Two daughters cross Murray's angle together

On a symmetric crown whose every branch takes the larger of the radius flow asks for and the radius bending stress asks for, the fork-angle trend stops at Murray’s angle rather than turning. The forks rise through the stress-sized generations near the trunk, meet 74.93° at the generation where flow takes over, and hold it all the way to the twigs. They turn back past Murray’s angle only when the twigs are thinner than the transport cost wants, spending more than a threshold share of their upkeep on pumping.

Every fork in that account had two equal daughters, so it had one angle. The question it left was the lopsided fork, which has two. The angle the cost chooses puts the thin daughter further from the parent’s line than the thick one, and on a crown sized two ways the two daughters of one fork can be sized by different requirements. Does each daughter meet its own Murray value at a different generation? Can the thin daughter overshoot while the thick one does not? Does the flat run survive at all?

A crown whose thin side is shallower #

A lopsided fork needs a reason to be lopsided, and on a real crown the reason is that one daughter feeds less than the other. The crown here builds that in through depth. Every branch has an order, its height in forks above the twigs. A branch of order $h$ forks into a thick daughter of order $h - 1$ and a thin daughter of order $h - 1 - \Delta$, so the thin side’s subtree is $\Delta$ orders shallower than its sister’s. An offset of nought is the even crown again. An offset of one is the Fibonacci tree, whose twig counts run through consecutive Fibonacci numbers: fourteen orders of it carry 987 twigs, against 16,384 on the even crown of the same height.

A branch’s length is a fixed fraction $\lambda$ of its parent’s for each order it sits below the trunk, so every twig has one length and a thin daughter is $\lambda^\Delta$ the length of its sister. Every twig carries one unit of flow and one unit of load. The flow radius goes as the cube root of the twigs a branch feeds; the stress radius as the cube root of the sum of those twigs’ lever arms, measured on the crown as drawn. The two are scaled equal at order seven, and each branch takes the larger.

Because a branch’s whole subtree is fixed by its order, every fork of one order is the same fork. A fourteen-order crown is read as fourteen forks, trunk to twig, with two angles at each.

The handover becomes a band #

On the even crown there is one fork where the requirements disagree: the fork one order above the handover, whose parent is sized by stress and whose daughters, at the handover order, are sized by flow. Every fork above it is stress-sized throughout, every fork below it flow-sized throughout.

A lopsided crown has more. The thin daughter of a fork sits $1 + \Delta$ orders below its parent, so it crosses into the flow-sized orders while its parent and its sister are still sized by stress. There are forks where the parent and the thick daughter are stress-sized and the thin daughter is not, and they run for $\Delta$ orders above the fork the even crown already had. Counted on all four crowns at every length ratio and at twig shares of a half and two, the band of mixed forks is exactly $\Delta + 1$ wide and always ends at the fork one order inside the handover.

So the first answer is structural. The handover, which on a symmetric crown is one generation, is on a lopsided crown a band whose width is the offset plus one. A tree with very lopsided forks has its two requirements disagreeing over a wide stretch of the crown.

The flat run survives, fork by fork #

Below the band every fork has three flow-sized branches. On the symmetric crown that was where the constant cancelled: a flow-sized branch spends the same share of its upkeep on pumping wherever it is, so each branch’s weight in the angle’s minimisation is its radius squared times one number, and the fork opens at Murray’s angle.

The lopsided crown keeps that. Each daughter’s weight is its radius squared times one plus the twigs’ share, and the radii are in Murray’s ratio, so each daughter opens at the angle Murray’s law gives it for that fork’s own daughter ratio. Across 560 flow-sized forks on sixteen crowns at five twig shares, the largest departure is under a ten-trillionth of a degree.

What changes is that the angle is not one number. On the Fibonacci crown the twig counts at a fork stand near the golden ratio, so the thick daughter opens at 30.58° and the thin one at 44.52°, a total of 75.1° rather than 74.93°. Near the twigs the ratio is a small-number one — two twigs against three, one against two — and the Murray angles move with it: 31.6° and 43.4° at order three, 27.7° and 47.6° at order two, and an even 37.47° each at the last fork, whose daughters are both single twigs. The flat run survives as a run of each fork’s own Murray angles.

Both daughters with cost-optimal twigs #

With the twigs at the radius the transport cost itself prefers — spending exactly half their upkeep on pumping — the Fibonacci crown at a length ratio of 0.794 does what the even crown did, twice over. The thick daughter opens at 10.0° at the trunk and rises through 11.6°, 13.9°, 16.8°, 20.5° and 25.0° to 29.0° at the last mixed fork. The thin daughter rises from 15.9° to 42.0°. Each then sits at its Murray value from order seven down.

Neither daughter, at any fork, opens wider than its own Murray angle. That holds at every offset and every length ratio measured, which is the even crown’s result carried across: with cost-optimal twigs the trend still stops rather than turns. And both daughters reach their Murray values at the same fork — the first with three flow-sized branches — because at that fork both are fixed by the same three weights.

Moving the slider lifts both curves together. At a share of a quarter the fork just inside the handover has the thick daughter 5.7° inside its Murray value and the thin one 9.1° inside; at three quarters both are past, by 1.0° and 1.7°; at one, by 2.8° and 4.8°. The thin daughter is always further from its Murray value than the thick one, in whichever direction, and the two change sign together.

Why they cross together #

The question was whether the thin daughter could overshoot alone. It cannot, and the reason is a property of a triangle.

The first fork that can overshoot is the one at the end of the band: its daughters are both flow-sized and its parent is not. Both daughters’ weights there are exactly what Murray’s law would give them, since they are flow-sized branches with the twigs’ share. Only the parent’s weight differs. The two angles come from one weighted triangle — the cosine of each is set by the three weights through the law of cosines — and with the two daughter sides fixed, lengthening the parent’s side narrows both angles at once, while shortening it opens both.

So there is one value of the parent’s weight at which both daughters sit on their Murray angles, and it is the weight Murray’s law gives the parent. Above it both are inside; below it both are past. The thick and thin daughters cross their Murray values at one twig share, by construction.

The thin daughter moves faster with the share — further inside at small shares and further past at large ones — but it crosses zero at the same place. At a length ratio of 0.794 that is a share of 0.638 for an offset of one, 0.631 for two and 0.638 for three. Found by bisection separately for each daughter on all sixteen crowns, the two thresholds agree to within a billionth.

A threshold that barely notices the offset #

The even crown had a deep closed form for this threshold, $\rho^* = (\lambda^{-2/3} - 1)/(1 - \lambda^{4/3})$: 0.702 at the planar ratio, 0.628 at 0.794 and 0.574 at 0.87. The measured even crowns of fourteen orders sit a little above it, at 0.710, 0.643 and 0.599, because the arms near the twigs have not converged — the same gap the earlier crowns showed.

The lopsided crowns sit beside them. At the planar ratio they read 0.706, 0.701 and 0.710 for offsets of one, two and three; at 0.794, 0.638, 0.631 and 0.638; at 0.87, 0.593, 0.584 and 0.589. The offset moves the threshold by under two hundredths at every ratio, and not monotonically.

That makes sense of the identity above. The threshold is decided at the one fork whose daughters are flow-sized and whose parent is not, and what decides it is the parent’s weight against the weight flow would give it. A lopsided parent’s stress radius depends on its own subtree’s arms, which the offset changes a little; the rule that decides the crossing does not change at all.

Past the threshold, the daughters peak apart #

Above the threshold the two daughters still cross together, but they do not reach their largest overshoots together.

With twigs spending twice their upkeep on pumping, the thin daughter’s largest overshoot is always at the fork one order inside the handover, the first whose daughters are both flow-sized: 11.6° past its Murray angle at an offset of one, 13.1° at two and 16.1° at three, on the crown at 0.794. The thick daughter’s is further towards the trunk — at order nine for an offset of one and order ten for two and three — by 10.0°, 9.1° and 8.1°.

So the even crown’s single peak, at 8.9° on both daughters, splits in two. The thin daughter’s grows with the offset and stays at the end of the band; the thick daughter’s shrinks and moves into it. At a length ratio of 0.87 with an offset of three, the thick daughter peaks three orders further in than the thin one; at the planar ratio with the same offset, the thin daughter overshoots by 20.6° and the thick by 7.9°.

Where the thick daughter peaks #

The thick daughter’s largest overshoot sits, on every lopsided crown measured, at a fork inside the band: a fork whose thin daughter is flow-sized while the thick one is still sized by stress. That is the disagreement the question asked about, and it is where the thick daughter does its own turning.

At such a fork the three weights are three different kinds of thing. The parent and the thick daughter are thicker than flow asks, so they spend less than the twigs’ share on pumping; the thin daughter spends exactly the twigs’ share. When that share is large, the thin daughter weighs more, relative to its sister, than Murray’s ratio gives it, and a heavier daughter stays nearer the parent’s line while the lighter one swings out. At order nine of the Fibonacci crown with twigs spending twice their upkeep, the thin daughter weighs 0.832 of its sister against Murray’s 0.726, and the thick daughter opens 10.0° past its Murray angle. With cost-optimal twigs the same fork has the thin daughter at 0.707 of its sister and the thick one 5.6° inside. What the measurement fixes is where the effect lands: inside the band, on every crown with an offset.

The order a field study would read #

That gives a lopsided crown two readings an even one does not have. On a crown whose twigs are cost-optimal, the two daughters’ angles rise together and reach their Murray values at the same fork, which is where the band of mixed forks ends. On a crown whose twigs are thin, the thin daughters’ angles peak at that same fork and the thick daughters’ peak above it, by as many orders as the band is wide less one, or close to it.

Neither reading needs a radius. A photograph of a lopsided crown with its twig counts known gives every fork’s two Murray angles; the angles themselves say where each daughter turns. A crown on which the thin daughter peaks above the thick one is a crown this account does not describe.

A lopsided trunk that does not fork #

One more thing changes, and it is the one that was not asked about. On the symmetric planar crown — a length ratio of $2^{-1/2}$, the ratio at which a crown fills a plane — the stress-sized forks close towards nothing near the trunk without quite getting there: 4.6° at the trunk of the fourteen-order crown, conserving an exponent of 1.999, a hair inside two.

A lopsided planar crown goes past it. With an offset of one, every fork from order ten up to the trunk has a parent outweighing both daughters together, the weight triangle cannot close, and the cost puts no fork there at all: the branch point runs to the end of the parent, as it did for an exponent of two exactly. Those junctions conserve an exponent of about 1.74. The trunk fork forms above a length ratio of 0.704 on the even crown, 0.775 with an offset of one, 0.806 with two and 0.822 with three.

The converse does not hold: the fork at order nine of the same crown conserves 1.736 and does form, because its thin daughter is flow-sized and pumps more, which lifts its weight. Every fork that fails to form conserves an exponent below two, but not every junction below two fails to form, once the pumping share is in the weights.

What Da Vinci’s rule looks like from here #

Which junctions say anything found that Da Vinci’s rule and Murray’s law differ most at even forks and least at twigs. A crown sized by stress near its trunk conserves something close to Da Vinci’s two there, and a cube law with a lever arm gave the stress rule’s exponent as a function of the length ratio alone.

The lopsided crown adds that the shape of the subtree enters too. At 0.794 the even crown’s trunk junctions conserve 2.24, the Fibonacci crown’s 2.02 and the crown with an offset of two 1.87. The deeper the asymmetry, the further below Murray’s three and the nearer to — or past — the boundary at which the angle stops being an angle the trunk end of the crown sits.

What the drawing shows #

Drawn with every daughter at the angle the cost predicts from its fork’s three radii, the crown with cost-optimal twigs closes tight at the trunk, its thin daughters always further out than their sisters, and opens towards the twigs. The crown with thin twigs fans wide in the band and closes back below it. As with the trees drawn at no angle, the drawing is a picture of the prediction and nothing in it was chosen to look like a tree.

What the lopsided crown assumes #

That the crown is a Horsfield tree with one offset at every fork, that a thin daughter is shorter than its sister by the same factor per order as every branch is shorter than its parent, and that every twig carries equal flow and equal load. The lever arms come from one drawn geometry, with the daughters at thirty degrees either side for the purpose of measuring arms. And, carried from every earlier fork essay, that the transport cost sets the angles on radii set by the larger of the two requirements. An optimum too flat to reach is the standing caution: a per cent of cost buys tens of degrees of angle, and nothing here says a real crown pays it.

The lopsided crown in four sentences #

On a lopsided crown sized by the larger of flow and stress, the handover is a band of $\Delta + 1$ forks. Every fork past it opens each daughter at the Murray angle for its own daughter ratio, exactly. The two daughters cross their Murray angles at one twig share, because at the first fork inside the handover only the parent’s weight differs from Murray’s — a share within two hundredths of the even crown’s, at every offset and length ratio. Past it the thin daughter peaks at the end of the band and by more; the thick daughter peaks inside it. A lopsided crown near the planar ratio has trunk forks that do not form.

A daughter past its angle alone would end it #

A fork with three flow-sized branches whose daughters sit off their own Murray angles. A twig share at which one daughter overshoots and the other does not. A crown with cost-optimal twigs carrying either daughter past its Murray angle anywhere. A band of mixed forks of any width but the offset plus one. Each of these is checked on every crown measured above.

Still open: a crown whose forks are lopsided by different amounts #

Every crown here has one offset at every fork, so the band of mixed forks is a clean stretch of orders and the threshold is decided at one fork. A real crown is lopsided by different amounts at different forks, and its thin daughters cross into the flow-sized orders at different heights. The identity that makes both daughters cross together is local — it holds at any fork whose daughters are both flow-sized and whose parent is not — but a crown with mixed offsets has many such forks at different orders, each with its own parent weight.

The measurement is a crown whose offset is drawn at random at each fork, from nought to three: whether the forks that first overshoot are still the ones whose daughters have just become flow-sized, whether the crown’s threshold becomes a spread of thresholds or stays one number set by its worst fork, and whether the thin daughters’ peaks still mark where each part of the crown hands over.

Measured, mixed forks turn back at the cost’s own share: with the offset drawn afresh at each fork, the first fork to turn back is still one whose daughters have just become flow-sized, and each such fork turns back at $q^4/(1 + q^2)$ of its parent’s thickness over flow; the threshold becomes a spread over the crown, and the crown’s own falls to within a hundredth or so of the cost’s one half, because some parent at the handover is only just thicker than flow asks.