Springs that disagree do not offer a choice

The fold count sets the spring ends by naming an assumption and guessing what happens without it. Its hinges rest flat and are identical; a hinge that rests part-folded shifts the zero of its energy, and — the guess — hinges that rest at different angles no longer act as one spring, because a set of springs that disagree about where rest is can hold a structure in more than one place.

That is a reasonable thing to expect and it is wrong. The disagreement produces an average, not an argument, and it does so for a reason that has nothing to do with hinges. A corrugation does have two places to rest. They come from somewhere the guess did not look, and the thing standing between them turns out to be the flat sheet.

What a remembered crease does to the energy #

Paper remembers what it has been folded to is the observation this rests on, and it is the ordinary property of a creased sheet rather than an exotic one: a crease relaxes toward the angle it was made at rather than toward straight.

Give each hinge a rest angle of its own. Hinge $i$ stores $c_i(\psi_i - \psi_{r,i})^2/2$, where $\psi_i$ is its turn and $\psi_{r,i}$ the turn it remembers. The corrugation has one freedom, so every hinge turns by the same amount up to its sense: with $s_i = \pm 1$ recording whether hinge $i$ is a mountain or a valley, $\psi_i = s_i\varphi$ for the single number $\varphi = \pi - 2\theta$. Writing $a_i = s_i\psi_{r,i}$ for what each hinge remembers, read in the sheet’s own sense, the whole energy is

$$E(\varphi) = \sum_i \frac{c_i(\varphi - a_i)^2}{2}$$

and that is the entire model. It has one variable in it, because the pattern has one freedom, and the rest of this essay is what follows from an expression of that shape.

Disagreement averages #

Expand the sum and the $\varphi^2$ terms collect, the $\varphi$ terms collect, and the constants collect. A sum of parabolas in one variable is a parabola. It has one stationary point, it is a minimum, and it sits at

$$\varphi^* = \frac{\sum_i c_i a_i}{\sum_i c_i}$$

the stiffness-weighted mean of what the hinges remember. There is no second minimum available at any disagreement, however wide, because a quadratic has no room for one.

Four hinges remembering 0.4, 0.8, 1.6 and 2.4 radians settle the sheet at 1.300 radians, which is not one of the four and is not near any of them. Every hinge is wrong about where the sheet should be, and every hinge is pulling; the pulls add, and the sheet sits where they cancel. A hinge creased sharply and a hinge barely creased do not fight for two outcomes. They negotiate one.

This is the failure mode of an argument by analogy, and it is worth naming because it is a good analogy. Springs that disagree do give multiple equilibria in plenty of systems — a buckled strut, a magnet in two fields, a snap fastener. What those have and a corrugation does not is more than one freedom, or an energy that is not quadratic in the freedom they have. The corrugation has one angle and pays quadratically for departing from each rest. Both halves of that are what kills the guess, and either one alone would not.

The second place is the mirror #

So there is exactly one resting place — on this branch. The branch is the thing the guess did not look at.

Reversing every crease of the corrugation gives a different sheet with the same panels: every mountain becomes a valley, every $s_i$ changes sign, every $a_i$ becomes $-a_i$, and the resting place moves to $-\varphi^*$. That is a second state the sheet can occupy, it is as stable as the first, and it is reached from the first only by passing through $\varphi = 0$ — the flat sheet, the one configuration the two branches share.

Laid on a single signed axis the two branches make a double well. The barrier is the energy at $\varphi = 0$, which for a sheet creased uniformly to $\psi_r$ is

$$E(0) = \frac{C\psi_r^2}{2}, \qquad C = \sum_i c_i$$

and that is not merely the height of the barrier but the whole of it. Flattening the sheet is the transition. There is nothing else in the way and nothing further to pay: a corrugation that has been pressed flat is at the top, and which way it falls off is a free choice.

Eight folds creased to 1.6 radians, on a sheet ten long with hinges of radius 0.05, store 179 at flat. That is the cost of changing the sheet’s mind, and it is the same number as the cost of flattening it.

Where a creased corrugation sits when nothing holds it #

The resting place has an immediate consequence that the flat-resting model cannot produce at all: a creased corrugation left entirely alone is part open.

Creased to 0.4 radians the sheet rests at 98.0 per cent of its span in a band 0.25 deep. Creased to 0.8, 92.1 per cent and 0.49. Creased to 1.6, 69.7 per cent and 0.90. Creased to 2.4, 36.2 per cent and 1.17. Nothing is holding any of these; the sheet is simply where its creases want it.

How far open is a question about the grip found every partly open state to be the equilibrium of some restraint, so that how far out a corrugation sits reads off its container. This adds one state to that picture that needs no container at all, and it is the state the container was probably built around. A bud holding a leaf at the angle the leaf was creased to is doing no work. A bud holding it anywhere else is.

The container’s job changes sign #

That single free state divides the motion in two, and either side of it the container is doing opposite things.

Inside the remembered angle — the sheet held further shut than its creases want — the container pushes in, as in the flat-resting model. Outside it, the sign reverses: the container has to pull the sheet open against creases that want to close it. A wing case that stows a wing tighter than its own memory and a latch that holds a wing flatter than its memory are mechanically opposite devices, and the drawing that tells them apart is one crossing of an axis.

The flat-resting model has no such crossing, because its rest angle is at the end of the motion rather than inside it, and so it makes every container a restraint and never a spreader. That is a qualitative difference rather than a numerical one, which is the kind of difference an idealisation is most likely to hide.

The last of the flattening cannot be bought #

The far end of that curve is not merely large. It is unbounded, and this is where the model finally agrees with everybody’s hands.

Near flat, write $\varepsilon = \pi/2 - \theta$. The turn is $\varphi = 2\varepsilon$ and the cosine of the angle is $\sin\varepsilon \approx \varepsilon$, so the force is

$$F \approx \frac{2C(2\varepsilon - \psi_r)}{S\varepsilon} \longrightarrow -\frac{2C\psi_r}{S\varepsilon}$$

which grows without limit as the sheet approaches flat. The numerator does not vanish, because the creases still want their angle, and the denominator does, because there is no leverage left. And the span is $S\cos\varepsilon$, so the angle left is $\varepsilon \approx \sqrt{2(1 - s/S)}$: the pull goes as one over the square root of the span still to come.

Reaching 90 per cent of the span takes 44.9; 99 per cent takes 262; 99.9 per cent takes 947; 99.99 per cent takes 3,115. The ratios between consecutive rows are 5.83, then 3.62, then 3.29, settling toward $\sqrt{10} = 3.162$ as the approximation takes hold. Flat is never reached.

Anybody who has tried to press a folded sheet of paper back into a flat sheet knows this and knows it as the last part being the hard part. How far open is a question about the grip says the opposite — its restraint falls all the way to flat, ending at the finite value $4kc/S$ — and the disagreement is now located exactly. It is the rest angle and nothing else. A lamina that grew flat and was folded afterwards behaves as that essay says; a sheet that was creased behaves as this one does; and the two are the same equation with one term moved.

What it costs to change the sheet’s mind #

The barrier is worth one more look, because its dependence is the simplest in the whole subject and it says something about commitment.

Four folds cost 89.7, eight cost 179, sixteen cost 359 — 22.4 a hinge on every row, exactly. Flattening turns every hinge through its own remembered angle and no hinge turns further than another, so the whole cost is one hinge’s times the count, with none of the sublinearity that the propped state introduced into the stored energy. The propping matters at the shut end of the motion and flat is the other end.

So a finer corrugation is not more committed per crease. It is more committed because it has more creases — which means a sheet’s reluctance to be reversed is a property of its pattern’s size rather than of its sharpness, at a fixed crease angle. Four materials, four optima has each material folding best at a count inversely proportional to its hinge radius, and putting the two together: a material with a narrow hinge folds at a high count, so its best corrugation is also the one most committed to the sense it was folded in.

What it does to the convergence argument #

The four structures that arrive at this corrugation are not in the same position with respect to any of it, and the division is the one just drawn rather than the one their biology suggests.

The same corrugation in four places gathers a leaf, a wing, a crushed cylinder and a solar array. A leaf grows flat and is folded by confinement; its hinges have no memory, so its container is always a restraint and flat is always cheap. A wing is folded and refolded over a lifetime and its hinges are a protein chosen for elasticity, which is a memory deliberately kept small. A crushed cylinder is a sheet whose creases were made by the crushing, so its memory is total and its rest state is somewhere in the middle of its own motion. And an array’s hinges are whatever a designer specified, which is to say the memory is a free parameter and is usually set to zero on purpose.

So the convergence on the pattern is real and the convergence on the mechanics is not. Four finders, one option argues the four had little to choose between when they chose the geometry; this says that having chosen it, two of them got a structure with one resting place and two got a structure with two, and no examination of the crease pattern distinguishes them. The census returns one closed the geometric question by finding a single pattern reaching a stated packing ratio. The mechanical question does not close the same way, because the quantity that decides it is not in the pattern.

It also qualifies what a container is for. The bud chooses the pattern has the container selecting among geometries the mathematics leaves open; for a sheet with memory the container has a second job, which is to set the memory, since a hinge relaxes toward wherever it is held. A bud is then not only choosing a pattern but writing a rest angle into it, and the structure that emerges is one the bud has partly designed.

The sheet that remembers nothing #

The contrast is worth drawing at the same scale, because the flat-resting case is not a straw man. It is what a growing lamina is.

A leaf lamina grows flat and is folded by being confined; its hinges are tissue that has never been creased, and the flat-resting model is the right one for it. A wing that has been stowed for weeks, a paper model, and an array that has sat folded in a fairing through a launch campaign are all on the other side of the line. The same pattern, the same panels and the same stiffness give two qualitatively different objects depending on a history the geometry cannot see.

That is a real difficulty for reading a structure from a photograph. Nothing in a body folds on a line gives a hinge its width from the material, and the crease has a radius gives the same quantity to a sheet of paper; the rest angle needs the history as well, and a hinge that has been held bent for a season has a rest angle somewhere between the two models — creep moves it, and moves it toward wherever the structure has been sitting.

What the wells cannot show #

The figures draw an energy and not a measurement, and three things about real hinges are outside them.

They cannot show creep. A plant hinge held bent for a week relaxes toward the angle it is held at, so its rest angle is not a constant of the material but a slow function of where the structure has been. The double well is drawn for a fixed memory; a real one drifts, and it drifts toward removing whichever barrier the structure has been sitting on one side of.

They cannot show what happens at the top. The barrier is computed as the energy of the flat sheet within the same quadratic model, and a sheet actually being pressed flat is a sheet whose creases are being locally unbent through a large angle — where the material may yield rather than store, which would lower the barrier without changing anything else in the picture.

And they cannot show whether the mirror pattern is reachable at all. Nothing here says a sheet can be turned inside out without tearing; it says only that if it can, the flat sheet is what stands in the way, and how much that costs. For a lamina attached along a midrib, or a wing hinged to a thorax, the mirror may be geometrically impossible for reasons that have nothing to do with energy.

The memory the model assumes #

A hinge’s energy is quadratic about its own rest angle. That makes each hinge a linear spring with a shifted zero, and it is what makes the total a parabola. A hinge with any other law could produce a second minimum on one branch, and the argument above would not apply to it.

The corrugation has exactly one freedom. Equal panels turning through one angle is what collapses the sum to a single variable. A pattern whose panels differ in length still has one freedom; a pattern with more than one does not, and multiple freedoms are the ordinary source of multiple equilibria.

Every hinge is creased to the same angle wherever a single memory is quoted. Where they differ, the weighted mean replaces it and every statement about the resting place survives with $\psi_r$ read as that mean.

And the panels neither bend nor touch except where the hinge props them. The propped state is included; contact anywhere else is not, and a stack pressing on itself near flat would add exactly the kind of steeply rising term the divergence is about.

How the numbers were checked #

The single minimum is found rather than assumed. The sum of wells is scanned at 601 states across the sheet’s whole range and its turning points counted; there is one, and it is required to agree with the weighted mean to within the scan’s own resolution. An error in the algebra would show as either a second turning point or a minimum in the wrong place.

The barrier is required to equal the flat state’s energy to within a part in a trillion, which is the claim that flattening is the whole transition rather than a step in it.

The divergence is checked as a rate, not as a size. The ratios between consecutive rows of the flattening table are required to fall, and the last of them to sit within five per cent of $\sqrt{10}$ — so the square-root law is verified where it applies and claimed nowhere else.

And the sign change is checked on every remembered curve: each must end negative at the flat end while the flat-resting curve ends positive. A model that had lost the rest angle somewhere would fail that immediately.

Still open: a hinge whose rest angle is moving #

The model’s own weakest assumption is a constant, and it is the one a season removes.

A hinge held bent relaxes toward where it is held. So the rest angle is not a parameter but a slow variable, driven by the structure’s history — and a structure sitting in a bud for a season has a rest angle migrating toward the bud’s own angle, which raises the free resting span toward where the container was holding it and lowers the force the container has to supply. A leaf that has been packed long enough is holding itself.

That is a computation with a definite shape: a fast elastic variable and a slow rest angle, coupled, with the slow one relaxing toward the fast one. It would say how long a structure has to be held before its container can be dispensed with, in units of the material’s own relaxation time, and it would say whether the barrier to the mirror grows or shrinks while that happens. Neither is guessable from here, and the second decides whether a long-stowed structure becomes more committed to its fold or less.

Sideways from here, the double well is the shape a self-folding sheet needs and cannot easily get. Paper that folds itself finds the hard half of that problem to be choosing which way an actuated crease goes, because the two outcomes are equally downhill; a remembered crease is precisely a bias between two wells, and the barrier computed here is what an actuator would have to beat. Whether a pattern can be creased so that its two branches differ enough to decide the outcome, without being creased so hard that it will not flatten, is the design question those two facts make possible.

The habit worth carrying is about where multiple outcomes come from. Count the freedoms before predicting a choice. Disagreeing preferences within a single freedom always average; a second outcome needs either another freedom or a term that is not quadratic, and looking for it in the disagreement is looking in the one place it cannot be.