A shared supply is the same bill

Reach taxes the extra lengths charged a tube’s fin lining for how far its supply has to carry. Every unit of surface paid channel in proportion to its distance from the lumen, and the charge fell hardest on the lining with the most lengths of fin, because the short fins that fill the circumference stand near the wall. Where supply was cheap, the extra lengths fell from being worth two thirds again to being worth half again.

That essay ended on the obvious objection to its own model. It had charged each unit of surface for its own journey, as if every square millimetre of fin had a private pipe from the middle of the tube. Nothing alive is plumbed that way. A gill filament, an intestinal villus, the lamellae of a fish’s gill all carry a vessel that runs out along them and feeds everything it passes, so each stretch of the vessel is shared among all the surface beyond it. The essay predicted the direction of the change: a shared supply would charge the outermost surface least, and might be the arrangement in which elaboration and reach stop fighting.

The prediction is wrong, and not by a little. With channels sized in proportion to the flow they carry, a shared supply costs exactly what the private pipes cost, for every lining and every way of branching. Sharing can only pay when a channel’s width grows more slowly than its flow, and when it does, it pays back an amount it was itself about to charge.

The same integral, read from the other end #

The model is the one the tube essays have used since in a tube the standing members lose set fins inside a cylinder. The tube has radius one and a lumen of radius $\rho$ kept clear down its middle. Fins stand on the wall and reach inward to rings of tips, and the number of fins at each radius, $N(r)$, is set by the circumference at the last ring of tips inside it, so the surface standing between $r$ and $r + dr$ is $dS = 2N(r)\,dr$, two faces to each fin. The reach charge was

$$\lambda \int_\rho^1 (r - \rho)\, dS(r),$$

with $\lambda$ the extra channel per unit of reach. That is a sum over the surface: each piece pays for its own distance.

A branching supply reads the same lining from the other end. Its channel at radius $r$ has to carry everything that stands further out, $F(r) = \int_r^1 dS$, and if a channel’s cross-section is proportional to what it carries, the total channel is $\lambda \int_\rho^1 F(r)\,dr$. That is a sum over the radius: each stretch of vessel pays for the traffic through it.

The two are the same number. Integrating by parts, $\int (r - \rho)\,dS = \big[(r - \rho)(-F)\big]_\rho^1 + \int F\,dr$, and the bracket vanishes at both ends, because nothing stands beyond the wall and nothing has been carried any distance at the lumen. The figure above draws both integrands for the supplied lining of three lengths at a channel of one sheet thickness per unit of surface and a reach cost of two. The per-unit charge is zero at the lumen and largest at the wall; the tree’s channel is largest at the lumen and zero at the wall. They cross once and enclose 0.1170 each, which is also what the lining’s own bill says the reach costs it.

Nothing in the argument depends on how the tree branches. A vessel that runs out along one fin and forks into a neighbour, a vessel per fin, a single trunk splitting into a hundred twigs — every arrangement whose paths run outward without doubling back carries $F(r)$ across radius $r$ in total, and if each channel’s size is proportional to its own flow, only the total matters. Sharing a channel sized to its flow is bookkeeping. The per-unit charge was never the expensive way to supply a lining. It was the bill of every radial tree, written down one unit of surface at a time.

Why sharing ever pays #

That leaves the question the reach essay was really asking: what would make a shared supply cheaper? The answer is in the word “proportional”. A pipe carrying twice the flow need not be twice as wide.

For laminar flow down a round tube, Poiseuille’s law makes the flow at a given pressure gradient grow as the fourth power of the radius, so the cross-section needed grows only as the square root of the flow. Cecil Murray asked in 1926 what radius a blood vessel should have if the body pays both to pump blood and to keep the blood volume, and found the cost least when the flow grows as the cube of the radius, which makes the cross-section grow as the flow to the power two thirds. That is Murray’s law, and it has been found to hold approximately in many arterial trees and in the water-conducting tissue of some plants. Both are economies of scale: a cross-section growing as $q^\alpha$ with $\alpha < 1$, so one vessel carrying everything is narrower than two carrying half each.

With $\alpha$ below one, the total channel at radius $r$ is $\sum q_v^\alpha$ over the vessels crossing it, and that does depend on how the flow is divided. The sum is smaller the fewer and fuller the vessels. A supply that keeps the short fins’ traffic in separate pipes all the way from the lumen pays more than one that merges it into the long fins’ vessels.

The drawing shows a sixth of a tube with fins of two lengths, the long ones starting at a lumen of three tenths of the radius and the short ones at six tenths. On the left every short fin’s vessel crosses from the lumen to its tip on its own, through the narrow gap between two long fins; on the right each short fin is a fork from a long neighbour, and the long fin’s vessel carries both fins’ supply until the fork. At Poiseuille’s square-root sizing the forked supply needs 15.9 per cent less channel than the separate one, 1.259 against 1.498 per unit of fin count. At Murray’s two thirds it needs 11.9 per cent less. At an exponent of one the two bills meet, as the identity says they must.

There is a third arrangement the drawing leaves out because it cannot exist under economies of scale: the private pipe per unit of surface that the reach essay charged. Split a flow into ever smaller pieces and $\sum q_v^\alpha$ grows without limit when $\alpha < 1$. A supply with economies of scale is forced to share. The question is no longer whether branching pays but how much, and to whom.

What a vessel carries #

Branching changes what each vessel carries, and that is the quantity economies of scale act on.

In a branching supply every fin present at a radius carries an equal share of the surface beyond it. That is not an assumption about the plumbing but a consequence of the lining’s symmetry: all the fins present at radius $r$ have the same surface of their own beyond $r$, and the fins born further out are shared among them equally, so each carries $F(r)/N(r)$. For a lining of one length, which has no fins born further out, that is just a fin’s own surface beyond $r$, $2(1 - r)$ in units of fin faces — the dashed line. For unlimited lengths it is $(1 - r^2)/r$, which is the same at the wall and larger toward the lumen, by a factor of $(1 + \rho)/2\rho$ at the lumen’s edge: 2.17 times a fin’s own surface at a lumen of three tenths.

A lining of a few lengths does something the two extremes do not. Its long fins carry the whole of the lining’s future until the next ring of tips, so their vessels are fattest just inside each ring and drop sharply as the new fins take their share. Two lengths’ vessels carry 2.22 at the lumen, more than unlimited lengths’ do there, and eight lengths’ carry 2.83. The sawtooth is what elaboration looks like from inside a vessel: the fewer the rings, the more each fin is a trunk.

That makes the direction of the effect look settled. Economies of scale favour fat vessels, the elaborate linings have fat vessels near the lumen, so branching ought to relieve them. It does relieve them, measured against a supply that does not branch. Measured against the flow-sized channel that the reach essay used, it barely moves them at all.

Priced the same, the lengths are worth the same #

A fair comparison needs one decision first, because a channel sized as $q^\alpha$ is not in the same units as a channel sized as $q$. Some reference flow has to be charged the same under both, and any choice favours one lining or another. The choice made here is the one that isolates routing: each sizing law is priced so that the single length comes out exactly where it did. The single length has no short fins, so it has nothing to branch and nothing to share. If its supply costs the same under every law, then any difference in the elaborate lining is the routing’s doing and not the price’s.

That price has a closed form. The single length’s reach bill under exponent $\alpha$, at its own supplied lumen $\rho_1$, matches the flow-sized bill when the reach cost is

$$\lambda' = \frac{\lambda\,(\alpha + 1)\,(1 - \rho_1)^{1-\alpha}}{2},$$

which is $\lambda$ again at $\alpha = 1$. It pins a price only where the single length’s supply actually binds — past the free lumen that a lumen costs nothing until it does found every lining keeps for nothing. Below a channel of $1/(2 + \lambda/2)$ per unit of surface the single length is supplied for nothing, its bill says nothing about what a unit of reach is worth, and the comparison has no anchor; the figures below start where it does.

The branched curves sit on the dashed line. At a reach cost of eight, over every channel at which the single length’s supply binds, unlimited lengths fed by a branching supply of Poiseuille’s vessels hold at most 0.17 points less than with flow-sized channels, and with Murray’s vessels 0.08 less. Fed by a vessel per fin, they lose up to 1.71 points with Poiseuille’s vessels and 1.06 with Murray’s, worst where the channel is narrowest. At a reach cost of two the losses are 0.95 and 0.60.

Set that beside what reach itself costs the same lining. With reach free, unlimited lengths hold $1/(1 + k)$ of the ceiling at a channel of $k$ sheets per unit of surface, the crossing supply decides whether lengths pay found; with reach at eight, they lose up to 19.6 points of it, at a channel of about three tenths. Routing moves the answer by less than a tenth of that, and the best routing moves it by less than a hundredth.

The table gives the same comparison at four channels. At a channel of a fifth of a sheet per unit of surface, unlimited lengths hold 83.3 per cent of the ceiling with reach free and 64.5 with flow-sized channels; Murray’s vessels branched give 64.4 and per fin 63.5; Poiseuille’s branched 64.3 and per fin 62.9. At one sheet the five figures run from 35.5 down to 34.9, and at five sheets they agree to a tenth of a point at 13.5, because there the lining is a thin ring at the wall with almost no reach inside it, which is the same reason the reach cost itself vanished there.

So the answer to the reach essay’s closing question is a negative with a number on it. The tax that reach levies on elaboration is set by where the surface stands, not by how it is fed. A supply with no economies of scale cannot change it by sharing. A supply with economies of scale can avoid making it worse, which is what branching does, and cannot make it better.

Why the economies cancel #

The near-exact agreement of the branched curves with the flow-sized one needs explaining, because two things are happening and they push in opposite directions.

The first is the one the vessel figure showed. Unlimited lengths’ vessels carry more flow than a single length’s near the lumen, so under economies of scale each unit of their flow there is cheaper. That favours the elaborate lining.

The second is shown here. Add one more unit of surface to the single length at radius $s$ and every stretch of vessel between the lumen and $s$ carries a little more. With flow-sized channels each stretch charges the same per unit of extra flow, so the cost of the added surface rises in a straight line, $1 + \lambda(s - \rho)$. With economies of scale a stretch’s charge per unit of extra flow is $\alpha (q/q_0)^{\alpha - 1}$, which is largest where the vessel is thinnest, and a vessel is thinnest at its last stretch before the wall. The cost of an added unit becomes $1 + \lambda'\left[(1 - \rho)^\alpha - (1 - s)^\alpha\right]$: slow at first, steep at the end. Halfway from the lumen to the wall the flow-sized charge has risen half its full amount, and the economies-of-scale charges only $1 - 2^{-\alpha}$ of theirs — 37 per cent with Murray’s vessels and 29 with Poiseuille’s.

Priced so that the single length’s total bill is unchanged, economies of scale therefore move the charge toward the wall, and the wall is exactly where the elaborate lining keeps its extra surface. That disfavours it. The discount on its fat inner vessels and the surcharge on its outer surface come within a fifth of a point of cancelling at every channel measured, and the cancellation is measured rather than derived: nothing in the argument above says the two must be equal, only that they push opposite ways. What is derived is the reason a vessel per fin loses: without branching, the short fins’ supply crosses from the lumen in pipes of its own, adding vessels at radii where the fins themselves are absent, and economies of scale charge every additional vessel for being thin.

The tree that designs a crane #

The same arithmetic runs through the oldest design method in folding, in the opposite limit. In the tree method of uniaxial design, a base is drawn as a stick figure whose leaves are flaps and whose internal edges are body segments, and every pair, not every circle states the condition the sheet must meet: every two flaps must be separated on the paper by their distance through the tree. A body segment shared by twenty flaps appears in every one of their pairwise distances, and it costs its length of paper once — a river in the circle packing a flap costs a circle describes, whose width is fixed whatever traffic passes through it.

That is a supply with exponent zero: a shared segment’s cost does not grow with what it serves. It is why a designer merges flaps into a common trunk wherever the subject allows, and why the tree method rewards sharing so strongly that the art of uniaxial design is largely the art of finding shared branches. A vessel sized to its flow is the other limit, exponent one, where sharing buys nothing at all. Murray’s and Poiseuille’s vessels sit between, and the tube lining shows how little of the difference reaches the answer once the surface’s position has been fixed.

Leonardo’s notebooks hold the other well-known rule about branching, observed on trees: at every height, the branches’ cross-sections taken together equal the trunk’s. That is exponent one. If Leonardo’s rule is how a tree sizes its limbs, then by the identity at the top of this essay a tree pays for its canopy’s reach exactly as if each leaf had a pipe of its own, and its branching is there for some other reason — mechanical support, light, the way a meristem divides — and not to save plumbing.

What an exponent and a fork leave out #

Any real vessel’s exponent. Murray’s law and Poiseuille’s are idealisations of laminar flow in rigid round tubes. A lymphatic vessel, a xylem conduit with pit membranes, a gill’s blood sinus and the gut’s capillary bed all depart from them, and the exponent that best describes any of them is a measurement this essay does not make.

What the supply is carrying. Flow is measured here only as the surface it serves. A supply that delivers oxygen, absorbs nutrients and removes heat at once would size its channels for whichever is limiting, and the limiting one may change along the radius.

A fork’s own cost. The branched supply joins a short fin to a long neighbour at the ring of tips, and the short stretch across the gap between them is not charged. Whether a folded sheet can branch at all without a cut, and what the vertex at a fork costs in paper, is the question how much surface fits in a body leaves open, and it is not settled here.

The free regime. Where a single length is supplied for nothing, the pricing that isolates routing pins nothing, and every comparison here starts above it. A different anchor, such as pricing a unit of flow at one standard vessel size, would extend the curves below that point and would make their position there depend on that standard.

Slabs, radial vessels and an equal split #

The fins are radial slabs standing on the wall, their count at each radius treated as a real number set by the circumference at the last ring of tips, exactly as in the wedge belongs to one length and every tube essay after it. Every vessel runs radially, from the lumen outward, and never doubles back or runs round the tube; a vessel that ran round would carry flow a longer way and break the identity’s condition that paths run outward.

The branched supply divides each radius’s traffic equally among the fins present there. For the linings measured that is exact, because fins present at the same radius are interchangeable. A tree that loaded some fins more than others would, under economies of scale, be slightly cheaper still, since the sum of $q^\alpha$ is least when the flow is most unequal; the equal division is the best a tree can do without favouring particular fins, not the best possible.

The pricing normalises a vessel per unit of fin count carrying both faces of a whole fin to the flow-sized charge, and then scales the reach cost so the single length’s bill is unchanged. That second step removes the first one’s arbitrariness for every comparison drawn; it does not remove it for absolute totals, which is why every figure here is a comparison.

Checked at exponent one against the reach model #

At exponent one both routings must reproduce the reach essay’s supplied linings — one length and unlimited lengths at channels of a fifth, one and five, at reach costs of two and eight — to a part in a million. They do, which is the identity checked against an independent solver rather than trusted.

The two integrands of the first figure must enclose the same area as the lining’s own bill, computed from its rings of tips by the formula the reach essay used, to two parts in ten thousand. All three agree at 0.1170.

The wall integral is checked against a closed form. The per-fin supply of unlimited lengths has one, and the numerical integration used for the branched supply, which has none, reproduces it to $10^{-7}$ at Poiseuille’s exponent, where the integrand’s corner at the wall is sharpest.

At every channel a vessel per fin must hold less than a branched supply, every routing and sizing must stay within a fifth of the reach tax of the flow-sized lining, and the halfway share of the added-surface charge must equal $1 - 2^{-\alpha}$ exactly. The forked sector must cost less than the separate one at every exponent below one and the same at one.

Still open: what a fork costs in paper #

The identity removes one reason to expect branching to matter and the measurements remove another, so what is left to price is the fork itself. A fin that splits partway to the wall is the continuous limit of adding lengths, and the branched supply above assumed the split was free. In a folded sheet it is not: a fin that branches without a cut needs a vertex where three walls meet, and the paper that vertex consumes, in the circle-and-river sense the tree method uses, is a charge on exactly the elaboration that supply has now been shown not to tax further. Whether that charge is per fork or grows with the fork’s depth from the wall would decide how many lengths a folded lining should have.

The second open direction is a supply whose exponent varies. Murray’s law describes the large vessels; the smallest ones, where flow is slow and walls are leaky, behave more like flow-sized channels. A supply whose exponent rises from two thirds near the lumen to one near the wall would combine the two limits measured here, and the cancellation found above — discount near the lumen, surcharge near the wall — would then be lopsided in a direction that could be predicted before it is measured.

Sideways from here, the channel grows with what it feeds priced supply on a flat base, where combs of standing walls beat stacks of plies by eight until supply took the margin back. On a flat base every wall is the same length, so there is nothing to branch, and the identity says a flat comb’s supply can be routed any way at all. A nest pays four a level is the flat-base arrangement where branching could matter, since each level of a nest serves everything inside it.

The habit worth carrying is about expecting a network to save something. Before crediting a shared structure with economy, ask whether its cost per unit grows more slowly than what it carries. If it does not, the sharing is bookkeeping: the same bill read along the radius instead of across the surface. If it does, the saving is real but goes first to undoing the price of having many thin channels, and only after that, if anything is left, to the design that chose to share.