How far open is a question about the grip

The fold count sets the spring prices a corrugation at one state: at a half-angle $\theta$ the zigzag needs a clearance of $(S/k)\cos\theta$, holding it there takes $2kc\psi/(S\cos\theta)$, and the two multiply to a number with no fold count in it. That is exact, and it is a photograph.

A leaf in a bud, a wing under its case and an array in a fairing are not photographs. Each of them travels — once, in one direction, from packed to flat — and along that journey the number above is not one quantity but a curve. Reading the curve turns out to answer a question the single state cannot even ask: not how hard a structure is to hold, but what decides how far open it is.

The sheet is loaded when it is shut #

The sign is the whole of it, and it is easy to read the wrong way round.

Each hinge is a compliant region that rests flat — the state a lamina grows in, before anything folds it — and stores an energy proportional to the square of how far it has been turned from that rest. At a half-angle $\theta$ every hinge has turned by $\psi = \pi - 2\theta$, so the corrugation holds

$$E = \frac{k c \psi^2}{2}$$

which is largest when the sheet is shut and zero when it is flat. A folded corrugation of this kind is a loaded spring. Left alone, it opens. Opening with nothing to pull is the kinematic half of the same statement — the span rises at every step, so a single slowly changing quantity can run the motion — and this is the energetic half: nothing has to run it, because the structure is already trying to go that way.

So the force in the held-state calculation is not a drive. It is a restraint: what the bud wall, the wing case, the surrounding tissue or the launch tie-down has to supply to stop the sheet going further. Read that way,

$$F(\theta) = \frac{2kc}{S}\cdot\frac{\pi - 2\theta}{\cos\theta}$$

and the expression separates into two brackets that have nothing to say to each other. The first carries everything about the structure — sheet length, fold count, one hinge’s stiffness. The second carries everything about the angle, and every corrugation ever folded meets the same one.

Two things fall at once and one falls faster #

The shape is not obvious, because the two effects inside it pull opposite ways.

As the sheet opens, each hinge has less turn left to give up: $\psi$ falls linearly to zero, so the torque each exerts falls linearly too, and that pushes the restraint down. But the leverage falls as well. The span is $S\sin\theta$, so a fixed change of angle buys less and less width as the angle grows — near flat the panels are almost along the span and turning them further hardly widens anything — and dividing by $\cos\theta$ is what that costs. That pushes the restraint up, and without bound.

Writing $\varepsilon = \pi/2 - \theta$ for the angle left to flat settles the race. The turn is $\psi = 2\varepsilon$ and the cosine is $\sin\varepsilon$, so the ratio is $2\varepsilon/\sin\varepsilon$, which tends to $2$ — not to zero and not to infinity. The unbending wins, and by a bounded margin.

The same substitution settles the direction. Differentiating puts $2(\varepsilon\cos\varepsilon - \sin\varepsilon)$ over a positive denominator, and that numerator is $-2\varepsilon^3/3$ to leading order: negative for every $\varepsilon > 0$, with no interior turning point anywhere. Sampled at two thousand steps from shut to flat, the restraint falls at every one of them.

So the two ends are $\pi$ and $2$, and the whole motion is priced inside a band of exactly $\pi/2$. Not approximately, not for a particular material, not at a particular fold count: 1.5708, for every corrugation there is or could be.

Where the shut state actually is #

The value at $\theta = 0$ is a limit no sheet occupies, and using it would overstate what a container has to do.

Adjacent panels of length $S/k$ tilted by $\theta$ separate by $2(S/k)\sin\theta$ at their far ends, so a hinge of radius $\rho$ props them apart when that separation reaches $2\rho$ — putting the packed state at $\sin\theta_0 = k\rho/S$. The finer the corrugation, the earlier it stops, because its panels are shorter and the same hinge takes a larger share of the angle to make room. Nothing in a body folds on a line is where that radius comes from, and it is doing a second job here: it sets the width of a crease, and it also decides how shut the shut state is.

At sixteen folds on a sheet of ten, hinges of radius 0.02 prop at 0.032 radians and need 432 to hold; radius 0.05 props at 0.080 and needs 168; radius 0.1 at 0.161 and 80.1; radius 0.2 at 0.326 and 36.8. Each is short of the doubled-back limit — 98.0 per cent of it at the narrowest hinge, 83.7 per cent at the widest — because a wider hinge is a softer spring and a longer prop, and both corrections push the same way.

The real band across a real motion is therefore a little under $\pi/2$ rather than exactly it. That correction is a few per cent, and it only ever removes requirement from the shut end.

How far open is decided by the container alone #

Here is what the falling curve is really for.

Suppose a container supplies a constant restraint $R$. Since $F$ falls strictly from its shut value to its flat value, there is at most one angle at which $F(\theta) = R$, and it is stable: open a little further and $F$ drops below $R$, so the container pushes back; close a little and $F$ rises above $R$, so the structure pushes out. Every partly open state of the sheet is the equilibrium of exactly one restraint, and every restraint in the band has exactly one state. How far out a corrugation sits is not something the pattern decides, or the material, or the fold count. It is a reading of the grip.

And because the band is only $\pi/2$ wide while the span runs the whole way from nothing to everything, the reading is extraordinarily sensitive. At eight folds on a sheet of ten with hinges of radius 0.05, a grip ten per cent weaker than the one that holds the sheet shut leaves it 23 per cent of the way out; twenty per cent weaker leaves it 47 per cent out; a third weaker leaves it 92 per cent out. A bud that loses a third of its strength has effectively opened.

The span is an instrument for the grip #

Sensitivity runs both ways, and the direction that looks like a weakness is a measurement.

The span responds violently to the grip, which means the grip is read very precisely by the span. Inverting the curve at eight folds: a sheet found 23 per cent out is being held at 90 per cent of its shut requirement, one found half out at 80 per cent, one found nine tenths out at 67 per cent. The whole range of grips that hold anything maps onto the whole range of spans, so the span resolves the grip to a per cent or two along its entire length — which is an unusually good instrument, and it is made out of the same steepness that makes the structure’s position so precarious.

What it needs to be usable is one number, and it is a number a photograph carries. The reading is a ratio, so the hinge stiffness, the sheet length and the fold count all cancel: the same measurement works on a leaf, a wing and a metal array without knowing any of their materials, provided the shut requirement is known or the same structure can be seen shut. That is a rare shape for a biological measurement, where almost everything needs a stiffness nobody has.

What it cannot do is work outside the band. A sheet found completely flat says only that the grip is below two thirds, and a sheet found completely shut says only that it is above the shut requirement. The instrument has a floor and a ceiling and nothing in between them is out of reach, which is the opposite arrangement from most measurements of a living structure.

That is a different account of a leaf’s timing from the one a pattern suggests. It says the shape of the opening curve through a season is the shape of the bud’s decline, compressed: a container losing strength steadily lets the leaf out in a rush near the end, not because anything about the leaf changed, but because the last third of the grip covers nine tenths of the span.

Below the flat-state value there is nowhere to stop #

The band has a floor, and the floor is what makes the release complete.

A restraint below the flat-state value $4kc/S$ solves $F(\theta) = R$ nowhere. There is no equilibrium at any angle, so the sheet runs to flat and stays there. On the figure that floor is 65.3 per cent of the shut requirement — a little above $2/\pi$, because the hinge props the panels short of doubled-back — and it divides the outcomes into three, not two. A container stronger than the shut requirement holds the sheet packed. A container between the two values holds it at one determined partial state. A container below the floor holds it nowhere.

So a corrugation cannot be released a little. Whatever fraction of its strength a container loses, either it still holds the sheet somewhere in that narrow band, or the sheet is flat. There is no long tail of nearly-open states available to a weak grip, because a weak grip is not weak enough to be interesting: everything below two thirds is the same outcome.

This is the property a mechanism would be designed for and an organism gets for nothing. Folding that gets built collects structures whose whole requirement is to be large in use and small in transit, along a path nobody has to trust to chance, and a deployment with no partial equilibria below a known threshold is exactly that path. The failure it rules out is the one a linkage near a singular configuration suffers: stopping somewhere nobody designed, with no force left to finish.

The release tracks the span almost exactly #

If the restraint falls by half as much again from end to end, the early span should be conspicuously where the energy goes. It is where the energy goes, and it is not conspicuous.

Half the stored energy is gone by the time 45.7 per cent of the span is out, and the curve of energy released against span delivered never leaves the diagonal by more than 4.4 percentage points. The near-cancellation is the earlier race seen from the other side: the restraint falls by $\pi/2$, the span bought per unit of turn falls by the same cosine that put the divisor there, and integrating recovers almost exactly the straight line a constant force would have given.

That is worth keeping because a rough argument gets it wrong in both directions. Told the restraint falls by fifty-seven per cent, a reader expects the first half of the span to carry well over half the energy; told the release tracks the span, a reader expects a constant force. Neither is right, and the truth is much nearer the second — so the energy a structure dumps into its surroundings arrives in rough proportion to the width it has gained, which is a gentler schedule than the force curve alone implies.

A finer corrugation holds more and reaches no further #

Along the whole motion the accounting for a finer pattern is nearly the same as at one state, and the small discrepancy says something.

Four folds are propped at 0.020 radians, need 43.5 to hold and 28.0 at flat, and let go of 337. Eight folds: 0.040 radians, 85.9 and 56.1, and 657. Sixteen: 0.080 radians, 168 and 112, and 1,246. The forces double exactly with the count. The energy does not — per fold it falls 7.6 per cent from four folds to sixteen, and the shortfall is exactly the square of the turn each hinge still has at its own propped state, since a finer corrugation starts less shut.

Everything else is proportional, and the span reached is ten on every row. A lineage that folds more finely needs a proportionally stronger container, releases proportionally more energy when that container fails, and arrives at exactly the same width. Four finders, one option argues that the four lineages had little to choose between when they chose the corrugation; this is what follows from having chosen it, and it is not a free parameter either.

The count that packs best is the one that lets go hardest #

The fold count is not chosen in isolation. Four materials, four optima finds a hinge radius $\rho$ implying a best packing count $k^* = S/2(\pi-2)\rho$, and that count is now also the most violent release the material can be asked for.

The restraint at the packed state is proportional to $k$ and to $c$, and both go as $1/\rho$, so a material folded at its own packing optimum needs a container whose strength goes as $1/\rho^2$. A material with a tenth of the hinge radius packs ten times as finely, packs ten times as small, and needs a hundred times the grip. The census returns one found a stated packing ratio forcing the pattern; the ratio also forces the count, the count forces the restraint, and the restraint has to come from a bud, a case or a tie-down that had no say in any of it.

It is also the reading a material made of creases would give. The paper supplies one number and the pattern supplies everything else, so two corrugations of one sheet at different counts are two materials — with different grips required, different energies stored, and the same reach.

What the curves cannot show #

The figures draw a model spring inside a model container, and neither half is measured.

They cannot show whether a restraint is constant. A bud wall is not a fixed force; it is tissue with its own stiffness, so the true problem is two springs meeting, and the equilibrium is where their curves cross rather than where one curve meets a horizontal line. That changes the arithmetic of the band, though not its existence: a container that stiffens as the structure pushes is a rising line crossing a falling one, which still has one intersection.

Nor can they show what else resists. A leaf opening inside a bud rubs against it, a wing slides out from under a case, an array unlatches — and an unlatching force can be much the largest number in the whole sequence and appears nowhere here. A contact force that rises as the structure opens could put an interior maximum back into a curve this essay spent its length showing has none.

And nothing in the figures is a rate. A restraint curve says where a structure sits, not how fast it gets there; for a leaf, which opens over days, the rate is the quantity a reader would most like and the one a purely elastic model cannot supply.

The springs the model assumes #

The hinges rest flat and store energy as the square of their turn. That is a uniform elastic strip at small strain, and it is what makes the shut state the loaded one. A hinge that rests part-folded moves the whole argument, because it moves where the energy is least.

Every hinge turns by the same angle. A corrugation of equal panels does, which is why one angle describes the structure and why the restraint separates into a structure factor and an angle factor at all.

The panels are rigid and touch only where the hinge props them. A panel that bends stores energy the model does not count; faces that touch add friction that rises with the number of contacts.

And the restraint acts on the span. The quantity computed is a force at the two ends. A container pressing over the whole surface supplies the same restraint in total and distributes it differently, and the neat band is a statement about the idealised pair of forces.

How the numbers were checked #

The restraint is sampled at two thousand steps from shut to flat and required to fall at every one, which is the monotonicity claim itself rather than an inspection of a drawn curve.

The two ends are required to be $\pi$ and $2$ to within a part in a billion, so the $\pi/2$ band is measured rather than taken on trust.

The equilibrium for a given grip is found by bisection, which is only valid because the function is strictly falling — and the figure requires the resulting spans to be monotone in the restraint, so a second solution anywhere would break it.

And the energy per fold is checked against $c\psi(\theta_0)^2/2$ to within a part in a trillion, which is what attributes the 7.6 per cent shortfall to the propped state rather than to arithmetic.

Still open: whether a corrugation can have two places to rest #

Every result here rests on one assumption, and it is the assumption everyday experience contradicts.

The hinges rest flat, so the stored energy is a single well with its lowest point at the flat sheet, and the motion is a slide down one side of it. A creased sheet’s hinges do not rest flat — paper remembers what it has been folded to — and a hinge that rests part-folded puts the lowest point of its own energy somewhere in the middle of the motion. A set of hinges resting at different angles is a set of springs that disagree about where the structure should sit, and a disagreement of that kind can in principle have more than one resolution.

Whether it actually can is the next computation, and it decides what a leaf that has been packed for a season is. If the answer is one resting place, a remembered crease shifts the whole band and changes nothing structural. If it is two, a corrugation has a snap in it — a state it stays in, another state it also stays in, and a threshold between them — and every statement above about a single equilibrium per grip needs qualifying.

Sideways from here, the same question is the one the insect wing asks at a single vertex, where the geometry is small enough to enumerate and the answer is known. A wing is not a corrugation, and whether the vertex’s answer survives being repeated along a sheet is exactly what is unclear.

The habit worth carrying is about signs before sizes. Before pricing a force, find where the energy is least, because that decides which way the structure is trying to go. A quantity computed correctly and read as a drive rather than as a restraint gives the right number attached to the wrong story, and every conclusion drawn from it will be about a structure that does not exist.