Supply decides whether lengths pay

A lumen costs nothing until it does drew a frontier for a tube lined with inward fins. For each share of the cross-section kept clear as a lumen down the middle, it found the most surface fins of a given number of lengths can hold: flat out to a free lumen of radius $R/(m + 1)$, where the fins would have stopped anyway, and bending down past it toward the straight line $1 - \rho^2$ that unlimited lengths reach. It called the frontier a menu and named the thing that would pick from it. A lining has to be supplied, the channel grows with what it feeds, and so the lumen a lining needs rises with the surface it holds. Put the need against the menu and the problem closes.

It closes in a way that splits in two. When a channel is narrow for the surface it serves, supply is free — the lining meets its demand on the flat stretch and holds exactly what it would hold unsupplied — and more lengths of fin are worth a great deal. When a channel is wide, every lining meets its demand near the full lumen, where the frontiers of every number of lengths crowd together, and the lengths are worth almost nothing.

The lumen a lining needs #

The channel essay set the rule on a flat base. A wall of height $c$ carries two faces, so it serves $2c$ of surface for each unit of its run, and whatever supplies that surface has to be sized for what it serves: a channel $\beta c$ wide runs beside it. So a channel takes $\beta/2$ of cross-section for each unit of surface it serves.

Carry that into the tube. A lining holding a share $s$ of the tube’s ceiling holds $s \cdot 2\pi R^2/\tau$ of surface for each unit of the tube’s length, where $\tau$ is the sheet’s thickness, and so it needs a lumen of area $(\beta/2)\, s\, 2\pi R^2/\tau$. As a share of the cross-section $\pi R^2$ that is

$$x = k\,s, \qquad k = \frac{\beta}{\tau}.$$

The one number that matters is $k$: the channel’s width for each unit of surface, measured in sheet thicknesses. A channel whose width per unit served is a fifth of the sheet is $k = 0.2$; the comb essay’s own numbers, a channel five per cent of what it serves on a sheet a hundredth of the clearance, give $k = 5$.

In the plane of lumen against surface that demand is a straight line through the origin with slope $1/k$. The frontier is the most surface for each lumen; the demand is the lumen each surface needs; the supplied lining sits where they meet, since to its left the lining would hold more surface than its lumen supplies and to its right its lumen is wider than its surface needs.

At $k = 0.2$ the line meets the one-length frontier on its flat stretch. A single length of fin holds half the ceiling and leaves a free lumen half the radius across; it needs a lumen of only $0.2 \times 0.5$, a tenth of the cross-section, radius 0.32 — well inside the half it leaves anyway. Its supply costs nothing. Three lengths hold 72.7 per cent and meet the line just past their flat stretch, with a lumen 0.38 of the radius; unlimited lengths hold 83.3 per cent. The whole run from one length to unlimited is worth two thirds again, and most of it is still there to be had.

When supply is free #

The meeting is on the flat stretch exactly when the demand at the flat stretch’s end is no more than the lumen it leaves. With $m$ lengths the flat stretch holds $m/(m + 1)$ of the ceiling and ends at a lumen of $1/(m + 1)^2$ of the cross-section, so supply is free when

$$k\,\frac{m}{m + 1} \le \frac{1}{(m + 1)^2}, \qquad\text{that is}\qquad k \le \frac{1}{m(m + 1)}.$$

One length has free supply up to $k = 1/2$, two lengths up to $1/6$, three up to $1/12$, ten up to $1/110$. More lengths hold more surface and leave a smaller free lumen, so the channel they can supply for nothing is narrower. While the channel is no wider than half a sheet per unit of surface, a lining can always be made free by using fewer lengths; what it gives up is surface.

The table reads down as the channel widens. At $k = 0.02$ every lining up to five lengths is free, and the surface climbs from 50 per cent with one length to 83 with five and 98 in the limit. At $k = 0.1$ three lengths are just past free, at 74.9 per cent against their unsupplied 75. At $k = 0.5$ only one length is free, and the column from one to unlimited runs from 50 to 67. At $k = 1$ none is free, and the column runs from 44.4 to 50. At $k = 5$ it runs from 16.5 to 16.7.

The most lengths a channel supplies for nothing #

The free rule can be read the other way round, as a design rule. Given a channel’s width $k$, the largest number of lengths whose lumen is still free is the largest $m$ with $m(m + 1) \le 1/k$, which is close to $1/\sqrt{k}$. A channel a hundredth of a sheet wide per unit of surface supplies nine lengths for nothing, since $9 \times 10$ is ninety; a channel a twenty-fifth of a sheet supplies four; a channel a fifth of a sheet supplies one.

That lining holds $m/(m + 1)$ of the ceiling, the figure the wedge belongs to one length found for fins with equally spaced tips, and it falls short of the supplied limit $1/(1 + k)$ by about $\sqrt{k}$. So a cheap channel buys free supply and a shortfall of the square root of its own width: at $k = 0.01$ nine lengths hold 90 per cent of the ceiling with their supply free, against 99 for unlimited lengths supplied at a price. The remaining nine points are there to be had, and every one of them is bought with lumen the lining would not otherwise have given up.

The square root is the useful part. It says the free regime is generous in lengths and stingy in surface: halving the channel’s width per unit of surface adds only about forty per cent more free lengths, and closes the shortfall by only about thirty per cent. A lining in the cheap regime therefore always has a choice between stopping at the free number of lengths and paying for more, and the table above is the price list for that choice.

Two regimes, and the formula between them #

The run of each row, from one length to unlimited, is what extra lengths are worth, and it has a closed form.

With one length on the bend of its frontier, the lining holds $2\rho(1 - \rho)$ with a lumen of radius $\rho$, and the demand says $k \cdot 2\rho(1 - \rho) = \rho^2$. So $\rho = 2k/(1 + 2k)$ and the surface is

$$s_1 = \frac{4k}{(1 + 2k)^2}.$$

With unlimited lengths the frontier is $1 - x$ and the demand $x = ks$, so $s_\infty = 1/(1 + k)$. Their ratio is

$$\frac{s_1}{s_\infty} = \frac{4k(1 + k)}{(1 + 2k)^2} = 1 - \frac{1}{(1 + 2k)^2}.$$

Everything that any number of extra lengths can add over a single length is $1/(1 + 2k)^2$ of the supplied limit — once supply is dear enough that one length is off its flat stretch, at $k$ past one half. Below that, one length is free at a half and the limit is $1/(1 + k)$, so the lengths are worth $1 - (1 + k)/2$, approaching half again as supply becomes free.

The picture shows the two regimes and the short passage between them. On the left, where a channel is a hundredth of a sheet wide for each unit of surface, one length holds barely more than half of what unlimited lengths hold, and each added length closes a steady share of the gap. On the right the curves have all run up to within a whisker of one. The passage is about a decade of $k$ wide, from a tenth to one: at $k = 1$ one length holds 89 per cent of the limit, at $k = 2$ 96 per cent, at $k = 5$ 99.2 per cent. At the comb essay’s own channel, fins of one length hold 16.5 per cent of the ceiling and fins of unlimited lengths 16.7.

Why dear supply flattens every difference #

The reason is visible in the frontier picture. Every frontier ends at the same corner: a full lumen and no surface. As the demand line steepens toward the horizontal axis — as $k$ grows — its meeting points slide down every curve toward that corner, and near the corner every curve is close to the straight limit. A lining whose supply takes most of the cross-section has only a thin ring of fins left, and in a thin ring near the wall there is no room for fins of different lengths to differ: they all stand in the last sliver of the radius.

At $k = 0.2$ the two cross-sections still look like the unsupplied linings. The single length stops at half the radius, as it always would, and its lumen is more than it needs. The three lengths would stop at a quarter of the radius if they could, and instead stop at 0.38 to leave the lumen their surface calls for; they give up 2.3 points of the ceiling to do it, 72.7 against 75.

At $k = 2$ both linings are a thin ring of fins round a lumen eight tenths of the radius across or more, and they hold 32.0 and 32.9 per cent of the ceiling. The lengths that separated them by a quarter of the ceiling unsupplied separate them by less than a point. What made a tube worth lining with many lengths was the room near the axis, and a dear supply is exactly what takes that room first.

What this says about a tube that is lined #

Nothing grown has a seam is the reminder that a lining is grown and not placed, and nothing here claims that a gut, a gill filament or a heat exchanger is fins of a few lengths optimising surface under a lumen constraint. What the two regimes offer is a reading of how much a lining’s complexity is doing.

A lining whose supply is cheap should be many-lengthed, because in that regime each length closes a steady share of a large gap, and the first few lengths may even be supplied for nothing. A lining whose supply is dear should be simple, because the lengths are worth under a per cent and each of them is a thing that has to be grown, folded or cut. The dividing line is a channel about one sheet thickness wide for each unit of surface served, give or take a factor of three. On either side of that, the answer to “how many lengths?” changes from “as many as can be made” to “one”.

That is the same shape of result standing up beats lying down by eight and its supplied sequel found on the flat base, where a comb of walls beat a stack of plies by eight until supply took the advantage back. There the architecture was decided by the clearance; here the elaboration of one architecture is decided by the channel. In both, a cost that grows with what it serves turns an advantage that looked structural into one that holds only while the cost is small.

The idealisations underneath #

The channel’s rule is carried from the flat base to the tube. On the flat base a channel $\beta c$ wide serves a wall of height $c$; here the lumen is taken to need the same cross-section per unit of surface, $\beta/2$, wherever the surface is. A real supply might need less for surface near the lumen and more for surface at the wall, where it has further to go, and a demand that grew with the fins’ reach would bend the demand line rather than keep it straight.

The lumen is one disc on the axis, as in the essay that drew the frontier, and supply is only room: flow, pressure and the distance from lumen to surface are not in the model at all.

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. The frontier and every crossing here inherit those assumptions unchanged.

What this cannot show #

Which regime any real tube is in. $k$ is the channel’s width per unit of surface in sheet thicknesses, and reading it off a specimen needs a supply channel’s width, the surface it serves and the thickness of the sheet, none of which is measured here. The comb essay’s value of five is its own illustrative choice, not a measurement of anything grown.

Whether a lining would rather have a larger lumen than it needs. The meeting point is the lining with the most surface its supply can sustain, which is the right target if surface is what is wanted. A lining that also needs to carry a load down its middle — food along a gut — needs more lumen than its own surface calls for, and would sit to the right of its meeting point on the same frontier.

And what a length costs. The two regimes say what extra lengths are worth; they do not say what they cost, and a lining should add lengths only while the worth exceeds that cost. Pricing a length — in folds, in growth, in the vertices a fin of a new length needs where it starts — is outside anything measured here.

How the numbers were checked #

The crossings are found by halving on the demand, for each number of lengths and each $k$, and at every $k$ drawn the one-length crossing is required to equal $4k/(1 + 2k)^2$ past a half and exactly one half below it, and the unlimited crossing to equal $1/(1 + k)$, to nine figures.

The free rule is required at five numbers of lengths for every $k$ tabled: a lining is on its flat stretch at the crossing exactly when $k \le 1/(m(m + 1))$, and a single lining on the wrong side of that line would stop the table.

The regimes curve for one length is required to follow $1 - 1/(1 + 2k)^2$ at every $k$ past a half across three and a half decades.

Still open: the demand that grows with the reach #

The straight demand line is the simplest the channel essay’s rule allows, and the most natural refinement is the one named above. A supply that must carry further to reach surface further out needs more channel per unit of surface at the wall than near the lumen, which bends the demand upward at large surface. That would move every meeting point toward the lumen side and shift the passage between the regimes to smaller channels, and the question worth answering is whether the closed form $1 - 1/(1 + 2k)^2$ survives with $k$ reinterpreted or breaks.

The branching fin is the other direction, carried from the essay before. A fin that splits partway to the wall is the continuous limit of adding lengths, and the regimes here say when it is worth growing: only in the cheap-supply regime, where lengths close a large gap. A branching fin under dear supply would be elaboration that buys nothing, and whether a folded sheet can branch without a cut at all belongs with how much surface fits in a body.

Sideways from here, a nest pays four a level priced a hierarchy of scale on a flat base, and in a tube the standing members lose found the tube’s convergence working against its fins; both would change under supply in the way this essay’s lining did, and both are candidates for the same two-regime reading.

The habit worth carrying is about menus and demands. A frontier says what is possible; it does not say where to stand until the thing being served has been put on the same axes. The frontier here had a free stretch and a bend and a limit, and all three looked like features worth exploiting. Which one mattered depended on a single ratio the frontier did not contain.