The fold count sets the spring

Four materials, four optima prices the closed corrugation and then turns to the open one. A leaf in a bud is opening; a wing is being stowed or deployed; an array unfolds once and then stays. On the way open, the essay says, the fold count does not change the span at all — sixteen folds and four sweep out the same width at the same angle — and changes only the depth the zigzag needs while it does it. A finer pattern is shallower, and it pays nothing for the privilege, because its hinges were already paid for when their surface was taken out of the sheet.

Which means the trade-off computed there is the whole of the argument against a finer pattern. Nothing else pushes back.

The first half of that is exactly right and is worth keeping. The second half is not. Something does push back, it is the hinges again, and it pushes back in exact proportion to what the finer pattern gains.

The span does not know the fold count #

Take a sheet of length $S$ and fold it into $k$ panels of equal length, $S/k$ each. Open the corrugation to a half-angle $\theta$, measured from shut, so that every panel is tilted by $\theta$ from the direction the sheet closes along. Each panel covers $(S/k)\sin\theta$ of span, and $k$ of them cover

$$\text{span} = S\sin\theta$$

whatever $k$ is. The first figure draws that curve for three fold counts and they coincide, because they are the same function.

The clearance is a different matter. A zigzag occupies a band as deep as one panel’s height, $(S/k)\cos\theta$, so four folds need four times the room sixteen do at every angle. That is the gain the earlier argument identified, and it is real: a bud or a wing case that can hold the open-ish structure is a container with a depth, and a finer corrugation fits a shallower one.

Opening with nothing to pull adds the property that makes this usable in an organism. The span rises monotonically as the angle runs, so a structure that only ever grows or only ever relaxes can drive the opening from one end to the other without anything needing to reverse. The fold count does not disturb that either.

Every hinge is a spring #

Nothing in a body folds on a line gave the crease a width. A hinge in an organism is a compliant region of arc $(\pi - 2)\rho$, and that region is where the bending happens: the panels either side stay flat and the hinge curves.

A region that bends stores energy. For a strip of uniform stiffness bent through an angle $\psi$ over an arc $a$, the curvature is $\psi/a$ and the energy is proportional to $\psi^2/a$, so each hinge behaves as a rotational spring whose stiffness is inversely proportional to its own arc. A fat hinge is a soft spring and a narrow one a stiff spring, which is the ordinary experience of bending a thick strap and a thin wire through the same angle and finding the wire harder.

Take the corrugation’s rest state to be flat — the state a leaf lamina grows in, before anything is folded — and each hinge stores an energy proportional to the square of how far it has turned away from flat. At a half-angle $\theta$ every hinge has turned by $\psi = \pi - 2\theta$. With $k$ hinges the corrugation stores

$$E = \frac{k\,c\,\psi^2}{2}, \qquad c = \frac{1}{(\pi - 2)\rho}$$

in units of the strip’s own bending stiffness. That is $k$ times one hinge’s energy, because every hinge has turned by the same angle.

The force needed to hold the corrugation at a span is the slope of that energy along the span. Differentiating both with respect to $\theta$ and dividing,

$$F = \frac{2kc\psi}{S\cos\theta}$$

which is proportional to the fold count at every angle. The figure checks the closed form against a difference quotient of the energy along the span, and the two agree to a part in a hundred billion.

In series along the paper, in parallel along the span #

The proportionality has a structural reason that is worth pulling out, because it is the opposite of what the arrangement looks like.

The hinges of a corrugation sit one after another along the paper. Springs one after another are ordinarily springs in series, and springs in series are softer than any one of them: a chain of ten identical springs stretches ten times as far under a load as one spring does, because each spring takes the load and the deflections add.

A corrugation’s hinges do not behave like that. When the span changes by a small amount, every hinge turns by the same small angle — the span is $S\sin\theta$ and every panel shares the same $\theta$ — so no hinge takes up more of the motion than another, and adding a hinge adds another spring turning through that same angle. That is how springs in parallel behave. The hinges lie in series along the sheet and act in parallel along the span, and so the stiffness of the whole grows with the number of folds instead of shrinking.

A concertina’s bellows is the familiar object with this property, and the property is why a bellows with fine pleats feels stiffer than one with coarse pleats of the same material.

What a finer fold saves in depth it pays in force #

Put the two quantities side by side and the trade-off is exact.

The clearance is $(S/k)\cos\theta$. The holding force is $2kc\psi/(S\cos\theta)$. Multiply them:

$$\text{clearance} \times \text{force} = 2c\psi = \frac{2\psi}{(\pi - 2)\rho}$$

and the fold count has gone. So has the sheet’s length, and so has the cosine. What is left depends on how far the hinges have turned and on how stiff one of them is.

At a half-angle of half a radian, with hinges of radius 0.05 on a sheet ten long, the product is 75.04. Four folds need a clearance of 2.19 and a force of 34.2; eight need 1.10 and 68.4; sixteen need 0.55 and 137; thirty-two need 0.27 and 274. Every row multiplies to 75.04.

That is the correction. A finer corrugation is not a free improvement on the way open; it is a different point on a curve of constant product. A structure can have a small clearance or a small holding force, and the fold count decides which, but it cannot decide to have both.

What a bud with a force limit decides #

The product being fixed has a consequence that removes the fold count from a question it looked essential to.

Suppose a structure has to sit, part-open, inside a container of depth $c_{\max}$, and whatever holds it there — the bud’s wall, a wing case, the turgor of surrounding tissue — can supply at most a force $F_{\max}$. The container demands a clearance of at most $c_{\max}$, which is a floor on the fold count: $k \ge S\cos\theta / c_{\max}$. The force limit demands a holding force of at most $F_{\max}$, which is a ceiling: $k \le F_{\max}\,S\cos\theta / 2c\psi$.

A fold count satisfying both exists exactly when the floor is below the ceiling, and cancelling the common factors, that is

$$c_{\max}\,F_{\max} \ge 2c\psi$$

which has no fold count in it. The container and the force together decide whether the state is reachable at all, and if it is, a band of fold counts works and the band’s position is set by the sheet. Whether a partly open corrugation can be held in a given bud is a question about the hinge and the bud, not about the pattern.

That turns a design question over. A lineage cannot solve a tight container with a weak hold by folding more finely, because every fold that saves depth costs force in the same proportion. It can only change the hinge — a softer spring, from a wider or more compliant region — or change how far the hinges have to turn.

It also sharpens what the bud chooses. That essay found the container selecting among patterns the geometry left open; this condition says the container selects nothing among fold counts of one hinge, because for a given hinge every fold count inside the band is equally held and every count outside it fails for one of two reasons that the product makes exchangeable.

The best fold count makes the stiffest spring #

Four materials, four optima found that a hinge radius $\rho$ implies a best fold count for packing, $k^* = S/2(\pi - 2)\rho$, inversely proportional to the radius. Fold each material at that count and ask what spring results.

The fold count goes as $1/\rho$. Each hinge’s stiffness goes as $1/\rho$. The holding force is their product, so it goes as $1/\rho^2$. Halve the hinge radius and the material’s best corrugation is a spring four times as stiff.

At a half-angle of half a radian on a sheet ten long, hinges of radius 0.2 fold best at 21.9 folds and need a force of 46.8 to hold; radius 0.1, 43.8 folds and 187; radius 0.05, 87.6 folds and 749; radius 0.02, 219 folds and 4,681. The clearances run the other way, from 0.40 down to 0.04, and every material’s clearance times force is its own constant.

So the packing optimum has a mechanical shadow. The material that packs best — narrow hinges, many folds — is also the material whose packed corrugation is hardest to hold part-open and springs hardest when released. Those can be the same property seen from two sides, or two costs, depending on what the structure is doing.

The same count, read as a material #

There is a reading of the product that belongs to a different field, and it is worth making because it changes what kind of claim the result is.

A material made of creases argues that a folded sheet’s mechanical properties belong to its pattern rather than to its paper — that a Miura’s response to being squeezed is set by its angles, and the paper contributes only a scale. The spring computed here is the simplest instance of that claim. The paper supplies one number, a hinge’s stiffness, and the pattern supplies everything else: how many hinges there are, how far each turns for a given change of span, and therefore how stiff the whole corrugation is along the span. Two corrugations of one paper at different fold counts are two different materials, with stiffnesses in the ratio of their counts.

What the product adds is a constraint on that material that the metamaterial reading does not usually state. A folded sheet’s stiffness along its span and its thickness across it are not independent properties that a designer tunes separately. For a corrugation they are tied by the fold count, and their product is fixed by the hinge. A thin stiff corrugation and a thick soft one of the same hinge are the same design at different counts; a thin soft one is not available at all.

The fold count that the census of corrugations found to be forced by a stated packing ratio therefore forces the spring as well. A lineage that has settled on a ratio has settled on a stiffness, and only a change of hinge can move it.

That also qualifies the paper-folder’s experience, which runs the other way. Paper remembers the angle it has been creased to and relaxes toward it, so a creased hinge’s rest angle is not flat and drifts over time; the spring computed here is the elastic part of that behaviour and not the whole of it. And a crease has a radius that a sharper crease reduces, which in this model makes each hinge stiffer and every corrugation of that paper stiffer by the same factor, without changing the product’s independence from the count.

Energy that closes is energy that opens #

The spring is not only a cost, and it would misrepresent the result to present it as one.

Energy stored by closing a corrugation is available when it is released. A structure pressed into a bud stores $kc\psi^2/2$ in its hinges, and if the bud opens, that energy drives the lamina toward flat with no growth required. A finer corrugation, packed to the same span, stores proportionally more and opens with proportionally more force. For a structure whose opening has to happen quickly, or against something, the stiffness is exactly what it wants.

Which of the two readings applies depends on how the structure changes state, and the model does not know. A leaf that expands by growth does work against its own hinges the whole way, and a stiffer spring is resistance. A wing or a seed pod that is held closed and then released is using the same stiffness as a drive. The fold count sets the spring; whether a spring is wanted is a fact about the organism.

What the model does insist on is that the two are one quantity. A structure cannot have fine folds that pack well and open gently, any more than it can have a small clearance and a small holding force, because the same k multiplies the same hinge.

What the curves cannot show #

The figures draw a model spring and not a measured one, and they cannot show whether any organism’s hinge is a spring at all.

Plant tissue at a fold is not an elastic strip. It is hydrated, it relaxes under a sustained load, and it grows into the shape it is held in, so a hinge held bent for a week in a bud may store very little energy by the time the bud opens. A wing hinge made of resilin — an unusually elastic protein — is much nearer the model. A mechanical joint in a deployable array may have no stiffness at all, or a deliberately fitted spring. The same curve cannot represent all three, and nothing in the figures says which one a given structure is.

Nor can the figures show what the force acts against. A holding force is a force on the ends of a corrugation, and a real structure is held by its surroundings over its whole surface. The argument survives that — the energy stored is what it is, however it is held — but the neat product of clearance and force is a statement about one idealised pair of forces, not about the stress in a bud’s wall.

The spring the model assumes #

The hinges are linear. Each stores energy in proportion to the square of its turn, which is the small-strain behaviour of a uniform strip and a poor description of a hinge bent nearly shut, where the material may yield, buckle or begin to touch itself.

The hinges are identical and rest flat. A hinge that rests part-folded shifts the zero of its energy, and hinges that rest at different angles no longer act as one spring — which is a subject in its own right, because a set of springs that disagree about where rest is can hold a structure in more than one place.

The panels are rigid and the stiffness goes as one over the hinge’s arc. That is what makes the optimum’s spring go as the inverse square of the radius. A hinge whose thickness changed with its width would change the exponent.

And nothing else stores energy. No panel bends, no contact pushes, no fluid resists. Each of those adds a term, and none of them removes the one computed here.

How the numbers were checked #

The force is checked against the energy it is the slope of. A central difference of the stored energy along the span agrees with the closed form to a part in a hundred billion, so the force is not a formula written down beside the energy but its derivative.

The span is summed panel by panel rather than written as $S\sin\theta$, at every fold count drawn, and agrees to a part in a billion.

The product is required to be constant across the fold counts to a part in a trillion, and $\rho^2$ times the force at each material’s own optimum is required to be constant across the radii to the same precision. A slip in either exponent would break one of the two by a large margin at the counts and radii drawn.

Still open: whether a structure sits where its bud and its hinge meet #

The feasibility condition above is a prediction that could fail, which is rare in this subject, and it says what would test it.

For a structure held part-open in a container, the product of the container’s depth and the force holding it should be at least twice the hinge’s stiffness times its turn, and the fold count should not matter to whether that holds. A lineage that was close to the edge of the condition would be one whose buds are tight and whose hinges are soft together; a lineage far from it would have slack in one or the other. Measuring that needs a hinge stiffness and a bud force, neither of which can be computed from the geometry alone.

The direction the computation can take is the one the model’s own assumptions point at. Hinges that do not rest flat, or that rest at different angles, are springs that disagree, and a set of springs that disagree can have more than one position in which nothing pushes. Whether a corrugation can hold two such positions, and what it would take for a folded structure to stay both open and shut without anything holding it, is a question about the shape of the stored energy rather than its size, and it is the one the insect wing asks.

The habit worth carrying is the one the product makes explicit. When a single count makes one quantity smaller and another larger, multiply them before calling either a gain. If the count cancels, the count was never the lever; it was only choosing where on a fixed curve to stand.