A parent sized for its daughters

A base that gives resized a buckling crown for the bases its parents really give and found three failures. With every load along its branches the junctions alternate about two from generation to generation, 0.878 at the first fork and 17.85 at the next. Under gravity, 226 branches of a thirteen-generation crown turned 20° at every fork have no radius that holds them, and 3,904 above them are left undetermined. And joined with bending, the division of the crown by direction — an upright core sized against buckling, a spreading shell sized for bending — holds at some balance angles and not at others.

All three had the same cause, and the essay said so at the end: the rule asked each branch to repair a margin that was not its own. A daughter turning back toward vertical off a steep parent carries more compression than its parent’s tilt allotted, and once that compression times the daughter’s length passes the stiffness of its parent’s tip, no thickening of the daughter can help. The requirement belongs to the parent. The measurement here puts it there.

The ceiling, and who owns it #

A column of stiffness $EI$ and length $L$ standing on a rotational spring of stiffness $k$ buckles at $EI\varphi^2/L^2$ with $\varphi\tan\varphi = kL/EI$. On a soft base the load falls towards $k/L$, the load at which a perfectly rigid rod on the same spring tips over, and it can never pass it.

That ceiling does not contain the column’s own stiffness. It contains the spring’s and the column’s length. For a branch in a crown the spring is its parent’s tip, which turns by $M L_p / (EI_p)$ under a moment $M$, so the parent gives $k = EI_p / L_p$ and the daughter’s ceiling is $EI_p/(L_p L_d)$. Hold a daughter to a share $\kappa$ of its buckling load and it can be sized at all only if

$$\kappa \cdot \frac{\pi^2}{4} \cdot N_d \, L_d \;<\; \frac{EI_p}{L_p},$$

which is a statement about the parent’s radius, the daughter’s compression $N_d$ and both lengths. Nothing about the daughter’s own radius appears in it. That is why the 226 branches in the resized crown were stuck: the rule reached them with their parents already fixed.

A base term, with a margin #

So give the parent a second term. Beside the radius its own column needs, it takes

$$r_p^4 \;\ge\; \mu \cdot \kappa \cdot \frac{\pi^2}{4} \cdot L_p \cdot \max_d \left( L_d \, N_d \right),$$

with $\mu$ a stated margin over the rigid-rod ceiling, and each branch is sized, from the trunk outward, by the larger of the two. Under gravity a third term enters when bending is joined in, and the branch takes the largest of the three. The crowns are the ones every essay on this thread has read: thirteen generations, each branch $2^{-1/2}$ of its parent’s length — the planar crown, where stress and stiffness agree on two — forks turned 15°, 20° or 30°, equal loads on the tips.

The base term has one property that decides everything below. It reads only loads and lengths, and a crown’s shape fixes both before any radius is chosen. The column term on a giving base depends on the parent’s radius, which is what made the resized crown’s generations push each other about. The base term does not, so wherever it governs, the feedback that caused the alternation is cut.

Sized for its daughters, a crown has a radius everywhere #

The first answer is the simplest and it is exact. On the parent’s base, at any margin over one, no branch of any crown read here is stuck. The ceiling is built into every parent before its daughters are sized, so every daughter stands on a base that can hold it, and the column solve always finds a radius. At a margin of 1.05, barely over the ceiling, the crown turned 20° that left 226 branches unsizable leaves none.

Drawn at a margin of 2.5, the colours say where each term matters. The base term sizes the interior — 249 of the nine-generation crown’s 511 branches — and the tips, which have no daughters, are sized by their own columns. Bending takes the few branches near level at the flanks. At a margin of 1.25 the same crown is mostly sized by its own columns, with the base term on 71 branches: parents tilted far from vertical, whose more upright daughter carries more compression than they do, which are the forks the closed form in the turn had already marked as the weakest.

The alternation stops, at a closed form #

The alternation was the axial crown’s defect, so the axial crown is the place to test whether sizing the parent removes it.

With no base term the junctions run 0.878, 17.85, 1.100, 5.353 and on down to 1.640 at the eleventh. At a margin of 1.5 the swing is smaller — 3.178 at the second junction, 1.503 at the third — but it is still a swing. At a margin of 2 it has nearly gone, 1.951 and 2.044, and still a hundredth off two at the eleventh. At 2.5 every interior junction conserves two, to the ninth decimal place.

The margin at which that happens has a closed form, and it comes from the essay before. There, an axial crown sized as if clamped kept 0.561 of its load on its parent’s base at every branch, and thickening every branch alike by $1/0.561$ in $r^4$ repaired it exactly. That uniformly thickened crown is a fixed point: every branch carries its load at the intended share and nothing needs to move. The base term reaches it for an interior branch when

$$\mu^* = \frac{4}{\pi^2} \cdot \frac{2}{\lambda} \cdot \frac{1}{m_0}, \qquad m_0 = \left(\frac{2\varphi_0}{\pi}\right)^2,\ \ \varphi_0\tan\varphi_0 = \frac{2}{\lambda},$$

which is 2.0431 at the planar ratio. Above it the base term sizes every interior branch at a fixed multiple of its clamped $r^4$, and a crown whose radii are all one multiple of the clamped crown’s conserves the clamped crown’s exponent — two, here — at every interior junction. Below it the base term only props up the low side of the zigzag, which is why the swing at 1.5 and 2 is damped rather than gone.

The last junction is the exception, at 1.601 when the margin is 2.5. Its daughters are tips, and a tip has no daughters to demand a base, so it is sized by its own column on a parent that was sized generously for it. The junction under the tips reads the gap between the two terms rather than either one — the same shape of edge effect the crown sized for how far it bends found near its tips for a different reason. The outermost junctions are the ones a survey can reach, so a crown sized this way would show a field worker its one anomalous generation first.

Under gravity, the clamped form one fork up #

Gravity was where the closed form broke last time. Here it comes back, and it comes back as a shift.

The base term makes a parent’s $r^4$ go as $\lambda L_p^2$ times the compression of its most upright daughter. For a parent tilted by at least one fork’s half-angle, the more upright daughter points one half-angle nearer vertical than the parent does, so every base-sized branch is sized as the clamped crown would size a branch one half-angle more upright. The junction therefore conserves exactly what the clamped buckling crown conserves one fork further up: with forks of 20°, 1.914 at a parent tilt of 20° where the clamped crown gives 1.914 at an upright parent, 1.908 at 40°, 1.881 at 60°, 1.751 at 80°. Every junction the base term sizes whole, three or more generations above the tips, sits on that form to $10^{-14}$.

The upright parent is the one case the shift does not cover, because both its daughters lean outward by the same angle and its most upright daughter is still a half-angle off vertical. It conserves 2.094 with forks of 20° — above two, where the clamped crown’s upright junction conserves 1.914 — and 2.051 with forks of 15° and 2.232 with forks of 30°. The trunk and the core above it are a little thinner, in exponent terms, than the flanks around them.

The shift also moves the edge. The clamped crown’s form ran out at 70° with forks of 20°, where a daughter reached level and stopped carrying compression. The base-sized form runs out only as the parent itself approaches level, because a parent at 80° still has one daughter at 60° that needs a base. A branch pointing nearly level, which buckling alone could not size at all, is now sized — by what the branch above it needs.

The whole path wants more #

The base term here is written for the parent’s give alone. When the whole path to the ground gives, a daughter’s base is the sum of every ancestor’s compliance, and a term sized for one ancestor underpays.

On the parent’s base the count is zero at every margin over one. On the whole path’s base it is 402 at a margin of 1.05 and 358 at 1.2, falls to 66 at 1.4 and stays there to 2, drops to 14 at 2.2 and reaches zero only at 2.4. The 402 is more than the 322 stuck with no base term at all, and the reason is bookkeeping rather than weakness: without the base term 4,960 branches were never tested, because they stood above a stuck one, and now they are reached. The ancestors’ compliances fall off geometrically toward the ground — the axial crown’s settled to $1/(1-\lambda/2)$ of the parent’s own share in the giving-base reading — so one fixed multiple of the parent’s term covers most paths at a modest margin. Which paths hold the plateau of 66 through the middle of the range was not traced; it is the one feature of this figure the argument here does not explain.

So the margin that makes a crown sizable depends on how much of the ground’s give the rule counts. On the parent’s base anything over one does. Counting the whole path, a margin of 2.4 does here. A rule that is honest about the whole path would write the base term against the path’s compliance rather than the parent’s, and that term would read the radii below it again — reintroducing a dependence the base term was chosen to avoid.

The cone of directions, back and wider #

The division by direction was the clamped crown’s most visible finding and the one the giving base broke. Joined with bending, the crown now takes the largest of three terms: its column on its real base, the base its daughters need, and the bending rule balanced against buckling at a stated tilt on the trunk.

Without a base term, balanced at 30°, the deep crown is sized by buckling to 60° and by bending from 40° — overlapping, with 212 branches unsized. At a margin of 1.25 the overlap shrinks to one shared tilt, 60°, and nothing is stuck. At 2.5 the crown divides cleanly: buckling to 80°, bending from 100°. Balanced at 50° and at 70° the division holds at both margins, and at 2.5 and a balance of 70° the buckling core reaches past level, to 100°.

So the cone comes back, and it is not the clamped crown’s cone. On clamped bases, balanced at 30°, the core stopped at 20° and bending started at 40°. Sized for its daughters, the core reaches 80°. The branches that widen it are not upright columns at all: they are nearly level branches carrying little compression of their own, sized to be stiff bases for the daughters that turn back up off them. A buckling-sized branch at 80° from vertical is a statement about the branch above it. The core is a set of branches whose sizes are owed to something upright, which is a different thing from a set of branches that point up.

The junctions inside it are closed forms again. Balanced at 50° with a margin of 2.5, the 403 deep junctions whose three branches are all base-sized sit on the same shifted form as the gravity crown, 1.881 to 2.094, to $10^{-14}$. Where a base-sized parent carries a bending-sized daughter, the junctions read 1.867 to 1.983; where bending sizes the parent and one daughter, 2.556 to 2.619. The straddling junctions read above two, as they did on the clamped crown, and for the same reason: the two criteria are the same power law with different constants, and a junction whose members obey different constants reads their ratio as well as their law.

The cheapest margin is below the threshold #

A margin is a free constant, and a free constant invites the question of what it costs. The wood in a branch goes as $r^2 L$, so the wood in a crown is $\sum r^2 L$ over its branches.

Each curve has a minimum, and none of the minima is at the threshold. With every load along its branches the cheapest crown sits at a margin of about 1.7. Under gravity with forks of 20°, and joined with bending at a balance of 50°, it sits at about 1.6. At 2.043, where the alternation stops, the axial crown carries 0.9 per cent more wood than at its cheapest; at a margin of 3 it carries 22 per cent more, and the curves climb steadily from there, because above the threshold the base term thickens every parent and a larger margin only thickens them more.

Below the cheapest margin the curves climb the other way. A small margin leaves the column term in charge of most branches, the column term on a soft base asks a daughter to be thick, and the thick daughter is a stiff base for thin grand-daughters — the zigzag, damped but present. With no base term at all the axial crown carries 1.150 times the cheapest crown’s wood.

Two readings follow, and they point in different directions. A crown that stops alternating is not the cheapest crown: the cheapest, at a margin of 1.72, still swings from 2.682 at its first junction to 1.633 at its second, and is an eighth off two at its eleventh, 2.124 beside 1.873 one generation in. So if trees are sized for least wood against buckling on real bases, the alternation is not an artefact to be designed out; it is what the cheapest crown does. And against the crown sized as if clamped — which did not hold its loads on the bases it had — the cheapest crown that does hold them carries 1.32 times the wood. That factor is the price of standing on a parent that bends, paid in the cheapest way this rule allows.

What a survey would read #

A branching exponent fitted across a crown sized this way reads two in the interior of an axial crown above the threshold, and a closed form in the tilt under gravity. Neither is new: two is what the clamped planar crown conserves, and the shifted form is the clamped crown’s own. Three rules, one exponent found that an exponent names a sizing rule only with the length ratio beside it; here, sizing the parent for its daughters and sizing each branch for itself as if clamped give the same interior exponent. What separates them is the tip junction at 1.601 and the upright junction at 2.094 — the two places where a branch has no daughter to size it, or two daughters leaning away alike.

At the cheapest margin the interior does not read two at all. Its outer junctions alternate 1.873 and 2.124, a difference of a quarter between neighbouring generations — larger than the scatter the ordinary two per cent error in each radius puts on a single junction, and of the shape a sample that is confidently wrong warned a pooled fit would average into one number that belongs to no junction. So the two readings of the margin disagree about what a survey should expect: the crown that stops alternating says two everywhere, the cheapest crown says a zigzag with an amplitude of an eighth.

What this does not establish #

That a real parent is sized for its daughters, or that anything sizes a crown against buckling at all. That a fork is a rotational spring and nothing else: two daughters load one tip together, and a crown buckles as a whole, in a mode that couples every branch to every other, while each branch here is checked on its own spring. That the margin is one number across a crown — a tree could hold its twigs to one margin and its limbs to another, and the cheapest crown under that freedom is a different optimisation. And that the ground is rigid.

What would overturn it #

An axial crown sized for its daughters above the threshold whose interior junctions depart from two. A gravity junction sized wholly by the base term that leaves the clamped form one half-angle up. A branch stuck on its parent’s base at a margin over one. Each would mean the arithmetic here is wrong rather than incomplete; the whole-crown mode is the way it is most likely to be incomplete.

Still open: the crown buckling as one #

Every margin here is a branch’s own, on a spring standing in for everything below it. A crown is one structure, and the load at which it buckles is the lowest eigenvalue of the whole frame — every branch’s stiffness, every joint’s rotation, every tip’s load at once. The next measurement assembles that frame for the crown sized for its daughters and for the crown resized without a base term, and asks three things: whether the whole crown buckles at the load its branches were sized for, above it or below it; which branch the lowest mode concentrates in, and whether that is the branch the spring model marks weakest; and whether the cheapest margin of about 1.6 to 1.7 still holds when the check is the whole crown rather than each branch on its spring.