The number is the angle

A wing that folds into nothing opens this ladder with a number: a beetle stows a wing longer than its body under a case a fraction of that length, and the ratio is the whole engineering problem. Every rung since has computed ratios — for four geometries, for a fold count, for a hinge with a radius — and every one of those computations has been made at a fold closed all the way.

A stowed wing is not closed all the way. It cannot be: the hinges have a radius, the membrane has a thickness, and the panels come to rest against one another at whatever angle those allow. So the state the ratio was computed at is a state the animal does not reach.

That would be a footnote if the gap were small. It is not a footnote, because of what the ratio does as the fold approaches the closed state.

The ratio has no ceiling in the model that computes it #

Take a corrugation of panels of unit length and close it to a half-angle θ measured from shut. Each panel tilts by θ, so the span it covers is sin θ of what it covered when the sheet was flat. The footprint shrinks in that proportion, and the packing ratio is therefore

ratio = 1 ⁄ sin θ

which is one at a flat sheet, two at thirty degrees, and unbounded as θ goes to zero.

Nothing in the pattern bounds it. A corrugation of five panels and a corrugation of five hundred both have this same law: the ratio at a given angle is the same, and the ratio at a fully closed fold is infinite for both.

That is a startling thing to notice about a quantity the whole field quotes, and it is not a paradox. It is what zero thickness means. A sheet with no thickness can be folded shut, and a fold that is shut occupies no width at all, so the ratio it achieves is the ratio of something to nothing.

The consequence for reading the literature is immediate and uncomfortable. A quoted packing ratio is a report of how far the fold closed. Fifteen times is a fold at some particular angle; forty times is the same fold closed further. Neither is a statement about a crease pattern, because the crease pattern sets neither number.

What the pattern does contribute, which is an exponent #

The pattern is not doing nothing. It is doing exactly one thing, and it is worth isolating.

A corrugation closes in one direction. Its footprint shrinks by sin θ along one axis and not at all along the other, so its ratio is 1 ⁄ sin θ.

A Miura closes in both at once. The pattern’s two crease directions shorten together as its single degree of freedom runs, so the footprint shrinks by sin θ along each axis and the ratio is 1 ⁄ sin²θ.

So the geometry enters as a power. At any angle whatever, a Miura’s ratio is the square of a corrugation’s — which sounds like an enormous advantage and is a much stranger thing than that.

Read it the other way. A quoted ratio of fifteen means a fold angle of 3.8° if the pattern is a corrugation and 14.9° if it is a Miura. Those are not similar states. The first is a fold pressed almost completely shut; the second is a fold that has stopped at a quarter of a right angle and is visibly open. A reader given the number fifteen and no pattern has been told nothing about which of those the animal is doing, and the difference is the whole mechanical question.

Where the ceiling comes from, and it is the hinge #

The angle is not free. Nothing in a body folds on a line: a crease in an organism is a compliant region with a radius, and a region with a radius will not close past the angle at which its own material meets itself.

Take that limiting angle as given — call it the closing floor — and the ceiling on the ratio follows at once. A geometry that reads 1 ⁄ sin θ is capped at 1 ⁄ sin(floor); one that reads 1 ⁄ sin²θ is capped at the square of that.

At a floor of six degrees the corrugation is capped at about ten and the Miura at about ninety-two. At ten degrees the numbers are six and thirty-three. The floor is a property of the material and the exponent is a property of the pattern, and the ceiling is the two of them together.

This is the reading a quoted ratio makes impossible and it is the one that matters. The ratio a structure can reach is the material’s closing floor raised to the pattern’s power. Two structures with the same hinge and different patterns have wildly different ceilings; two with the same pattern and different hinges have ceilings in the same proportion, squared or not squared according to the pattern.

Two charges, and they are not the same charge #

The hinge appears twice in this ladder and it is worth being careful about which appearance is which, because they act on different quantities and they are easy to run together.

The surface charge. Every fold spends (π − 2)ρ of the sheet on its own hinge, so a pattern with many folds loses much of its area to them. That is the charge the rung below this one prices, and the one that decides how many folds to use. It acts on the numerator: there is less paper to pack.

The closing charge. Every hinge stops the fold at some angle, so the pattern cannot reach the state its ratio was computed at. That is this rung’s charge and it acts on the denominator: the footprint is larger than the idealisation says.

They come from the same physical fact and they are independent as constraints. A structure could have narrow hinges that will not close — a stiff thin membrane — or wide hinges that fold shut easily. Nothing ties the radius to the closing floor except through a material model, and this collection does not have one.

So the honest statement of what a hinge costs a folded wing has two numbers in it, and the published ratios contain both, added together and unlabelled.

Which of the two the beetle is up against #

The ladder’s first rung computed the four geometries at a fixed angle and found them far apart. This rung says the angle was the interesting variable all along, so it is worth asking what happens when the four are compared the other way round.

A fan and a roll have no fold angle in them. A fan of k sectors packs to two over k because the sectors sweep through a half-turn as it closes — an accounting the shrink census makes for every corrugation here — and the sweep is complete or it is not; a roll of k turns packs to one over k because the length is wrapped k times, and a partly-unwrapped roll is a roll of fewer turns. Both are indexed by a count rather than by an angle, and both therefore have ratios that are bounded by the count and unaffected by a closing floor.

That is a real structural difference and it points the opposite way from the one the ladder’s first rung found. A corrugation and a Miura have unbounded ratios in the model and are limited by their hinges. A fan and a roll have bounded ratios in the model and are not. A lineage facing a hinge it cannot close is better served by a geometry whose ratio never depended on closing — and the two geometries that pack best on paper are precisely the two that do.

What a factor of five between the ceilings is worth #

The Miura’s exponent looks like a decisive advantage and the closing floor is where it becomes one, so it deserves a number rather than an adjective.

At a floor of ten degrees the corrugation reaches six times and the Miura thirty-three. At six degrees the corrugation reaches ten and the Miura ninety-two. In both cases the Miura’s ceiling is the corrugation’s squared, which is what the exponent means, and squaring a number above one is a widening rather than a constant advantage: the finer the hinge, the further ahead the two-direction pattern gets.

That has a consequence for what a lineage should do about its hinges, and it is not the obvious one. Improving a hinge — making it narrower, or making it close further — buys a corrugation a linear improvement and buys a Miura a quadratic one. So the return on the material depends on the pattern, and a lineage using a two-direction pattern has more to gain from a better hinge than one using a corrugation has.

Which is a testable asymmetry, and it points at something worth looking for. Structures that close in two directions ought to have invested more in their hinges than structures that close in one, because the same investment is worth more to them. Nothing in this collection can check that, and it is the kind of claim that a morphological survey could settle.

The other side of the same arithmetic is less encouraging. A Miura at a coarse hinge is punished quadratically too. At a floor of thirty degrees a corrugation reaches two and a Miura four, and the gap that was a factor of five at ten degrees has collapsed to a factor of two. The two-direction pattern’s advantage is not a property of the pattern; it is a property of the pattern together with a good hinge, and a lineage with a poor hinge gets very little for the extra crease direction.

What the ladder has been computing, restated #

Four rungs is enough to be worth summarising, and the summary is shorter than the rungs.

The first computed what four geometries pack to, at a fold angle chosen for the drawing. The second found the condition that makes a deployment possible with muscles only at the base, which is a count of freedoms rather than a ratio. The third gave the crease a radius and found that the radius consumes surface in exact proportion to the fold count, which puts a ceiling on how fine a pattern can usefully get.

This rung says the first of those was reporting a state rather than a pattern. That does not undo it: the four geometries really do differ, at any fixed angle, and the differences the first rung found are real differences at the angle it used. What changes is what the numbers are of. They are not properties of the four patterns; they are the four patterns evaluated at one point of a family, and the family is indexed by the quantity the material controls.

Read that way the ladder has a shape it did not obviously have. Every rung has been a correction in the same direction — the real thing is worse than the model and the gap grows with the fold count — and this one is a correction of a different kind. It does not say the model is optimistic. It says the model’s answer is unbounded, so any finite answer taken from it is a report of where the reader stopped evaluating.

Which theorem was checked and how #

The Miura is built at each sampled angle rather than assumed to follow the law. The pattern is constructed at three angles and its interior vertices are counted, and the figure refuses if the count changes — which is what would happen if the construction were quietly producing a different pattern as the angle ran.

The divergence is asserted rather than drawn. At a hundredth of a radian both laws are required to read above forty, so a figure whose ratio stayed bounded as the fold shut would refuse. That is the claim the essay turns on, and it is the one a wrong sign would hide.

The angle a quoted ratio implies is computed by inverting each law, and the two answers are required to differ by more than a factor of one and eight tenths. A pair of geometries that implied nearly the same angle would make the essay’s central point false, and the figure would say so.

And the closing floor is a stated parameter rather than a measurement. No hinge in any organism is measured here. What is computed is what a floor of a given size does to each geometry’s ceiling, which is arithmetic on the laws above.

Where the model stops #

The corrugation law assumes every panel tilts by the same angle, which is what a single degree of freedom means and what a real compliant structure only approximates. A wing whose hinges have different radii closes unevenly, and its footprint is set by whichever hinge closed least.

The Miura law assumes both directions close together. They do, for the rigid Miura with its one freedom; a membrane with compliant hinges has more freedoms than the panel model gives it, and the two directions can come apart.

The closing floor is treated as a single angle and it is a distribution. Different hinges in the same structure stop at different places, and the footprint is decided by the worst rather than by the mean — the same asymmetry a corrugation with unequal panels has, where the longest panel sets the pile.

And no ratio here is anybody’s measurement. The numbers 8, 15 and 30 are drawn as lines to be read against the laws, not as reports of any animal. What the essay claims is about what such a number can mean, given the laws.

What the picture cannot show #

The curves are laws and not observations, so nothing in the figure says where any structure sits along one. A wing might be anywhere on the corrugation curve, or on neither curve.

More sharply: a figure of ratio against angle cannot show which law applies, and that is the whole difficulty. Given a structure and a number, deciding whether it is a corrugation reading 1 ⁄ sin θ or a Miura reading its square requires seeing the crease pattern — and the crease pattern of a stowed wing is exactly what is hard to see, because it is stowed.

That is the reason the ratio became the reported quantity in the first place. It can be measured on a closed wing and an opened one with a ruler, and everything else about the fold needs the wing spread out and traced — including the count of freedoms the second rung turns on. The number that is easy to get is the number with the least in it, which is a shape this collection keeps meeting.

The idealisation, named #

Zero thickness is doing all the work here and it is worth saying exactly which work.

A sheet with no thickness folded shut occupies no width, so its packing ratio is unbounded. Give the sheet a thickness t and a corrugation of k folds has a footprint that cannot go below k·t however hard it is pressed, so the ratio is capped at the sheet’s extent over k·t — a bound that has nothing to do with the hinge and everything to do with the stack.

So a real structure has three ceilings on its ratio and this ladder has now priced all three: the surface its hinges consume, the angle its hinges will close to, and the depth its own layers fill. Which one binds is a question about the numbers, and for a thin membrane with wide compliant hinges it is the second — which is this rung — while for a stack of panels with narrow joints it is the third.

The essay is written in the zero-thickness model because that is the model the quoted ratios are computed in. Its point survives the correction: adding thickness makes the ratio bounded and leaves it a function of the angle, so the number is still a report of how far the fold got.

Where the ladder goes next #

This closes the wings ladder’s account of what its own numbers mean, and it leaves the ladder pointing at a measurement rather than at another computation.

The measurement is the fold angle. Every rung here has computed something that depends on it and no published description of a stowed wing gives it, because a wing is described by what it achieves rather than by the state it achieves it in. A single photograph of a cross-section through a stowed wing would supply it, and with it every ratio in this ladder becomes a statement about a mechanism rather than an arithmetic identity.

The other direction is towards what the fold angle costs elsewhere. A structure held at a fold angle short of shut is a structure storing elastic energy in its hinges, and that energy is what drives it open — so the angle is not only a limitation on the packing but the mechanism of the deployment. A wing that closed all the way would have nothing to open it, which turns this rung’s constraint into the ladder’s earlier finding about one degree of freedom and one thing to pull read from the other side.

The habit worth carrying is a question to ask of any performance number. What does this quantity do at the limit of the model that computes it? A number that diverges there is not describing the object; it is describing how close to the limit the object got, and the two are routinely reported in the same units.