Reach taxes the extra lengths

Supply decides whether lengths pay set a tube’s fin lining against the lumen it needs to feed itself and found two regimes and a formula between them. When the supply channel is narrow for what it serves, extra lengths of fin are worth up to half again and the first few are supplied for nothing; when it is wide, every lining is a thin ring at the wall and the lengths are worth under one per cent. Between the two, a single length holds exactly $1 - 1/(1 + 2k)^2$ of what unlimited lengths hold, with $k$ the channel’s width per unit of surface in sheet thicknesses.

That result rested on the simplest demand the channel rule allows: a unit of surface costs the same channel wherever it stands. The essay named the obvious correction as the thing to try next. A supply that has to carry further to reach surface further out needs more channel for surface at the wall than for surface beside the lumen, and the question was whether the formula survives with $k$ reinterpreted or breaks.

It breaks, and the reason is more useful than the formula was. A reach cost charges a lining for where its surface stands, and a lining of many lengths stands its extra surface exactly where reach is dearest.

A channel that widens with its reach #

The model keeps everything from the earlier tube essays and changes one term. The tube has radius one; a lumen of radius $\rho$ is kept clear down its middle; fins stand on the wall and reach inward in rings of tips, and the number of fins at each radius is set by the circumference at the last ring of tips inside it. A lumen costs nothing until it does drew the frontier this makes: fins of $m$ lengths hold at most $m/(m + 1)$ of the tube’s ceiling, the surface a lining of the sheet could reach if the whole cross-section were fins, and they need no lumen smaller than a free one of radius $1/(m + 1)$.

The supply rule came from the channel grows with what it feeds: a channel takes $\beta/2$ of cross-section for each unit of surface it serves. Here that unit cost is multiplied by a factor that grows with the distance from the lumen’s edge to the surface being fed,

$$w(r) = 1 + \lambda\,(r - \rho),$$

where $\lambda$ is the extra channel needed per unit of reach, measured in tube radii. At $\lambda = 0$ the model is the earlier one exactly. At $\lambda = 2$ a unit of surface halfway from the lumen to the wall of a narrow-lumened tube costs about twice what the same unit beside the lumen costs.

The consequence for the method is larger than the change to the formula. With the straight demand, a lining’s cost depended only on how much surface it held, so the best lining for each lumen could be read off the frontier and the supplied lining sat where a straight line met it. With a reach cost, two linings holding the same surface can cost different amounts, depending on where their fins stand, and the frontier stops being enough. Each lining has to be solved for directly: the lumen and every ring of tips chosen together so that the surface is as large as possible while the lumen is wide enough for the supply the surface needs.

For one length and for unlimited lengths the solution has a closed form. One length’s fins all start at the lumen and run to the wall in equal numbers, so its demand is $k(1 - \rho)(2 + \lambda(1 - \rho)) = \rho$ once its lumen is no longer free. Unlimited lengths fill the circumference at every radius, and their demand is

$$k\left[(1 - \rho^2) + \tfrac{\lambda}{3}(1 - \rho)^2(2 + \rho)\right] = \rho^2.$$

Both reduce to the earlier crossings when $\lambda$ is zero. Two and three lengths are solved numerically, and the solver reproduces both the reach-free crossings of the earlier essay and the one-length closed form to two parts in ten thousand.

Where the extra surface stands #

The picture that explains everything else is a picture of the two linings’ fins along the radius.

A single length starts every fin at the same ring of tips, so the count is the same at every radius and the surface is spread evenly from the lumen to the wall. Its average surface stands halfway out, at $(1 - \rho)/2$ from the lumen’s edge. Unlimited lengths add a fin wherever the circumference makes room for one, so the count grows in proportion to the radius and the surface piles toward the wall; its average stands at $(1 - \rho)(2 + \rho)/(3(1 + \rho))$, further out.

At a channel of a fifth of a sheet per unit of surface and a reach cost of two, the single length’s surface stands on average 0.25 of the radius from its lumen and the unlimited lining’s 0.285. That fourteen per cent of extra reach is the whole story. Every lining pays for its supply by where its surface stands, and the surface that more lengths add is surface near the wall — the short fins filling the space the long ones diverge from — which is exactly the surface a reaching supply finds most expensive.

So a reach cost is not a uniform tax that rescales the channel for everyone. It is a tax on elaboration, levied in proportion to how far toward the wall a lining’s extra surface stands, and a single length pays the least of it. It is the second thing the tube’s shape does against its fins: in a tube the standing members lose found the convergence of the walls toward the axis leaving less room for fins the further in they reach, and a reaching supply now charges the fins that stand furthest out. Between the two, the tube taxes its fins at both ends — the long ones where their tips crowd toward the axis, the short ones for how far their supply must carry — and the single length, whose fins all share one ring of tips, is the lining with no short fins for the second tax to fall on.

What the lengths are worth now #

The earlier essay’s headline was the ratio of what one length holds to what unlimited lengths hold, both supplied, because that ratio is what all the extra lengths together are worth.

With reach free the curve is the closed form, and at a channel of one sheet thickness per unit of surface one length holds 88.9 per cent of the unlimited lining’s surface. At a reach cost of two it holds 91.5 per cent, and at eight 94.3. If the reach cost were only a rescaling of the channel, the ratio would sit on the same curve at a different $k$; instead it rises everywhere in the middle of the range, because the unlimited lining is charged more per unit of surface than the single length is.

In the cheap-supply regime the effect is large. At $k = 0.2$ the lengths beyond the first were worth two thirds again with reach free: one length held half the ceiling and unlimited lengths 83.3 per cent. At a reach cost of two the unlimited lining falls to 76.1 per cent and the lengths are worth half again; at eight it falls to 64.5 per cent and they are worth under a third. The single length holds half the ceiling at the first two of those settings, because its lumen is still free and a free lumen costs it nothing, and 49.8 per cent at the third, where even its free lumen is just too narrow.

The passage between the regimes moves with it. The earlier essay put it at about a decade of $k$, between a tenth and one. One way to put a number on its position is to ask where one length first holds nine tenths of what unlimited lengths hold. With reach free that is at $k = 1.08$; at a reach cost of two it is at 0.87, and at eight at 0.55 — half the channel. Dear reach makes a lining simple at narrower channels, which is what the earlier essay predicted in words and can now say in numbers.

Where it does not matter #

The right-hand end of that figure is the other half of the result, and it is the reason the formula is not simply wrong.

At a channel five sheet thicknesses wide per unit of surface — the comb essay’s own figure — one length holds 99.17 per cent of what unlimited lengths hold with reach free, and 99.13 per cent at a reach cost of two or eight. The curves have come back together, and for dear supply the reach cost has hardly moved them at all; if anything, it has made the extra lengths very slightly more worthwhile.

The reason is the geometry that made dear supply flatten every difference in the first place. When supply is dear the lining’s own lumen takes most of the cross-section — nine tenths of the radius at $k = 5$ — and the fins are a thin ring at the wall. In a thin ring there is no reach to charge for. The supply never has to carry more than a tenth of the radius from lumen to surface, the weight $w(r)$ never rises far above one, and the reach cost multiplies a quantity that is already nearly a constant.

So the formula $1 - 1/(1 + 2k)^2$ breaks in the middle of the range, where the ring of fins is thick and different linings stand their surface in different places, and survives at dear supply with nothing reinterpreted at all. The regime where extra lengths were worth nothing is the regime a reach cost cannot touch, and the regime where they were worth a great deal is the one it taxes hardest.

The table, and the free regime narrowing #

The numbers for two and three lengths sit between the closed forms and show the same shape.

At $k = 0.02$ the first three columns are unchanged — every lining up to three lengths is still supplied for nothing — and only the unlimited lining falls, from 98.0 to 96.2 per cent. At $k = 0.2$ two lengths fall from 66.5 to 64.4 per cent, three from 72.7 to 68.7, and unlimited from 83.3 to 76.1, so each added length gives up more than the one before. At $k = 1$ every column falls by four to six points. At $k = 5$ every column falls by about one point together, and the columns stay within 0.2 of each other.

The free regime is the part of the table a reach cost changes most sharply. A lining is supplied for nothing when the free lumen its fins leave anyway is wide enough for the supply its surface needs. With reach free that happens for $m$ lengths up to $k = 1/(m(m + 1))$. With reach charged, the free lining’s own surface is charged too — its fins still stand at distances from the lumen — so every threshold falls.

A single length is free up to a channel of a half with reach free and up to a third at a reach cost of two. Two lengths fall from a sixth to 0.094, three from a twelfth to 0.044, and five from a thirtieth to 0.016 — under half of where they were. The more lengths a lining has, the larger the share of its free threshold a reach cost takes away, for the same reason as before: the lining with more lengths stands more of its surface near the wall, and even a lining that costs nothing to supply at the lumen’s edge has to be supplied out there.

The earlier essay’s rule of thumb — a channel supplies about $1/\sqrt{k}$ lengths for nothing — therefore needs a correction that grows with the count. At a reach cost of two, a channel a hundredth of a sheet wide per unit of surface supplies six lengths for nothing rather than nine.

The rings of tips spread toward the wall #

One more thing changes, and it is the only one that changes the shape of a lining rather than how much it holds.

With reach free, the best lining of three lengths spaces its rings of tips equally from the lumen to the wall, a result the wedge belongs to one length found for fins with nothing to supply and the lumen essay carried over. With reach charged the spacing stops being equal. At a channel of one sheet per unit of surface and a reach cost of two, the three gaps from the lumen outward are 0.082, 0.086 and 0.092 of the radius; at eight they are 0.058, 0.065 and 0.078. The rings spread toward the wall, which means the short fins near the wall — the ones that fill the last sliver of circumference — are the ones the lining economises on.

That is the second time the same shape has turned up in this subject for the same reason, and the first was in paper rather than tissue. The paper reads strain placed the starts of straight tucks on a gathered cap and found that judged by what the paper actually bears, they should spread toward the rim rather than crowd, because a millimetre of shortfall costs less strain on a long outer circle. Here a unit of surface costs more supply on the far side of a long reach. In both, a cost that varies along the radius decides the spacing of features along it, and neither the tucks nor the fins had any reason to be equally spaced once the cost was measured in the unit the material cares about.

What this picture cannot show #

What the reach cost is for any real tube. $\lambda$ is extra channel per unit of reach in tube radii, and nothing here measures one. A gut, a gill filament and a heat exchanger each carry something different through their supply, over different distances and at different pressures, and the model has one number standing in for all of it.

Whether a supply really does widen in proportion to its reach. A channel carrying flow at a fixed pressure drop needs a cross-section that grows with its length, but not linearly: Poiseuille’s law for a round pipe makes the area grow as the square root of the length. The straight-line weight is the simplest rule that charges for reach and was chosen for that reason. A square-root weight would charge long reaches less, and every effect here would be smaller in the same direction.

How a supply is actually routed. The model sizes one lumen for all the surface, with each unit charged by its own reach; a real lining has vessels inside its fins, branching as they go, and a branching supply shares its reach among the surface it passes. That is the direction how much surface fits in a body points, and none of it is here.

The idealisations underneath #

The fins are radial slabs of the sheet’s thickness, standing on the wall and ending in rings of tips, and their count at each radius is treated as a real number, as in every earlier tube essay. The lumen is one disc on the axis, and supply is only room: flow, pressure and time are not in the model.

The reach is measured from the lumen’s edge along the radius, as the shortest distance a supply could carry from the lumen to a point on a fin. A supply that ran along the wall or round the tube would measure reach differently.

And a lining’s lumen is never wider than its innermost ring of tips. A lumen wider than the fins leave would waste cross-section and leave every unit of surface further from it, so the solver never chooses one; the free regime is exactly the case where the fins’ own free lumen is already wide enough.

How the numbers were checked #

The lining solver is checked against both things already known about it, before any figure is drawn: with reach free, three lengths must reproduce the straight demand’s crossings from the earlier essay; with reach charged, one length must reproduce its own closed form. Both hold to two parts in ten thousand at channels of a fifth, one and five.

The ratio curve with reach free must follow $1 - 1/(1 + 2k)^2$ at every channel past a half, to six figures, so the new model is the old one exactly when its new term is zero.

At a channel of one sheet per unit of surface the ratio must rise with every step of the reach cost, which is the claim that a reach cost taxes the extra lengths rather than every lining alike.

The rings of tips must be equally spaced with reach free and must spread outward — every gap at least as wide as the one inside it — at each reach cost drawn. And in the table no lining may hold more with reach charged than with reach free, while more lengths must still hold at least as much as fewer.

Still open: a supply that branches #

The reach cost charges every unit of surface for its own journey from the lumen, which is the most expensive way to supply a lining. A supply that branches — one vessel running out along a fin and feeding everything it passes — shares each stretch of its reach among all the surface beyond it, and then the cost of standing surface far out depends on how much surface stands further out still. That is a different demand again, and it would change the result in a predictable direction: the unlimited lining’s extra surface, standing furthest out, would be charged least per unit by a shared supply and most by an unshared one.

The branching fin, carried from the two essays before this one, is the natural partner of a branching supply. A fin that splits partway to the wall is the continuous limit of adding lengths, and it puts its extra surface near the wall — exactly where the reach cost is heaviest. A branching fin fed by a branching supply might be the one arrangement in which elaboration and reach stop fighting, and whether it is would need the two built together.

Sideways from here, a nest pays four a level priced a hierarchy of scale on a flat base, and each level of a nest stands further from its supply than the one before; charged for reach, the four a level would fall with depth, and the question of how many levels pay would have an answer the flat-base essays could not give. Standing up beats lying down by eight is the flat-base comparison where the same cost would be charged to walls of different heights.

The habit worth carrying is about taxes on structure. When a cost is charged per unit of output, ask where the units stand before assuming it falls on every design alike. The straight demand charged surface wherever it was and could only rescale the answer; charging by reach found that the elaborate lining’s extra surface was the dearest surface it had.