A blend turns the twigs back first

Mixed forks turn back at the cost’s own share sized each branch of a crown by whichever of two radii was larger: the radius its flow asks for, and the radius the bending load on it asks for. On crowns whose forks were lopsided by different amounts, the first fork to open its daughters past their Murray angles was always one whose two daughters were exactly flow-sized and whose parent had only just been caught by stress, and the crown turned back at a twig pumping share of $q^4/(1 + q^2)$ — within fourteen thousandths of the cost’s own one half on every crown. The margin a regular crown had was gone.

That result leans on the switch. Under the larger of two radii a branch is either exactly what flow asks or thickened by stress, never a little of both, and the forks that decided the threshold were the ones the switch had only just reached. A branch that carries sap and bends in the wind is more likely sized by some blend of the two requirements, and under a blend no branch sits exactly on its flow radius. The question that essay left was whether a blend restores a margin by thickening every parent a little, how sharp the blend must be before the margin is lost, and whether the forks that turn back are still the ones nearest the handover.

The blend and the crowns it sizes #

The crowns are the same twelve: fourteen orders, each fork sending its thin daughter’s subtree nought to three orders shallower than its sister’s, the offset drawn afresh at each fork on a stated seed; every twig carrying one unit of flow and one of load; each branch 0.794 of its parent’s length; and the two radii scaled equal at the median branch of order seven. Only the rule that turns two radii into one changes. Each branch takes

$$r = \left(r_f^{\,k} + r_s^{\,k}\right)^{1/k},$$

the flow radius $r_f$ and the stress radius $r_s$ combined at an exponent $k$. At $k = 1$ the two are added; at $k = 2$ they combine in quadrature, as two independent requirements often are; and as $k$ grows the larger of the two takes over, so the earlier crowns are the limit $k \to \infty$. Nothing measured fixes $k$ for a tree. The family is used because it holds the sum and the larger-of rule at its two ends and is smooth between them.

No branch is flow-sized any more #

Under the larger-of rule, orders nought to seven are flow-sized exactly and the first thickening arrives at order eight, 1.085 times what flow asks. Under a blend the line never touches one. At $k = 4$ the twigs are 1.0076 times their flow radius, the branches that carry two twigs 1.0211 times, and the handover order 1.189 times; at $k = 2$ the twigs are 1.084 and the handover order 1.414, which is $\sqrt{2}$, because the two radii were set equal there. At $k = 8$ the twigs are thicker than flow asks by one part in ten thousand and the branches above them by nine.

That last number is the one the rest of this essay turns on. A blend thickens the twigs least, since at a twig the stress radius is only 0.42 of the flow radius and a small number raised to the power $k$ vanishes quickly. But it thickens the branch carrying two twigs a little more than either twig, because the stress on that branch is a larger share of its flow. So at every finite $k$ the forks of two twigs have a parent thickened slightly more than its daughters — a parent barely heavier, in the sense that matters for the angle, than the two branches it splits into.

The form, generalised to three thickened branches #

The angle the cost chooses sets each daughter’s angle from three weights, and a branch’s weight is its cost per unit length: its radius squared times one plus the share of its upkeep spent pumping. A branch $q$ times as thick as flow asks spends $\rho/q^6$ of its upkeep pumping, if a flow-sized twig spends $\rho$, so its weight is $q^2(1 + \rho/q^6)$ times what flow alone would have given it. The earlier essay had daughters at $q = 1$. Now suppose both daughters are thickened by the same $q_d$ and the parent by $q_p$. The daughters’ weights still stand in Murray’s ratio to each other, and both open past their Murray angles exactly when the parent’s factor is the smaller, which is when

$$\rho > \frac{q_p^{4}\, q_d^{4}}{q_p^{2} + q_d^{2}}.$$

At $q_d = 1$ this is the earlier $q^4/(1 + q^2)$. As $q_p$ and $q_d$ both approach one it approaches one half, from above, as $q_p^6/2$ does.

On every crown at every exponent up to eight, the first fork to open a daughter past its Murray angle is a fork of two twigs, and the measured share sits on the generalised form to within a millionth of itself — sixty crowns, five exponents. A fork of two twigs is exactly the kind of fork the form assumes: both daughters are order nought, thickened alike. The crown’s threshold at $k = 1$ is 5.23 on the first seed, at $k = 2$ 0.939, at $k = 3$ 0.623, at $k = 4$ 0.545, at $k = 8$ 0.5016.

Read strictly, every margin goes #

The strict reading is the one every earlier fork essay here used: a fork has turned back when either daughter opens past its Murray angle by any amount at all. Read that way, a blend is generous at small $k$ — the mixed crowns turn back at 4.91 to 5.23 when the radii are added and at 0.905 to 0.938 in quadrature — and then the margin drains away fast. At $k = 4$ the crowns turn back at 0.539 to 0.545, at $k = 6$ at 0.506 to 0.508, and at $k = 8$ at 0.5012 to 0.5016. From twelve on, every crown turns back within a ten-thousandth of one half.

The even crown makes the strangeness plain. Under the larger-of rule, a crown whose forks are all even keeps the margin the trend that stops at Murray’s angle found for it, 0.643 at this length ratio. Under a blend at $k = 8$ the same crown turns back at 0.5008. A blend is supposed to approach the larger-of rule as $k$ grows; read strictly, the margin it approaches is not the larger-of rule’s but none at all, and then it jumps back to 0.643 in the limit. The threshold is discontinuous at the very rule it was meant to approach.

The openings that make up a strict margin are invisible #

A threshold that jumps when the radii change by one part in ten thousand is measuring something about the instrument. What the strict reading counts is any opening at all, and the openings a blend produces near its threshold are as small as the thickenings that cause them. At $k = 8$, with the twigs pumping one per cent more than the first crown’s strict threshold, the furthest any daughter anywhere on the crown opens past its Murray angle is eight ten-thousandths of a degree; five per cent past it, two hundredths. At $k = 4$ the same two steps give a hundredth and seven hundredths. The larger-of rule has no such forks to open, because its twigs and their parents are exactly flow-sized and an all-flow fork sits on Murray’s angles exactly. The discontinuity is the difference between an opening that is tiny and an opening that is nought.

No protractor reads a thousandth of a degree, and no photograph of a fork reads a tenth. The fragile junctions are the informative ones showed how quickly an angle read off a real fork loses its last degree to the branch’s own taper and curvature. So the reading that asks what a field measurement could see is a different threshold: the twig share at which some daughter first opens past its Murray angle by more than a stated tolerance.

Read at a degree, the blend restores a margin #

Read at a degree, the picture turns over. The mixed crowns sized by the larger of the two radii turn back at 0.603 to 0.672 — already a margin above one half, which the strict reading hid. A blend turns them back later still, at every exponent tried: 1.12 to 1.15 at $k = 2$, 0.905 to 0.914 at three, 0.817 to 0.821 at four, 0.71 to 0.75 at six and at eight, 0.67 to 0.70 at sixteen, and 0.61 to 0.68 at forty-eight. On every one of the twelve seeds, a blend at two, four or eight turns back later than the larger-of rule does. The margin falls steadily as the blend sharpens and arrives at the larger-of rule’s from above, with no jump.

The even crown is above the mixed ones throughout — 1.08 at $k = 2$, 1.06 at four, 0.94 at eight and 0.72 under the larger — and it is not monotone: it turns back later at $k = 4$ than at $k = 3$, because the fork that first opens a degree changes order between the two, as the next section draws.

The margin as twig radii #

A share is abstract; a twig’s radius is not. A flow-sized twig spending $\rho$ of its upkeep on pumping has a radius $(1/(2\rho))^{1/6}$ times the cost’s own, so each threshold says how much thinner than the cost wants the twigs may be before some fork opens a degree past Murray’s angles. Under the larger-of rule that is 3.1 to 4.8 per cent. Under a blend at $k = 8$ it is about six per cent, at $k = 4$ about eight, and in quadrature about thirteen. Read strictly the same blends allow 0.05 per cent at $k = 8$ — a difference in twig radius no measurement of a twig could resolve, which is the strict reading’s verdict restated as a length.

So the earlier essay’s conclusion needs one qualification, and this is where it comes from. “The margin is gone” was true of the strict reading on the larger-of rule, where the first opening was decided by a parent 1.0003 times as thick as flow asks. Read at a degree, those same crowns hold a margin of a tenth or more of the share, and a blend holds a larger one.

A tenth of a degree is between the two #

A tenth of a degree is finer than a field instrument, and it sits between the two readings as it should. The larger-of rule’s mixed crowns turn back at 0.536 to 0.553; a blend at $k = 4$ at 0.586, at $k = 8$ at 0.555 to 0.559, at forty-eight at 0.536 to 0.547. The ordering survives up to $k = 8$ — no blend that soft turns back earlier than the larger-of rule on any of the twelve seeds — but it does not survive further: one seed in twelve turns back earlier at $k = 12$, three at sixteen and seven at forty-eight. At this tolerance the gap closes much sooner as $k$ rises, because the openings a sharp blend makes below the handover are worth a few tenths of a degree and no more.

The pattern across the three tolerances is the useful thing. The strict threshold is a statement about the first infinitesimal opening, and it is decided by whatever fork is closest to being exactly balanced. A threshold at a visible opening is decided by the fork whose daughters open fastest once they start, and those are different forks.

Which fork opens first #

Read strictly, the first fork is a fork of two twigs on every crown at every exponent up to sixteen. Only at twenty-four and above does it begin to climb, to order two or three, and at forty-eight to order four or five — because by then the thickening of the branches just above the twigs has become so small, falling as the $k$-th power of their stress ratio, that the forks a little higher, where the stress share is larger, are the closer to balance. Under the larger-of rule the first fork is the handover’s, order seven, as the earlier essay found.

Read at a degree, the first fork climbs steadily from the start: order one at $k = 2$, order three at four, five at eight, seven at sixteen, and seven or eight under the larger-of rule. The fork that decides a visible margin under a sharp blend is close to where the larger-of rule put it; under a soft blend it is at the twigs. So the third answer has two halves. The strict first fork is not near the handover at any finite $k$; the visible first fork moves to the handover as the blend sharpens, and is at the twigs only for blends softer than about three.

Where the crown opens when the twigs are thin #

With twigs pumping their whole upkeep — twice the cost’s own share — the whole crown can be read at once. Under the larger-of rule every fork below the handover sits on Murray’s angles exactly, and the openings start abruptly at order seven, peaking at 7.24° at order eight before the stress-sized orders close well inside. Under a blend at $k = 8$ the forks below the handover are open by fractions of a degree, the peak moves down to order six and shrinks to 2.75°. At $k = 4$ it is 2.31° at order four, and in quadrature the only fork open at all is the fork of two twigs, by 0.31°.

So a blend does two things to the opening at once. It spreads it down the crown, so that no fork is exactly on its Murray angles, and it flattens it, so that the largest opening anywhere is a third or less of the larger-of rule’s. A field study looking for daughters past their Murray angles on a blended crown would find many slightly open and none strongly open — the opposite signature from a crown sized by the larger, where the forks just above the handover open several degrees and everything below is exact.

Why the twigs, and why it fades #

The twig fork opens first under a soft blend for the same reason the barely thickened edge fork did under the larger-of rule: an optimum too flat to reach. A branch’s weight is flat in its radius at the cost’s own radius, so a parent a little thicker than its daughters, with twigs a little thinner than the cost wants, is lighter than its daughters expect. What the blend changes is where the least-thickened parent stands against its daughters. Under the larger-of rule it is a parent of order seven over two exactly flow-sized daughters; under a blend it is the branch carrying two twigs, over two twigs thickened very slightly less.

That pairing is close to balanced but not as close as it looks. Its threshold is about $q_p^6/2$ when the two thickenings are nearly equal, and $q_p - 1$ falls as the $k$-th power of the stress ratio of the branch carrying two twigs, which is 0.54: at $k = 4$ that branch is 1.0211 times its flow radius, and $(1 + 0.544^4)^{1/4}$ is 1.0212. So the strict margin vanishes geometrically in $k$, which is why it is 0.04 at $k = 4$ and 0.0014 at $k = 8$. A visible opening needs the parent to be appreciably lighter than its daughters, and at large $k$ only the forks near the handover, where the stress radius has caught up with the flow radius, have parents that differ from their daughters by enough.

Moving the reference changes nothing here #

The two radii are scaled equal at the median branch of order seven, and the earlier essay checked that moving that reference to either quartile of the order left the larger-of crowns turning back within a fortieth of one half. Under a blend the reference moves every branch’s stress ratio by a common factor, which shifts the twigs’ thickening and so the strict threshold at a given $k$ — but it moves the twigs and their parents together. Measured with the reference at either quartile, the first strict fork is still the fork of two twigs on every crown, and at $k = 8$ every crown turns back strictly between 0.5011 and 0.5019. The visible margin moves by a few hundredths and keeps its order: at a degree, 0.706 to 0.763 at $k = 8$ against 0.614 to 0.661 for the larger-of rule.

The same flatness, read at two resolutions #

One constant for every fork and the essays after it read the fork-angle trend as a sharp switch: below a twig share the daughters stay inside their Murray angles, above it they open. A blend shows that the switch is sharp only because the larger-of rule makes some forks exactly balanced. Remove the exactness and the strict switch moves to wherever the crown is nearest to balance, which is always within a hair of one half; the visible switch stays where the crown’s forks open quickly, which is a property of the crown’s shape and the blend’s softness.

This is the flatness that the cube law puts at the centre of the subject, met again from the side of measurement. A cost that is flat at its optimum makes small departures cheap, and so it makes the question “has the fork moved at all?” nearly always answerable with yes and nearly always uninformative. The answerable question is how far.

What a field study would read #

The earlier essay proposed reading a crown fork by fork: each fork’s angles, each branch’s radius and the twigs it feeds, giving each fork’s $q$ and a prediction per fork. A blend changes what to look for. If a crown is blended, its twigs and small branches are measurably thicker than flow alone asks — by nine per cent at the handover order at $k = 8$ and by nineteen at $k = 4$ — and thickening rises smoothly with order rather than switching on at one. That profile is the first thing to measure, because it says which sizing a crown has before any angle is read.

Then the angles. Under a blend, daughters past their Murray angles should be common and slight below the handover and largest a few orders below it; under the larger-of rule they should be absent below the handover and largest just above it. Which junctions say anything is the guide to which forks can carry that reading — the even forks — and the exponent an error moves is the warning about how much a radius error of a few per cent does to the $q$ any of this depends on.

What the blended crowns assume #

That each branch takes one blend of the two radii at one exponent throughout the crown; that the crowns are the earlier essay’s, offsets drawn evenly from nought to three; that every twig carries equal flow and load; that the transport cost sets the angles on radii fixed otherwise. The tolerances — nought, a tenth of a degree, a degree — are choices about an instrument, not about a tree. Twelve crowns are drawn. Nothing here measures a tree, and nothing says a tree blends at any particular $k$, or blends at all.

The blend in one paragraph #

Sized by a blend of flow and stress, no branch is flow-sized, and the forks of two twigs open first, on the closed form $q_p^4 q_d^4/(q_p^2 + q_d^2)$. Read strictly, that sends every crown’s threshold to within two thousandths of one half by $k = 8$, the even crown included, though the larger-of rule kept the even crown at 0.643. Read at a degree, a blend restores a margin at every exponent — 1.12 in quadrature, 0.82 at four, about 0.73 at eight, against 0.60 to 0.67 for the larger — and the fork that opens first climbs from the twigs to the handover as the blend sharpens. The margin was never only a property of the crown; it was a property of the crown read at a resolution.

What would overturn it #

A twig fork turning back off the generalised form. A crown at an exponent of eight or more whose strict threshold stands more than two thousandths above one half. A blend at two, four or eight whose one-degree threshold falls below the larger-of rule’s on the same crown. The first fork to open a degree sitting lower on the crown at a large exponent than at a small one. Each is checked on every crown measured above.

Still open: a blend that changes along the crown #

Every branch here is sized by one blend at one exponent. A tree’s twigs and its trunk are not built alike: a twig’s radius is set mostly by its water supply and its leaves, a trunk’s by wind and its own weight, and the way the two requirements combine may differ between them — closer to a sum near the twigs, where both are small and both are paid for, and closer to the larger near the trunk. The measurement is the same mixed crowns with the exponent running with order, from a soft blend at the twigs to a sharp one at the trunk, asking whether the visible margin is set by the soft end, where the twig forks open first, or by the sharp end, where the handover forks open most — and whether a crown blended that way shows the field signature of a blend below the handover and of the larger-of rule above it.