Holding a fold moves the force

Springs that disagree do not offer a choice gave the hinges of a corrugation memories. Each hinge stores energy as a spring about the turn it remembers, the corrugation has one freedom, and so the energy is a sum of parabolas with a single minimum at the stiffness-weighted mean of what the hinges remember — while the mirror pattern, every crease reversed, is a second well with the flat sheet as the barrier between them.

It named its own weakest assumption. The remembered turn is treated as a constant, and a season removes it. A hinge held bent relaxes toward where it is held, so a leaf in a bud for months, or a wing folded under a case, has a remembered angle that migrates toward the angle of its packing. That is a computation with a definite shape — a fast elastic variable and a slow rest angle, coupled — and it says two things nobody had derived: how long a structure has to be held before its container can be dispensed with, and whether the barrier to the mirror state grows or shrinks while that happens.

Both answers are short. The more interesting result is a third one, about where the force goes.

A rest angle that moves #

Write $\varphi$ for the corrugation’s turn and $\varphi^*$ for the turn its hinges remember. The energy is $C(\varphi - \varphi^*)^2/2$, as before, where $C$ is the stiffness of all the hinges together. Now let the remembered turn creep toward the actual one at a rate proportional to the gap:

$$\frac{d\varphi^*}{dt} = \frac{\varphi - \varphi^*}{\tau},$$

with $\tau$ the hinge material’s relaxation time. That is the simplest law in which a held hinge forgets its old rest and learns a new one, and it is the linear version of what creep in a polymer or a cell wall does.

Pack the corrugation at a turn $\varphi_h$, tighter than the $\varphi_0$ it was creased to, and hold it there. The actual turn is fixed, so the rest turn relaxes exponentially toward it:

$$\varphi^*(T) = \varphi_h + (\varphi_0 - \varphi_h)\,e^{-T/\tau}.$$

After one relaxation time the hinges have forgotten 63 per cent of the gap, after three 95 per cent, and they never forget all of it.

Where the force goes #

The container holding the corrugation at $\varphi_h$ must supply the torque the hinges exert back, $C(\varphi_h - \varphi^*)$, and that falls as the rest turn creeps up to meet it. This is the sense in which a long-packed structure holds itself.

Now release it and open it to a working turn $\varphi_o$ looser than it was creased to. The hinges resist with $C(\varphi^* - \varphi_o)$, and that has risen, because the rest turn has moved away from $\varphi_o$ toward the packed state. Add the two:

$$C(\varphi_h - \varphi^*) + C(\varphi^* - \varphi_o) = C(\varphi_h - \varphi_o),$$

which does not contain $\varphi^*$ at all. At every moment of the holding, the torque the container must still supply and the torque the opening will need sum to the same total. Holding a fold does not use up the force it takes to keep it folded; it transfers it, a little at a time, from the container to whatever has to open the structure afterwards.

For a corrugation creased to 0.8 radians, packed to 2.4 and later opened to 0.4, the total is $C \times 2.0$. At the start the container supplies 80 per cent of it and the opening would need 20. After half a relaxation time they are even. After one, the container supplies 29 per cent and the opening would need 71; after three, 4 and 96; after five, 1 and 99.

The slider changes how tightly the corrugation is packed. A tighter packing makes the container’s starting share larger and the transfer larger with it, but the shape is always the same exponential exchange, and the sum is always the whole.

How long before the container can go #

The question the relaxation was supposed to answer has a definite form now. The container can be dispensed with once its share is below whatever the structure can tolerate springing back, and that takes

$$T = \tau \ln\frac{(\varphi_h - \varphi_0)}{\varepsilon\,(\varphi_h - \varphi_o)}$$

to bring the share below $\varepsilon$. At 5 per cent that is about three relaxation times for the corrugation above, and the time is set by the material’s $\tau$ alone once the angles are fixed. A leaf that has been packed three relaxation times is a leaf that keeps its packed shape when the bud opens, which is the observation the earlier essay guessed at.

What the arithmetic adds is the price. The same leaf now needs 96 per cent of the total torque to unfold, and in a leaf that torque comes from growth or turgor. The bud’s work has been handed to the unfolding, and a plant that holds its leaves packed longer must push them open harder — or wait, since the rest angle relaxes toward the open turn once the leaf is held open, at the same rate.

The barrier grows while it is held #

The barrier between the corrugation and its mirror image is the energy of the flat sheet, $C\varphi^{*2}/2$, because the flat sheet is the only state the two branches share. As the rest turn moves from 0.8 toward 2.4, that barrier rises as the square of it: 3.19 times its creased value after half a relaxation time, 5.13 after one, 8.41 after three, 8.92 after five, approaching $(2.4/0.8)^2 = 9$.

So the second question has a clear answer. A long-packed structure becomes more committed to its fold, not less. The disturbance it would take to snap it through the flat sheet into the mirror state grows nearly ninefold over a season, which is the stability a folded wing or a stowed leaf would want while packed — and it is also the reason that such a structure, once opened, is more reluctant to be refolded the other way.

After the release, the same law runs backwards #

The exchange does not stop when the container is removed. Opened and held at the working turn $\varphi_o$, the rest turn relaxes toward $\varphi_o$ at the same rate it relaxed toward $\varphi_h$, and the torque the opening mechanism must keep supplying falls in exactly the way the container’s did:

$$\varphi^*(t) = \varphi_o + \big(\varphi^*(T) - \varphi_o\big)\,e^{-t/\tau}.$$

So the unfolding’s burden is also temporary, and a structure held open for three relaxation times comes to remember its open state and stay there unaided — which is, for a leaf, the difference between a blade held flat by turgor and one that has set flat.

The two stages together have a simple accounting. Every relaxation time spent packed moves the same share of the total torque from the container to the opening; every relaxation time spent open moves it from the opening mechanism back to nothing, as the rest turn settles where the structure now is. A structure that spends as long open as it did packed ends with its hinges remembering the open state as firmly as they remembered the packed one, and the history of its folds is written into its rest angles as a weighted average of where it has been, with recent positions weighted most.

That is the property the sheet remembers observed in paper as a static fact — a creased sheet relaxes toward its crease rather than toward flat — and the relaxation law gives it a clock. Paper that has been folded and unfolded remembers the fold; paper that has been unfolded for long enough remembers the unfolding.

What this says about a wing that snaps #

A corrugation has one resting state and the wall is the flat sheet found that a folded wing’s hinges, modelled as springs, give a single vertex two resting states separated by the flat sheet, and that the wall’s height is each spring’s stiffness times its rest angle squared, summed. Relaxation moves every term of that sum.

A wing held folded between flights has hinges whose rest angles creep toward the folded state, so the wall it must cross to open — the flat sheet’s energy — grows while it is stowed, exactly as the corrugation’s barrier does. A wing that has been folded a long time should be harder to snap open than one folded briefly, by the square of how far its hinges’ memories have moved. Conversely a wing flown for a long time relaxes toward open, lowers the wall on the folded side, and folds more easily.

Whether insects exploit that is a question about insects; resilin, the rubbery protein at many insect wing joints, is known for returning almost all the energy put into it, and a material that loses little to internal friction is usually one that also creeps little — which in this arithmetic would make the wall’s height nearly independent of how long the wing has been at rest. How slowly resilin’s rest angle actually moves under a sustained fold is a measurement this account does not have. A hinge that does not forget gives a mechanism whose effort does not depend on its history, and for a structure opened and closed many times a day that may matter more than any other property of the joint.

Hinges that relax at different rates #

A real fold is not made of one material. A leaf’s hinge tissue, a wing’s resilin joints and its stiffer veins relax at different rates, and a corrugation whose hinges differ remembers their stiffness-weighted mean — which is now a sum of exponentials rather than one.

Give half the stiffness a relaxation time of one and half a time of six. The container’s load halves in 1.51 time units, which a single relaxation time of 2.17 would also do. But after fifteen time units the mixed corrugation still needs 3.3 per cent of the total from its container, where the single time would leave 0.08 per cent — forty times as much. The fast hinges let go early and the slow ones keep pulling.

The reason is visible in the sum. Early in the holding the fast population dominates the change, because its exponential is steep; late in the holding it has finished and only the slow one is still moving, with half the stiffness and a sixth of the rate. A mixture of relaxation times has a half-life set by its fastest members and a tail set by its slowest, and no single number can report both.

That matters for reading a packed structure. A measured half-life of the holding force says almost nothing about how long it takes to release completely, because the tail is set by the slowest population of hinges, and a structure that seems to have taken its packed shape in days can still be springing back, a few per cent at a time, for months.

What it adds to the convergence argument #

This line of essays began with a leaf, a wing, a crushed cylinder and a solar array arriving at the same corrugation, and four materials, four optima separated them again by hinge radius. The fold count sets the spring showed that the fold count only chooses where on a fixed curve of force against clearance a structure sits, and how far open is a question about the grip that every held state lies in a narrow band of grips.

Relaxation adds time to that picture, and it separates the four finders. A solar array’s hinges are metal or composite and relax so slowly that its rest angle is effectively fixed; its container, the launch restraint, supplies the whole holding force for as long as it is stowed, and the deployment mechanism never inherits any of it. A leaf’s hinges relax in days to weeks, a beetle’s resilin joints in hours; for them the packing period is several relaxation times, the bud or the elytra hand most of the holding force to the unfolding, and the unfolding is where the organism must spend. The same corrugation, packed for the same season, is a different energy budget depending on whether its hinges forget.

What the model assumes #

Relaxation is linear. The rest turn creeps at a rate proportional to the gap, with one time constant. Real creep depends on how large the gap is and slows as a material hardens, and a biological hinge can grow into its new shape, which is relaxation with no return.

The packed turn is held exactly. A container that yields a little as the torque falls lets the corrugation creep further than the law assumes, and hands over even more of the force.

The hinges are springs about their rest turns and nothing else. Friction between layers of a packed structure, contact with the container and the surface tension of a wet leaf all hold a packing too, and none of them relax in the same way.

The stiffness does not relax, only the rest turn. A material under sustained strain can soften as well as creep, and a hinge whose stiffness fell while it was held would lower both the container’s torque and the opening’s, breaking the conserved sum in the direction of less work at both ends.

And the opening is to a fixed turn. A structure opened by growth opens to wherever its growth reaches, and the torque it needs is a force it may or may not have.

What the curves cannot show #

They do not treat the packed turn as a choice. Every figure packs to one turn and opens to another, fixed in advance. A structure free to choose how tightly to pack — a bud that could be looser — trades the barrier it gains while packed against the torque its unfolding needs, and the best packing turn for a given season is a small optimisation this account sets up and does not run.

They do not give a relaxation time. Every time here is in units of $\tau$, and $\tau$ for a leaf hinge, a resilin joint or a creased polymer is a material measurement no geometry supplies.

They do not include recovery after release. Once opened and held open, the rest turn relaxes back toward the open turn at the same rate, and a structure opened and repacked repeatedly has a rest turn that wanders with its history; the model can follow that, and the figures do not.

And they do not say what an organism does about it. Whether plants time their leaves’ packing against their hinges’ relaxation, or whether the unfolding torque a long packing demands is a real constraint on when leaves emerge, is a question for measurements of plants.

Still open: a hinge that relaxes toward the other branch #

The relaxation above keeps the corrugation on its own side of the flat sheet: packed tighter than it remembers, it learns to remember tighter. A structure held on the other side of the flat sheet — pressed into its mirror state — relaxes toward the mirror. Its rest turn crosses zero, the flat sheet stops being a barrier between two wells of different depth and becomes the rim of a single well that has moved, and there is a moment during the holding when the structure has no preferred branch at all.

That moment is computable from the same law and it is exactly the event paper that folds itself needs and cannot easily get: a pattern whose choice of branch is erased by holding and then set by releasing. Whether a self-folding sheet could be programmed by holding it — creased one way, held the other until its memory crosses, then released — is a question with a definite threshold in relaxation times.

Sideways from here, the conserved total suggests a rule for anything packed. Before crediting a packing with holding itself, ask who pays to unpack it. When the rest state moves toward the packed state, the force a container gives up does not disappear; it moves to the far end of the story, where the structure has to be opened, and a design that ignores the far end has simply moved its bill.