A chain of vertices switches all at once

The wall is the flat sheet found that a sprung degree-four vertex switches between its two resting states in only one way. Its configurations are two curves — the two branches of its motion — and the curves meet at the flat state and nowhere a sheet can pass, so every switch goes back to flat and out again. The flat state is also the top of that road on almost every setting of the springs, and its energy, each spring’s stiffness times its rest angle squared, depends on the springs and not on the vertex.

It ended on the obvious objection. A folding wing is not one vertex; it is a fan of vertices sharing creases, and shared creases tie the vertices’ branches together. Perhaps a sheet of vertices can switch one region at a time, climbing a small wall where one vertex changes branch while the rest stay put. Perhaps its walls add and it holds its states more firmly. The two possibilities make opposite predictions about how a many-vertex fold behaves, and a chain of vertices is the smallest place to tell them apart.

The chain switches all at once.

One number fixes the chain, once the branches are chosen #

A degree-four vertex that folds flat is geared. What the vertex does on the way solves its motion, and along each branch the tangents of the four half fold angles keep fixed ratios to one another: the whole branch is a single number times a fixed vector. The two branches have two different vectors, and at the flat state both numbers are nought.

Now chain the vertices. Each shares one crease with the next — the first crease of one is the third of the next — so the fold angle on that crease belongs to both. Choose a branch at every vertex. The first vertex’s number fixes its fold angle on the shared crease, which fixes the second vertex’s number, which fixes its fold angle on the crease it shares with the third, and so on down the chain. Once a branch has been chosen at every vertex, one number fixes every fold angle in the chain.

So a chain of $m$ vertices has $2^m$ curves of configurations, one for each way of choosing branches, and each is a one-parameter motion like a single vertex’s. That is the chain’s version of one crease decides the sheet: a single driven angle settles everything, but only after a discrete choice has been made at every vertex, and the choices multiply.

Every combination rests somewhere #

Give every crease a spring with a rest angle and a stiffness, exactly as for the single vertex, and draw the rest angles at random. The first figure counts resting states over two hundred settings for chains of one, two, three and four vertices, with four, seven, ten and thirteen creases.

The median setting rests in two, four, eight and sixteen states: exactly one for every combination of branches. The share of settings with at least one resting state on every single combination is 198 of 200 for one vertex, 193 for two, 181 for three and 134 for four; on the others some combination has no resting state of its own.

The second figure draws all four curves for a chain of two vertices with one particular setting of its seven springs. Each curve holds a resting state of its own, and all four cross in the middle, at the flat sheet.

That doubling is what the question about wings was really asking. A fan of vertices is not a switch with two positions; it is a switch with a position for every combination of branches, and on almost every setting of its springs it can rest in all of them.

The curves meet in one place #

The curves’ shape is what decides how the chain switches between those states, and the shape follows from the gearing.

Every curve passes through the flat sheet, because at flat every vertex’s number is nought whatever branch it is on. Could two curves meet anywhere else? If they did, some configuration would lie on two different combinations of branches at once. Take the first vertex where the two combinations differ: in that configuration the vertex would be on both of its branches at once, which a degree-four vertex is only when it is flat — or fully folded, a coincidence of positions a sheet reaches only by passing through itself. But if one vertex in the chain is flat, the crease it shares with its neighbours is flat, so they are flat too, and so on down the chain. Two combinations of branches can meet only where the whole chain is flat.

So a chain resting on one combination of branches that is to end up on another — even if the two differ at a single vertex — has one road: back along its own curve to the flat sheet, and out along the other. There is no way to change one vertex’s branch while its neighbours stay folded, because a neighbour folded keeps the shared crease folded, and a shared crease folded keeps the vertex on its branch.

The third figure draws that road for a chain of three vertices, between its two deepest resting states, which here lie on different combinations. The road runs from the deeper state down to flat and up to the next, and its highest point is the flat sheet.

The first figure’s third column checks that across the census. The road between the two deepest states crosses the flat sheet and tops out there on 197 of 198 settings with two or more states for one vertex, 200 of 200 for two, 199 of 200 for three and 200 of 200 for four. A chain of vertices switches all at once or not at all.

The walls add #

The road’s summit is the flat sheet, and the flat sheet’s energy is as easy to write down for a chain as for one vertex. Every fold angle is nought, so every spring is exactly its rest angle away:

$$E_{\text{flat}} = \sum_{\text{creases}} c\,r^2.$$

The sum runs over every crease of the chain. So the wall is every spring’s own flat energy added up, and adding a vertex adds three creases and three creases’ worth of wall. With rest angles drawn evenly from nearly shut one way to nearly shut the other, each crease contributes on average $(0.95\pi)^2/3 \approx 2.97$ units, and the fourth figure measures that: 3.10, 2.99, 3.01 and 2.93 units a crease on chains of one to four vertices, walls averaging 12.4, 21.0, 30.1 and 38.1.

The deepest resting states grow with the wall, because they too are sums over more springs; what changes is how far below the wall they sit. A single vertex, the earlier census found, holds its second state by a median of about five per cent of the wall — often barely at all. A chain holds its second state much more firmly.

The second state is held firmly #

The fifth figure measures the climb out of the shallower of the two deepest states, as a share of the wall. For a single vertex on this set of spring settings it is 2.9 per cent in the median. For two vertices it is 14.0; for three, 16.8; for four, 18.7.

In absolute terms the difference is larger again, because the wall itself grows. A single vertex’s second state is held by about three per cent of a wall of twelve units; a chain of four holds its second state by nearly a fifth of a wall of thirty-eight. The second state of a chain of four is held roughly twenty times as firmly as a single vertex’s, in the same kind of springs — a comparison of medians, so a rough factor rather than an exact one.

The reason is in the shape of the curves. On a single vertex the two branches are two curves, and the second-deepest state is usually a shallow dip on the other branch, not far below where the curve leaves the flat sheet. On a chain there are many curves, the two deepest states usually lie on different ones, and each curve’s resting state is the best that combination of branches can do for springs that disagree with it. With more combinations to choose from, the deepest two are both deep, and the road between them must still climb all the way to a flat sheet whose energy counts every spring.

One input, many outcomes #

The same structure that makes the chain switch all at once makes it hard to drive into a particular state, and the arithmetic of that is short.

Paper that folds itself found the difficulty at a single vertex: driven out of flat, a vertex has two branches running downhill and nothing in the torque decides between them. A chain has the same choice at every vertex. Leaving flat, the first vertex must choose a branch, and so must the second, the third and the fourth, and nothing in the one number that drives the chain says which; the hardest instant of driving a sheet is at flat for a related reason. A chain of $m$ vertices driven from flat has $2^m$ ways to fold, and one of them is the one wanted. If each vertex’s choice were settled by a coin — by whatever small asymmetry of the hinges happened to win — a chain of four would reach the intended combination one time in sixteen.

That is the price of the doubling the census found. Every combination of branches holding a resting state is what makes a chain a switch with many positions; every combination being reachable from flat by the same motion is what makes it a switch that cannot be told which position to go to. One crease decides the sheet showed that a single biased vertex settles the choice for a meshed sheet whose loops tie every vertex’s branch to its neighbours’. A chain has no loops, so a bias at one vertex settles that vertex and nothing else: a chain needs a bias at every vertex, and a wing built as a chain would need its hinges to be asymmetric all the way along.

The wall’s height has an equally short derivation. With every rest angle drawn evenly between nearly shut one way and nearly shut the other, between $-0.95\pi$ and $0.95\pi$, the average of a rest angle squared is a third of the largest one squared, $(0.95\pi)^2/3 \approx 2.97$. Each crease adds that much to the flat sheet’s energy on average, and a chain of $m$ vertices has $3m + 1$ creases, so its wall averages about $2.97(3m + 1)$: 11.9, 20.8, 29.7 and 38.6 for one to four vertices, against the measured 12.4, 21.0, 30.1 and 38.1.

Why the switch has to be global #

The result has a familiar shape from magnetism, and the comparison makes clear which feature of the chain produces it.

In a bar magnet, regions of aligned atoms called domains can flip one at a time: a boundary between a region pointing one way and a region pointing the other can form and move, and the energy cost of the switch is paid locally, at the boundary. That is how a magnet reverses without every atom having to point sideways at once. A chain of vertices has no such boundary. Two neighbouring vertices on different branches are not a domain wall in a folded state; they are a configuration that exists only when both are flat. The coupling through a shared crease is so tight that there is no intermediate arrangement in which part of the chain has switched and part has not.

That makes the chain the opposite of the corrugation that started this subject. A corrugation has one resting state because all its creases fold by one angle and its energy is a single parabola. A chain of vertices has a resting state for every combination of branches because each vertex adds a discrete choice — but it switches between them with the same all-or-nothing rigidity, because the choices can only be changed together, at flat.

For a wing — which has no motor in the fold — the reading is suggestive rather than decisive. A fan of vertices that locks open and shut with its hinges alone would, on this model, hold both positions firmly and would need to be driven through the fully open, flat configuration to change — which is also what a wing held open in flight is closest to. Nothing here measures a wing, and the earwig’s folding hindwing that prompted the question has vertices arranged in a fan rather than a single chain.

What the chain leaves out #

A chain has no loops. Each vertex shares a crease with the next and with nothing else, so the shared angles pass the chain’s one number along without ever having to come back to where they started. A sheet of vertices arranged round panels — a quadrilateral mesh — has loops, and a loop requires the numbers passed round it to agree when they return. For a general mesh most combinations of branches fail that requirement and do not exist at all; for the Miura, one of each vertex’s two branches is a plain fold about a straight crease. How many combinations survive the loops, and whether the all-at-once switch survives with them, is not computed here.

The springs are linear and the panels rigid, as for the single vertex, and nothing collides. A chain folded far enough brings panels of different vertices into contact, and contact is exactly the kind of constraint that could hold part of a chain folded while another part moves.

And the chain is a line of vertices chosen for the census, four particular degree-four vertices that fold flat, each sharing one crease with the next. A different set of sectors changes the energies and not the argument: the gearing makes every branch a single number times a vector, and that is what makes the curves meet only at flat.

How the census was taken #

Each vertex’s branch vectors are measured, not assumed. The tangents of its half fold angles are read off its solved motion at two different points of each branch and required to have the same ratios at both, so the gearing the chain relies on is checked on every vertex used.

Every combination’s curve is sampled at 1,200 values of the first vertex’s driven angle, and a resting state is a sample lower than both its neighbours. The flat sheet is tested separately along every curve on both sides.

The road between the two deepest states is taken explicitly, back along each state’s own curve to the flat sheet, and the flat sheet’s energy is computed from the rest angles alone and required to equal the road’s top. And the curves are required to meet nowhere but the flat sheet on the settings drawn.

Still open: whether a loop lets a region switch #

The chain answers the question the single vertex left, and leaves a sharper one. On a sheet with loops, can a region switch branches without the whole sheet going flat? A loop removes most combinations of branches, but it also creates a new kind of junction: a configuration in which the vertices around a panel sit where the loop’s requirement is met by more than one combination. If such junctions exist away from the flat sheet, a meshed wing could switch one region at a time and its walls would not simply add. The quadrilateral meshes already solved crease by crease are where to look, starting with a single loop of four vertices round one panel.

The other question is the wing itself. The census says that a hinged fan holds its states firmly and switches through flat. Whether an insect’s wing sits near that picture or relies instead on a panel that bends, or on contact between layers, is a measurement of wings rather than of models.

The habit worth carrying is a check on any claim that a large system can change one part at a time. Ask whether the parts’ configurations meet anywhere except at the state where everything is at rest. If they do not, the system switches all at once, however many parts it has, and its barrier is the sum of theirs.