The wall is the flat sheet

A corrugation has one resting state counts the configurations a sprung fold can rest in, and finds the difference a vertex makes. Give every crease a spring that wants some fold angle. A corrugation, whose creases all fold by one angle, has energy that is a parabola in that angle and exactly one place to rest, however the springs disagree. A single degree-four vertex has two branches to its motion, and the same springs give it two resting states on almost every setting — which is what a wing that locks both open and shut without anything holding it would need.

A count of resting states says nothing about how firmly each is held. A state held by a dimple a millionth of a joule deep passes that count and empties in the first gust. So the question left over is the barrier: how much energy the vertex has to be given to leave one resting state for the other.

That question turns out to have an answer that does not depend on the vertex. The road from one state to the other has to cross one particular configuration, the barrier is that configuration’s energy, and that energy is fixed by the springs before any sector angle is chosen.

One road between the branches #

What the vertex does on the way solves a degree-four vertex’s motion. Fix the fold angle of one crease and the other three follow, but they can follow in two ways: the vertex has two branches, and the configurations it can take are two curves rather than one, meeting as two mechanisms at one point do. On one branch the sheet folds with one crease the odd one out, on the other with a different crease odd, and the fold angles along each are geared by a fixed law.

The two curves meet at the flat sheet, where every fold angle is zero and both branches pass. They also meet at the two fully folded states, but only in the sense of having the same positions: arriving there on one branch and leaving on the other would turn two creases from fully mountain to fully valley, which paper does only by passing through itself. So for a sheet that cannot, the flat sheet is the one junction. A vertex resting on branch one that is to end up resting on branch two has no choice about its route: back along branch one to flat, then out along branch two.

That is a statement about the shape of the set of configurations, and it holds before any spring is chosen. The first figure adds the springs. With four springs whose rest angles are borrowed from a folded state on each branch — the setting the corrugation comparison used — the vertex rests at an energy of 12.40 on branch one and 17.65 on branch two, and the road between them rises from each to a single summit in the middle. The summit is the flat sheet, at 26.16.

The pass is the highest point, nearly always #

A single road through a single point does not yet make that point the barrier. The road could climb over a hump on one of the branches before it reaches flat, and then the hump would be the thing a switch has to get over.

The second figure asks how often that happens, over the same two hundred random spring settings the earlier count used. On the vertex with sectors of 60°, 90°, 120° and 90°, 198 settings give two resting states or more; for every one of them the road between the two deepest crosses the flat sheet, and for every one the flat sheet is its highest point. On the vertex with sectors of 45°, 100°, 135° and 80° the same holds on all 196. On the nearly square vertex, 80°, 95°, 100° and 85°, it holds on 177 of 191.

So for a sprung vertex the barrier between its two resting states is the flat sheet, with a small family of exceptions on vertices whose branches are nearly alike, which come back below. The switch a wing would make between locked open and locked shut is, in this model, a passage through the fully flat configuration of that vertex, and the cost of the passage is the height of the flat sheet above whichever state it starts from.

A wall the geometry cannot move #

The height of the flat sheet is the part worth dwelling on, because it can be written down.

At the flat sheet every fold angle is zero. A spring on crease $i$, stiffness $c_i$, rest angle $r_i$, stores $c_i(\rho_i - r_i)^2$, and with $\rho_i = 0$ that is $c_i r_i^2$. So the energy of the flat sheet is

$$E_{\text{flat}} = \sum_i c_i\, r_i^2$$

and nothing else. The sector angles are not in it. Neither is the gearing, nor which crease is odd on which branch, nor anything else about the vertex. The wall is set by what the springs want, and it is the same wall on every vertex those springs could be put on.

The third figure checks that directly. The same four springs, wanting $-2.87$, $2.20$, $-2.87$ and $2.20$ radians, are put on four vertices with different sector angles. The flat sheet’s energy is 26.160 on every one. The states below it are not: the deeper sits at 12.40, 13.08, 10.02 and 10.51 on the four, and the shallower at 17.65, 18.89, 16.88 and 17.60. The climbs out of them therefore vary from vertex to vertex, but only because the floors move. The ceiling does not.

This separates two design decisions that look like one. A designer who wants a folding structure to be hard to switch is choosing how far from flat its springs want to be. A designer who wants its two states held unequally, or equally, is choosing its geometry. Changing the sectors to make a stiffer lock does nothing to the wall and moves the states only a little; changing the rest angles moves the wall itself.

Why the flat sheet is never a place to rest #

The same arithmetic says something about the flat sheet that sounds like a separate fact and is not.

Four roads leave the flat sheet: both ways along branch one, and both ways along branch two. Near flat each road is a straight line in fold-angle space, and the energy along it changes at a rate set by how the springs pull on that line. Going one way along a branch reverses the rate going the other way. So whatever the springs are, each branch is downhill in one direction and uphill in the other, and the flat sheet has two downhill roads out of it and two uphill.

The fourth figure counts the roads on the census. On the first two vertices, every one of the two hundred settings has exactly two downhill roads out of the flat sheet; on the nearly square one, 197 do and three have one. None has zero.

For the flat sheet to be a resting state all four roads would have to be uphill, which needs the springs’ pull to vanish along both branches at once. That is two conditions on the rest angles, and a setting drawn at random satisfies both with probability zero. So the flat sheet is never a rest, and it is always the pass: the one configuration every switch goes through is the one configuration no sprung vertex stays in. A wing locked flat open by its hinges alone is, in this model, a wing whose springs have been tuned to a knife edge.

The second state is usually barely held #

Knowing the wall, the question the earlier count could not ask has a direct answer: how far below it does each state sit?

For the deeper of the two states the answer varies widely. For the shallower one it is usually small. The fifth figure takes each two-state setting in the census, measures the climb out of the shallower state, and expresses it as a share of the wall. In the median setting it is 4.8 per cent on the first vertex, 4.9 on the second and 5.8 on the third. A quarter of settings hold the shallower state by about one per cent of the wall; nine in ten hold it by less than a quarter.

That is the qualification the first count needed. A sprung vertex has two resting states almost always, and one of them is almost always nearly empty. The ratio of the two climbs spreads enormously: a median of about five, a quarter of settings at about two or less, a tenth over three hundred. A structure whose second state is meant to survive handling needs its springs chosen for it, not drawn.

Springs that want more make a higher wall #

If the wall is the springs’ own energy at flat, then springs that want angles further from flat make a higher wall, and the sixth figure shows how the two states share it as that happens.

The springs are taken from folded states on each branch, folded by a steadily larger amount, from 0.4 radians to 2.8. The wall rises the whole way, from 3.68 to 34.28, roughly as the square of how far the springs want to fold, which is what a sum of squared rest angles does — and how far a wing is folded is, as the number is the angle found, the quantity its packing depends on. The two states rise beneath it, and the climb out of the shallower one grows faster than the climb out of the deeper: at 0.4 radians the deeper state’s climb is ten times the shallower’s, at 2.8 only 1.16 times.

So for this kind of setting, a vertex whose springs want to be well folded holds both of its states firmly and nearly equally, and a vertex whose springs want only a little holds one state firmly and the other hardly at all. That is a sharper version of the earlier observation that a vertex buys its second resting state by being worse at the first. Here both states are worse at pleasing their springs than a corrugation would be, and what the vertex gets in return is two states under a wall the springs build.

For comparison, the seventh figure draws the energy over both branches for springs taken from states folded by 1.2 radians, the middle of the sweep. The two resting states are visible as the two low points, and the flat sheet is visible as the point where the two curves cross.

When the two states share a branch #

The exceptions on the nearly square vertex have a shape of their own, and it is worth seeing because it is the case in which geometry does set the barrier.

On a vertex whose sectors are close to a square grid, the two branches are nearly alike and each has room for more than one resting state. For nine of its two-state settings the two deepest states sit on the same branch and the same side of flat. The road between them never reaches the flat sheet; it runs along the branch from one state to the other and back down. The last figure draws one: states at 15.97 and 16.09, a hump between them at 19.36, and a flat sheet at 18.47 that the road does not visit.

There the barrier is a property of the branch: of how the gearing carries the four fold angles through the springs’ preferences between the two states. That is where a designer’s choice of sector angles would matter, and it is a small corner of the settings — nine of 191 on the one vertex where it appears, none on the other two.

A transition state fixed before the energy #

The shape of this result has a well-known relative, and the comparison says what is unusual about it.

In chemistry, a molecule changing from one stable arrangement to another passes over a barrier, and the arrangement at the top of the lowest route — the transition state — is found by searching the energy landscape for a saddle. The rate of the change depends on the barrier’s height exponentially, which is Arrhenius’s law of 1889, and finding the transition state is most of the work of predicting the rate, because the landscape could put the saddle anywhere.

A sprung vertex’s landscape cannot put it anywhere. Its configurations are two curves meeting at one admissible point, so the transition state between the branches is known from the topology of the motion before a single energy is computed, and its energy then follows from the rest angles by one line of arithmetic. The pass is fixed by the shape of the space and priced by the springs, and neither of those depends on the other. Chemistry’s saddle is found; this one is given, and it is the flat sheet.

That also restates what paper that folds itself found from the other direction, and what the Miura’s two ways of folding turn on. A self-folding sheet driven out of flat has to choose a branch with nothing to decide it, because both are downhill. The count of roads here says exactly that: two of the four roads out of flat go down, and a sheet sitting on the pass will roll down one of them. What makes that a defect for a sheet meant to become one model makes it a mechanism for a wing, and the wall says how much it costs to come back.

What the energy landscape leaves out #

The springs are linear, the panels rigid, and the vertex alone. Each crease stores energy as the square of its departure from a rest angle, nothing bends between creases, and nothing collides. A real hinge stiffens near its limits and may have a rest angle that shifts with use; a real wing is many vertices sharing creases, not one, with no motor out along its length to hold any of them.

The switch is taken as slow. The road and its summit describe a vertex moved gently enough that its energy is all in its springs. A fast snap carries kinetic energy and can overshoot a state, and whether a disturbance empties a state depends on how quickly it arrives as well as on how much energy it carries — neither of which an energy landscape contains.

The fully folded junction is closed. The branches also meet at the two fully folded states, and a sheet of zero thickness that could pass through itself would have a second road there. Paper cannot, and the figures treat that junction as no road at all.

And the figures cannot say what disturbs a wing, or how big a climb an insect’s hinge springs actually provide. The shares are shares of the wall, which is a unit the model supplies; turning five per cent of a wall into a load a wing survives needs a hinge stiffness, which no figure here measures. What they establish is the structure: where the barrier is, what sets its height, and how unequally the two states usually share it.

How the roads were measured #

Both branches are sampled at 1,440 values of the driven crease’s angle across their whole travel, and a resting state is a sample lower than both its neighbours. The flat sheet sits between the two middle samples of each branch and is tested separately on all four sides.

The road between the two deepest states is taken explicitly: along the branch of each to the flat sheet, or along the one branch between them when both are on one side of flat. Its highest sample is the pass, and the flat sheet’s energy is computed separately, from the rest angles alone, and required to equal the road’s top wherever the road crosses flat.

The wall’s indifference to geometry is tested rather than argued: four vertices with one set of springs are required to give the same flat energy to a billionth while their deeper states differ by more than a twentieth of the wall. And the spring settings are the same two hundred the earlier count drew, so the resting states measured here are the ones it counted.

Still open: whether a sheet’s walls add #

Everything above is one vertex, and a folding wing is a fan of vertices sharing creases. A crease shared by two vertices has one fold angle, so the branches of neighbouring vertices are tied together, and the configurations of the whole sheet are no longer two curves through one point.

Two things follow that are worth computing. The junction a switch passes through is still a flat-looking configuration — the sheet can pass between branch combinations only where their curves meet — but on a sheet of several vertices there may be more than one such junction, and whether the wall of a sheet is the sum of its vertices’ walls, or something lower that lets a sheet switch one region at a time, decides whether a many-vertex wing holds its states more firmly than one vertex or less. The first measurement is a strip of two vertices sharing a crease, the smallest case of one crease deciding a sheet, where the branch combinations can still be listed, and the second is a closed ring of four around a panel, where the quadrilateral meshes already folded crease by crease say most combinations cannot exist at all.

The other direction is the load. A climb of five per cent of the wall is a number that becomes a statement about a wing only once a hinge’s stiffness is known, and the census here says which quantity to measure: the flat-state energy of the hinge springs, $\sum_i c_i r_i^2$, which is also the energy a wing stores when it is held flat open against them.

The habit worth carrying is a question to ask before searching any landscape for its barrier. Does the space have a junction every route must cross? If it does, the barrier is there, and its height is whatever the energy is at that one place — which is often something that can be written down in a line.