The angle the eight does not know

Standing up beats lying down by eight put two ways of filling a clearance against each other and found a ratio with no size in it. Plies lying parallel to a base share the clearance and pay a quadratic penalty for the sharing; walls standing across it each have the whole height and compete only for somewhere to stand. The first reaches $c/4\tau$, the second $2c/\tau$, and the quotient is eight at every clearance and every thickness.

Every drawing of that argument has its walls square to the base, and the squareness has never been given a reason. It is the natural thing to draw and it is what a diagram of a comb looks like. Whether the factor needs it is a different question, and it turns out to have a short answer.

What leaning costs and what it buys #

Lean a member over to an angle $\theta$ with the base. Two things change and they change in opposite directions.

It has to be longer. To reach a clearance $c$ from the base at an angle $\theta$, a member must be $c/\sin\theta$ long, so its two faces carry $2c/\sin\theta$ of surface rather than $2c$. At forty-five degrees that is forty-one per cent more surface per member; at ten degrees it is nearly six times as much.

It has to take more of the base. A slab of thickness $\tau$ meeting the base at $\theta$ cuts a strip of width $\tau/\sin\theta$ across it, and two neighbours cannot stand closer than that without meeting. So the number of members per unit of base falls from $1/\tau$ to $\sin\theta/\tau$ — at ten degrees, a sixth as many.

Multiply the two and the reach is

$$\frac{2c}{\sin\theta}\cdot\frac{\sin\theta}{\tau} = \frac{2c}{\tau}$$

with no angle in it. The sine cancels before anything is evaluated, which means the result is not an approximation and does not depend on the numbers chosen. Checked against the geometry at ten angles from a right angle down to one degree, the reach comes out at the same 200.000000000 on all ten, while the member length runs over a factor of fifty-seven and the count over the same factor the other way.

The strip a leaned member stands on #

The footing is the half of the argument that is easy to get wrong, so it is worth doing slowly.

A member is a slab of thickness $\tau$. Standing square, it meets the base along a strip exactly $\tau$ wide, and two members are as close as they can be when those strips touch. Lean it to $\theta$ and the slab still has thickness $\tau$ measured across its faces, but the base now cuts it obliquely, so the strip it occupies is $\tau/\sin\theta$ wide — at thirty degrees, twice as wide as the slab is thick.

The constraint that matters is not the strip, though: it is that two neighbouring members must not pass through one another anywhere along their length, and two parallel slabs at pitch $p$ along the base are separated perpendicular to their faces by $p\sin\theta$. Requiring that to be at least $\tau$ gives $p \ge \tau/\sin\theta$ — the same number the strip gave, which is the check worth making rather than assuming, because the two conditions are about different things and only coincide because the members are parallel.

A comb of members at mixed angles has no such coincidence, and that is where the accounting here stops applying.

Ten angles, and the two columns that move #

The table underneath the first figure is the whole finding in numbers, and its interest is entirely in the columns that are not constant.

At a right angle each member is exactly as long as the clearance, and there are a hundred of them per unit of base out of a sheet a hundredth thick. At forty-five degrees each is 1.414 long and there are 70.7. At twenty degrees, 2.924 and 34.2. At six degrees, 9.567 and 10.5. At one degree each member is 57.299 clearances long and there are 1.745 of them — fewer than two members where there were a hundred, each nearly sixty times as long.

Every one of those ten arrangements holds 200.000000000 times its base in surface. The product is not approximately fixed and it is not fixed to within the precision of the measurement; it is fixed because the two factors are reciprocal, and the measurement is there to confirm that the geometry really does produce the two factors the algebra assumed.

Two surfaces in one box found a division that looked free turning out to cost, and this is the mirror of it: an operation that looks as though it must cost turns out to be free. Neither could be settled by looking at the drawing.

And the same under supply #

The channel grows with what it feeds took the comb’s advantage back by making its members expensive to reach. A wall of height $c$ serves $2c$ of surface, whatever supplies it must be sized for what it serves, and that channel runs beside the wall — charged against the pitch, which is the budget the comb was winning on. The supplied comb reaches $2c/(\tau+\beta c)$ and has a ceiling of $2/\beta$ that no clearance passes.

The leaning cancels there too, and for the same reason twice over. A member at $\theta$ serves $2c/\sin\theta$ and needs a channel in proportion, so both its footing and its channel are divided by $\sin\theta$ — and the pitch becomes $(\tau + \beta c)/\sin\theta$ while the count becomes $\sin\theta/(\tau+\beta c)$. Measured at four angles, the supplied reach is 33.333333333 at every one of them, against the square arrangement’s identical number.

So the ceiling a body cannot pass is not a ceiling on square construction. It is a ceiling on standing members of any inclination, which is a considerably stronger statement and costs nothing extra to make.

What the eight is about instead #

The cancellation says what the factor of eight is not about. It is worth saying what is left.

A stack’s plies share the clearance: each one that is added takes $\tau$ out of the depth every other ply has to work in, so the $k$-th ply is paid less than the first and the total turns over at $k = c/2\tau$. That turning-over is the oldest result on this line of argument — how much surface fits in a body found it with a corrugation rather than a stack, and found that a sheet of no thickness has no turn at all. A comb’s members do not share the clearance at all — every member has the whole height whatever the count is — and they compete only for footing, which is a budget in the other direction.

That is the whole of it, and the angle has nothing to do with it. A leaned member still has the whole clearance; it simply approaches it more slowly. Leaning changes how a member is arranged in the box and not what it is charged for, and a ratio between two architectures can only see what they are charged for.

Why nothing else in the model notices either #

A fair question is whether the angle survives anywhere in the accounting, and the answer is that it survives in every intermediate quantity and in no final one.

The pitch carries it: $\tau/\sin\theta$ unsupplied, $(\tau+\beta c)/\sin\theta$ supplied. The member length carries it. The surface per member carries it. The horizontal span of a member carries it, and that one never appears in a reach at all. What does not carry it is the surface per unit of base, which is the only quantity any of these comparisons has ever been about.

That is a narrow claim and it should be read narrowly. A nest pays four a level found a price per stage by exactly this kind of bookkeeping, and the price was not one; the surface has to be supplied found supply and sheet entering an optimum on identical terms, which again is a statement about what a budget is charged rather than about what an arrangement looks like. Every result on this line of argument has come from asking which budget pays for what, and the angle is the first parameter to be charged to two budgets that cancel.

The stack is not a comb leaned flat #

The invariance suggests a family with the stack at one end of it, and there is no such family.

As $\theta$ falls the members get longer and rarer, and at $\theta = 0$ there are none: the count $\sin\theta/\tau$ goes to zero, the footing of a single member goes to infinity, and one member lying flat on the base occupies all of it and reaches nothing. The limit is empty rather than being a stack. A stack’s plies do not stand on the base; they lie above it, one over another, and nothing in the inclined family ever produces a second layer.

That is worth stating because the two architectures are otherwise easy to imagine as extremes of one arrangement, and every ratio computed between them would then be a ratio between two points of a continuum. They are not. The comb is a one-parameter family whose parameter does nothing, and the stack is a separate object with its own count and its own penalty.

What a leaned comb does to the box #

One quantity the reach hides is worth putting a number on, because it is the reason a body could not lean its members as far as the algebra permits.

A member at $\theta$ rises the clearance $c$ while travelling $c/\tan\theta$ along the base. At a right angle that displacement is nothing; at forty-five degrees it is one clearance; at ten degrees it is 5.67 clearances, and at one degree it is 57.3. So the arrangement that holds the same surface with two members instead of a hundred needs those two members to lie across fifty-seven times the clearance of base each, and a box shorter than that holds no complete member at all.

The reach is a rate, and a rate is only collected where the pattern repeats. In a box of length $L$ the members that reach the clearance without running off the end number about $(L - c/\tan\theta)/p$ rather than $L/p$, so the leaned comb loses a fixed piece of the box rather than a fixed fraction — which matters when the box is a few clearances long and vanishes when it is a thousand. Every number here is the limit of a long box, and the limit is approached slowly at small angles.

That is the honest form of the caveat: the invariance is exact and the arrangement it describes gets harder to fit as the angle falls, for a reason that has nothing to do with area.

What the cancellation rests on #

A member is a slab of uniform thickness. Its footing is its own cross-section, $\tau/\sin\theta$, and nothing is added for a foot, a fillet or an attachment. A real member joined to a base needs more room where it meets it than it does further up, and that extra would not cancel.

The base is flat and the box is long. A member at $\theta$ spans $c/\tan\theta$ horizontally, which at ten degrees is nearly six clearances, so the arrangement only repeats cleanly in a box long enough to hold many of them. The reach computed is a rate per unit of base in the middle of such a box, and the ends of a short box would lose most of a leaned comb.

Nothing is asked to stand up. There is no stiffness here and no gravity: a member at one degree is a sheet nearly lying down that somehow holds its own shape and does not touch the neighbour it overhangs. Whether it could is a question about a material, and this account has none.

And the supply is a width beside the member. That is the model the channel grows with what it feeds set up, and leaning divides it by the same sine as everything else only because it is charged along the base. A channel charged against the clearance instead would not cancel.

What the invariance cannot decide #

It does not say the angle is free. It says the surface does not see it. Everything else about an inclined comb differs — the horizontal span, the shape of the space between members, where a fluid moving along the base would go, and how far a member’s tip is from its own foot — and a body choosing an angle is choosing among those rather than among surface areas.

It does not cover members of unequal angle. Every member in these arrangements leans the same way by the same amount. A comb of members at mixed angles has a packing problem rather than a rate, and nothing here bounds it.

It says nothing about members that are themselves folded. A member here is a flat slab with two faces. One that carried a corrugation of its own would hold more, and what a level inside a level costs is the price a nest pays rather than anything the angle decides — a question settled elsewhere and not reopened by this one.

It says nothing about curved members. A member that bends as it rises has a length and a footing that do not stand in the reciprocal relation the cancellation needs, and whether such a member does better or worse is not computed.

And it is a statement about a rate, not about a structure. Nothing grown has a seam is the standing warning that a surface in a body is reached and built rather than placed, and an arrangement that is equal on paper may be unreachable in practice for reasons this model has no words for. The same goes for the division of a clearance between several surfaces, which turns out not to be free once each of them has to be reached: an invariance in one variable says nothing about the others.

A reading, and its limits #

Nothing here measures an organ, and the standing rule on this line of argument is that a figure draws a geometry computed rather than a specimen. What the invariance does give is permission to stop reading one thing into a described structure.

A lining whose members are described as oblique, slanted or imbricate is not, by that fact, holding less surface than one whose members stand square. It may be holding the same, and the model says it is holding exactly the same if the members are parallel slabs reaching one clearance. So an angle observed in a described structure is evidence about something other than area — about how something moves past it, about how it grew, about how it packs when the tube around it contracts — and treating it as a compromise on surface is reading a cost the geometry does not charge.

The converse is the sharper half. A structure that stands its members square has not thereby bought anything, so squareness is not evidence of optimisation either. Both readings were available before this was computed and neither was licensed.

Still open: where the leaning would show #

The surface cannot see the angle, so the next question is which measurable quantity can.

The obvious candidate is the space between the members rather than the members themselves, and it is the space that a channel has to run in. In a square comb that space is a slot of constant width; in a leaned one it is a slot of the same width tilted, so anything moving along the base meets the members obliquely and anything moving up the slot travels further. A model with a flow in it would price that, and would give the angle its first consequence.

The second is the tip, and it is the one with a computation attached. A leaned member’s tip is displaced horizontally from its foot by $c/\tan\theta$, so a comb of leaned members has a free edge that is a different shape from the base it stands on. Where the box is not a box — where the base curves — that displacement stops being cosmetic, because the clearance a member reaches into is then not the same at its tip as at its foot. That is the one change of setting this model has not been given, and it is the setting a body actually presents.

Sideways from here, the cancellation is worth remembering as a shape of result rather than a fact about combs. Two quantities that are reciprocal in a parameter make their product blind to it, and a ratio computed from that product then looks like a fact about the arrangement when it is a fact about the budget. The check is one line: write the parameter into both factors and see whether it survives the multiplication.

The habit worth carrying is about drawings that settle things quietly. A figure has to draw some particular arrangement, and whatever it draws reads as part of the claim. Every picture of this comb stood its members square, so squareness joined the argument without ever being claimed, and the only way to find out was to vary it and watch nothing happen.