A corrugation has one resting state

The number is the angle ends with a remark that changes what a fold angle is. A wing stowed short of shut is a structure whose hinges are bent, and bent hinges store energy — so the angle is not only a limit on packing but the mechanism of deployment. A wing closed all the way would have nothing to open it.

Follow that one step further and a harder question appears. An insect needs its wing to stay open in flight and to stay folded at rest, and at the base of the wing, where the muscles are, there is nothing that can hold the folded part in either place. If the hinges alone are to do it, the energy they store has to have two configurations in which nothing pushes: one open, one folded, with a barrier between them.

Whether a folded structure can have two such states is a question about the shape of its stored energy rather than the size of it. And the answer turns out to depend on something the packing ratios never distinguished: whether the creases meet at vertices.

Springs that want different angles #

Give every crease a spring. A spring on crease $i$ wants some fold angle $r_i$ — its rest angle — and stores energy in proportion to the square of how far the crease is from it. The energy of the whole structure in a configuration is the sum over its creases, each weighted by its stiffness:

$$E = \sum_i c_i\,(\rho_i - r_i)^2$$

where $\rho_i$ is crease $i$’s actual fold angle. A resting state is a configuration from which every possible motion raises the energy: a bottom of the energy over the set of configurations the structure can actually reach.

If all the springs want angles belonging to one configuration, that configuration has zero energy and is a resting state. The interesting case is springs that disagree — rest angles that no single configuration satisfies at once — because a structure whose springs agree rests in exactly the state they agree on and nowhere else.

The question is therefore: when springs disagree, how many bottoms can the energy have? And that depends entirely on the set of configurations the energy is being minimised over.

A corrugation’s energy is a parabola #

A corrugation is a pattern of parallel creases, and with one degree of freedom — the property the no-motor argument turned on — every crease folds by the same angle, mountains one way and valleys the other. So each crease’s fold angle is $\pm\rho$ for a single number $\rho$, and the energy is

$$E(\rho) = \sum_i c_i\,(s_i\rho - r_i)^2, \qquad s_i = \pm 1$$

which is a sum of squares of linear functions of one variable. A sum of upward parabolas is an upward parabola. It has exactly one bottom, at

$$\rho^* = \frac{\sum_i c_i s_i r_i}{\sum_i c_i}$$

the stiffness-weighted mean of what the springs want, signed by which way each crease folds. However the springs disagree, the corrugation compromises between them at one angle and rests there, and every other angle costs more.

The figure’s springs make the compromise concrete. They want the angles −2.87, 2.20, −2.87 and 2.20 radians — the first and third taken from a folded state on one branch of a vertex and the second and fourth from a folded state on the other. On a corrugation the second and fourth creases fold the opposite way to the first and third, so their springs read as wanting −2.20, and the one angle the corrugation can take that pleases all four as well as it can is their mean, −2.535. At that angle every spring is a third of a radian from what it wants, the energy is small, and there is nowhere else to go.

Unequal stiffnesses do not rescue a second state either. Making one spring ten times stiffer moves the compromise toward what that spring wants and leaves the energy a parabola, because the weights only change the coefficients of the square. Nor does adding springs, or setting them to want angles outside the range the corrugation can reach. Every linear spring on a one-angle structure adds a parabola, and parabolas add to a parabola.

The census makes that an observation rather than a derivation. Two hundred sets of rest angles are drawn at random, each spring wanting anything between nearly shut one way and nearly shut the other. The corrugation’s energy is sampled across every angle it can take, and the number of bottoms is counted. It is one, two hundred times.

So a corrugation cannot hold a wing both open and shut on its own springs. Disagreeing springs do not produce a second state; they produce a compromise. A structure that snaps between two positions needs the set of positions it can take to be something other than a single line.

A vertex has two branches #

A degree-four vertex is that something. What the vertex does on the way solves its motion: fix one crease’s fold angle and the other three follow — one degree of freedom, as for the corrugation — but the other three follow along one of two branches. The branches cross at the flat state, where every fold angle is zero, and leave it in different directions: on one the sheet folds with one crease the odd one out, on the other with a different crease odd.

So the configurations of a vertex are not a line but two curves meeting at a point. And along each curve the fold angles are not proportional to one another — they are geared, with the tangents of the half-angles in a fixed ratio, which is a law that bends.

Put the same springs on it. The energy is still a sum of squares, but a sum of squares of curved functions of the driving angle, evaluated along two curves. There is no reason for that to have one bottom, and in general it does not. The springs can make a bottom on each branch, because each branch reaches configurations the other does not.

The flat state needs care, and the counting gives it care. It lies on both branches at once, so it is a resting state only if the energy rises along all four directions leading out of it — both ways along both branches. A test along one branch alone would call it a bottom whenever that branch climbed away from it, even while the other branch ran downhill.

Two resting states by default #

With that handled, the census counts bottoms across both branches for the same two hundred spring settings.

For the vertex with sectors of 60°, 90°, 120° and 90°: two resting states on 198 of the settings, one on the other two. For 45°, 100°, 135° and 80°: two on 196, one on four. For 80°, 95°, 100° and 85° — a vertex close to a square grid, whose two branches are nearly alike — two on 126, three on 62, four on 8, and one on only 4.

A sprung degree-four vertex has two resting states unless its springs are set specially. One resting state is the exception, and it happens when the springs agree about a configuration or are arranged so that one branch has nowhere to settle.

The contrast with a corrugation is the whole result. The springs are the same springs; the rest angles are the same angles; the count of freedoms is the same single freedom. What differs is the shape of the set of configurations, and a set with two branches supplies a second bottom that a line cannot.

The curse of a self-folding sheet is the gift of a wing #

This result has already appeared once, from the opposite side, and the two readings are worth putting together.

Paper that folds itself found that a self-folding sheet has to supply the fold and then choose what to fold into, because a degree-four vertex driven from flat can go down either branch and both are equally downhill. The two branches were the difficulty: no amount of torque decides between them, and a sheet that is meant to become one model may become another.

A wing wants exactly that ambiguity. Two branches, each able to hold a resting state, are two configurations a structure can occupy without an actuator at the fold. The property that makes a self-folding sheet unreliable is the property that lets a wing lock open and lock shut for free. Whether a vertex’s two branches are a defect or a mechanism depends only on whether two outcomes are wanted.

That is the connection this subject is best at supplying. Earwig hindwings are reported to fold and lock both open and closed without muscular action at the fold, and to do it with a fan of creases meeting at vertices rather than with parallel pleats (Faber, Arrieta and Studart, 2018). Nothing here measures an earwig. What the census says is that the geometry they use is the kind that makes two resting states ordinary, and the geometry they do not use cannot make two at all.

One freedom is not one outcome #

No motor in the fold established that a pattern with one degree of freedom can be deployed by one pull, and it counted a vertex and a corrugation as the same kind of object: one freedom, one driver. That count is still right. The census shows that it is not the whole description.

A corrugation has one freedom and its configurations are one line, so one freedom means one path and, with springs, one place to rest. A vertex has one freedom and its configurations are two lines through a point, so one freedom means one path per branch and, with springs, a place to rest on each. The number of freedoms is local — how many numbers it takes to say where the structure is near a configuration — and the number of resting states is global, a property of the whole set of configurations.

That is a distinction the Miura’s own two ways of folding turned on as well. A mechanism can be completely determined by one number near any configuration and still have more than one configuration for a given value of that number, and everything interesting about stability lives in that difference.

Which patterns can lock, read off the census #

The census turns into a prediction about patterns once it is read the right way round, and the prediction is testable on anything folded.

A pattern whose creases are all parallel is a corrugation in the sense used here, whatever its spacing: a leaf’s tapered corrugation, a fan of creases that never cross, a concertina. However its hinges are made, a structure of that kind has one resting state, and if it is found holding two it is holding the second by something other than its crease springs — friction, a latch, a panel that bends, contact between layers.

A pattern with interior vertices is not bound by that. Each degree-four vertex brings two branches, and springs that disagree across them make a second resting state the ordinary case. The Miura folds two ways for the same reason, and a Miura with hinge springs is therefore a candidate for a structure that stays put in more than one shape, where a pleated strip of the same paper is not.

That is a sharper statement than a count of freedoms makes, and it points at where to look. A folding structure observed to lock in two positions with nothing but its hinges has vertices, or it has hinges that are themselves switches. The first is geometry and the census covers it; the second is material, and none of the models used here covers it.

The energies are worth a sentence because they are not small differences. With the mixed springs, the corrugation’s single compromise holds an energy of 0.45 in units of one spring’s stiffness per radian squared. The vertex’s two resting states hold 12.4 and 17.6: the vertex cannot please its springs nearly as well as the corrugation can, because its angles are geared and cannot be adjusted one at a time. A vertex buys its second resting state by being worse at the first, which is a trade worth knowing before calling either arrangement better.

What the energy curves cannot show #

The curves show where a sprung vertex can rest and not whether it would stay there.

A resting state is a bottom of the energy, and a shallow bottom next to a low barrier is a state a gust empties. The figures mark bottoms and draw the energy around them, but they do not compare barrier heights to anything a wing experiences — air loads, the insect’s own motion, the weight of the wing — because none of those is in the model. A wing whose second state was a dimple a millionth of a joule deep would pass the census and fail in flight.

Nor can they show a wing. A real folding pattern has many vertices sharing creases, and the resting states of a sheet are not the resting states of its vertices counted independently: a crease shared by two vertices carries one fold angle, which ties their branches together. The count for a whole sheet could be as small as two or as large as two to the number of vertices, and it is not computed here.

What the figures establish firmly is the negative half. No corrugation, with any linear springs at its creases, rests in more than one place. That needs no further model to be true, because a sum of squares of one variable cannot have two bottoms.

The springs the census assumes #

Each crease is a linear torsional spring, storing energy in proportion to the square of its departure from a rest angle. Real hinges are stiffer near their limits and may be bistable themselves; a hinge that is its own two-state switch makes the whole question different.

The panels are rigid and meet only at the vertex. Nothing collides, nothing bends between creases, and the vertex follows the spherical linkage exactly.

The springs are equally stiff and their rest angles are drawn uniformly from nearly shut one way to nearly shut the other. Unequal stiffnesses move the bottoms and can merge or split them; the census reports what happens for one particular spread of rest angles and makes no claim about the rest angles of any organism.

The corrugation’s creases fold by one shared angle. That is the definition of the corrugation compared here, and a corrugation whose panels could bend would have more freedoms and more configurations.

How the bottoms were counted #

Both branches are sampled across the whole motion at 721 values of the driving angle, from folded shut one way to folded shut the other, and a bottom is a sample lower than both its neighbours.

The flat state is tested on all four sides, both directions along both branches, because it belongs to both branches and a one-branch test would report a false bottom there whenever the other branch ran downhill.

The corrugation’s single bottom is checked against its closed form, the stiffness-weighted mean of the signed rest angles, and has to agree to within one sample spacing. It does on every one of the two hundred settings, and on none of them does the corrugation show a second bottom.

The controls are drawn beside the finding. A vertex whose springs agree on one folded state rests there alone, so the second state in the disagreeing case is attributable to the disagreement across the branches.

Still open: how deep the second resting state is #

A count of bottoms says a vertex can hold two configurations and says nothing about how firmly. The natural next measurement is the barrier: the smallest energy the structure has to be given to leave one resting state for the other, which is the height of the lowest path between them along the branches through the flat state. A wing that locks usefully needs that barrier large against whatever disturbs it and small against whatever is meant to switch it, and the ratio of the two is a design quantity with a definite value for every spring setting.

The second direction is the sheet. A fan of vertices sharing creases ties its vertices’ branches together, and whether that multiplies the resting states or collapses them is a computation on the quadrilateral meshes already folded crease by crease. It would say whether a pattern with many vertices is a switch with many positions or with exactly two.

The habit worth carrying is a caution about counts of freedom. One freedom fixes how a structure moves near where it is, and says nothing about how many places it can be. When stability is the question, count the branches.