the handover 1.967; at 0.87, 2.453 and 2.341. The exponent jumps at the generation where the fork angle stops rising.

That is a useful redundancy. Two different measurements on a real crown, one of radii and one of angles, name the same generation, and a crown whose two readings put the handover in different places is a crown this account does not describe.

Twigs that are not the cost’s own #

Nothing forces a tree’s twigs to the transport cost’s own radius, and the calibration the earlier essays needed was in effect a statement about how far from it they sit. So the same crown is measured again with twigs spending a quarter, a half, three quarters, one and two times their upkeep on pumping.

The fork just inside the handover opens at 49.8° when the twigs spend a quarter, 66.3° at a half, 76.1° at three quarters, 82.8° at one and 96.9° at two. The last three overshoot Murray’s angle, and every fork from the handover outward is back at it. Those crowns turn their trend exactly once, at the handover.

A reversal needs thin twigs #

A twig spending more than half its upkeep on pumping is thinner than the cost wants: flow is being driven through too narrow a tube. The stress-sized branches just inside the handover are then thick relative to their daughters’ pumping share, the weights lean towards the daughters, and the fork opens past Murray’s angle before the flow-sized generations pull it back.

So the sign change the question asked about exists, and it is a statement about the twigs. It says nothing about the trunk, the wind or the handover’s position; it happens when, and only when, the twigs are sufficiently far on the thin side of the cost’s own radius.

How thin #

Setting the closed form’s fork at $k = 1$ equal to Murray’s angle gives the threshold share exactly: $\rho^* = (\lambda^{-2/3} - 1)/(1 - \lambda^{4/3})$.

It is 0.974 at halving lengths, 0.702 at the planar crown and 0.505 at 0.99. Found by bisection on thirteen-generation crowns handing over at generation six, it is 0.714 at the planar crown, 0.687 at 0.74, 0.649 at 0.794 and 0.606 at 0.87 — a little above the form at every ratio, by more on the slowly shortening crowns, for the same unconverged arms.

The margin closes as branches stop shortening #

The cost’s own twig spends one half, which is below the threshold at every length ratio, so a crown with cost-optimal twigs never turns back. But the margin is not uniform. A fast-shortening crown tolerates twigs spending nearly their whole upkeep on pumping before the trend reverses; a crown whose branches barely shorten reverses once its twigs spend a few per cent more than optimal.

That is the same crowns that an optimum too flat to reach would worry about most. A few per cent of cost buys tens of degrees of angle, and the crowns where Murray’s angle is closest to the stress tree’s range are the ones where a small departure from the optimum changes the shape of the trend.

Drawn #

Drawn with every fork at the angle the cost predicts from that fork’s own radii and tips, the crown with cost-optimal twigs closes tight at its trunk and opens steadily to Murray’s angle, then keeps it; the crown with thin twigs opens to 98° one generation inside the handover and closes back. The trees drawn at no angle is the reminder that a drawing is a picture of the prediction, and nothing in either was chosen to look like a tree.

What the stress crown’s outer forks were #

On the crown sized by stress alone, calibrated at a middle fork, the outermost generations opened to the 120° ceiling. That looked like a feature of the stress rule and it was a feature of the calibration: nothing stopped the constant’s share growing without limit towards the twigs. On a crown whose twigs are sized for flow the share stops at the twigs’ own value, and the outer forks sit at Murray’s angle.

So the ceiling at the tips was an artefact of asking one rule to size a whole crown, and the flat run of Murray’s angle is what replaces it once the rule that binds at the tips is allowed to bind there.

What a field study would read #

The fork angle generation by generation, from trunk to twig, with each generation’s radii and tip counts. On a crown this account describes, the angles rise and then flatten, the exponent drops below three at the same generation, and the angles in the rising part follow the closed form in the crown’s length ratio.

The reading that would say most is the one nobody would think to take: whether any fork just inside the flat run opens wider than the flat run itself. If one does, the twigs are thinner than the cost wants, and the closed form says by how much.

What it assumes #

That each branch takes the larger of the two radii, rather than a blend of them, and that the flow radius is proportional to the tips a branch feeds throughout. That the loads bending the crown are on its tips alone. That the transport cost sets the angles on radii set by other means, which is the combination the earlier fork essays accepted and is carried unchanged. And that every fork is even: a lopsided fork has two angles, and nothing here says how they share the trend.

The claim, reduced #

On a crown sized by the larger of flow and stress, stress sizes the trunk end and flow the twigs, and twigs at the cost’s own radius spend exactly half their upkeep on pumping, which fixes the constant. The fork angle then rises through the stress generations along a closed form, meets Murray’s 74.93° at the handover and stays there, with no fork anywhere wider; the trend turns back, once and at the handover, only when the twigs spend more than $(\lambda^{-2/3} - 1)/(1 - \lambda^{4/3})$ of their upkeep pumping.

What would withdraw it #

A flow-sized fork away from Murray’s angle. A crown with cost-optimal twigs carrying a fork wider than Murray’s anywhere. A reversal at a twig share below the measured threshold, or none above it. A stress region whose angles depart from the closed form by more as the handover moves away from the tips. Each is checked whenever the crowns are measured.

Still open: forks that are not even #

Every fork here has two equal daughters, so it has one angle and one weight triangle. A lopsided fork has two angles, and the angle the cost chooses puts the thinner daughter further from the parent’s line. On a crown sized two ways the two daughters of a lopsided fork near the handover can be sized by different requirements, one by stress and one by flow.

The measurement is the joined crown with a fixed daughter ratio at every fork: whether each daughter’s angle meets its own Murray value at a different generation, whether the thinner daughter’s angle can overshoot while the thicker one’s does not, and whether the flat run survives at all once a fork’s two branches are allowed to disagree about which requirement binds.