A loop takes choices away

A chain of vertices switches all at once answered the question a single sprung vertex left. A degree-four vertex rests in two states, one on each branch of its motion, and switches only through the flat sheet; chain vertices together by shared creases and every combination of branches becomes a resting state — two, four, eight, sixteen for chains of one to four — but every combination’s curve of configurations passes through the one point where the whole chain is flat, and meets no other curve anywhere else. The chain switches all at once, and its wall is every crease’s flat energy added up.

It closed on the objection a real wing raises. A wing is not a chain; its vertices sit round panels, and a panel’s vertices form a loop. A loop requires the fold angles passed round it to agree when they come back, which should remove most combinations — but it might also create something a chain cannot have: a configuration where the loop’s requirement is met by more than one combination at once, away from the flat sheet. If such junctions exist, a meshed wing could switch one region at a time.

They do not, and the reason is in the same arithmetic that made the chain global — applied once more, round a closed path.

What a loop requires #

On a flat-foldable degree-four vertex, each branch of the motion is one number times a fixed vector: the vertex is geared, and along a branch the tangents of its four half fold angles keep fixed ratios. In a chain, a shared crease hands the number from one vertex to the next, scaled by the ratio of the two vertices’ gears on that crease.

Close four vertices round a panel and the number goes all the way round and arrives back at the first vertex, scaled by the product of four gear ratios. A combination of branches is a real motion of the loop only if that product is exactly one; otherwise the only number consistent with itself is nought, and the loop is flat. So a loop does not merely restrict a chain’s sixteen combinations, it tests each of them against a single equation, and whether a combination passes depends on the geometry of the four vertices together.

The gearing is what turns the loop into arithmetic. Because a branch is a fixed direction scaled by one number, the question of whether a combination can move is not a question about any particular fold angle; it is a question about four fixed numbers multiplied together. That is why a loop’s survivors can be counted once and trusted at every angle, and also why the answer cannot depend on how far the face has folded: nothing in the product changes as it does.

It also says what kind of face keeps more than one. The product is a property of the four vertices’ sector angles taken together, and a face keeps several combinations when several choices of branch give the same product. Repeated vertices make repeated factors, and repeated factors make coincident products; a face of four different vertices generically has no two products equal, and if it folds at all — which is itself the requirement that one product be exactly one — it keeps that one and no other.

What survives, on two faces #

The census drives one crease of a three-by-three mesh — four interior vertices round one central panel — to a fixed angle, and counts every assignment of fold angles to the other creases that closes at every vertex with every crease folded.

On a face whose four vertices are all different, one combination survives. The face, solved to fold rigidly, folds in exactly one way: every one of its twelve creases, driven to any angle, leaves a single consistent assignment. Fifteen of the chain’s sixteen combinations fail the loop.

On a Miura face, four combinations survive. The Miura repeats one vertex, so its gear ratios repeat round the loop, and more products come out as exactly one. Driven at its most ambiguous crease, the face leaves four assignments; driven at its others, two or one.

And both counts are the same at every angle the face is driven to — 0.3, 0.6, 1.0, 1.4 and 1.8 radians — which is what the product condition predicts: the gear ratios are constants of the vertices, so a combination either closes along its whole motion or nowhere.

Why the survivors still meet only at flat #

A junction is a configuration two combinations share. Each surviving combination is one number times a fixed vector — the loop’s gears, round and back — and two such curves can share a point only where they give every crease the same fold angle.

That happens only where some vertex’s two branches meet. Two combinations differ somewhere, at some vertex taking one branch in the first and the other branch in the second, and at that vertex the two branches agree only at its flat state. A flat vertex has all four creases unfolded, and each of those creases is shared with a neighbour, so the neighbour has a crease at nought. A flat-foldable degree-four vertex with one crease at nought has all four at nought, because every other crease’s half-angle tangent is a fixed multiple of that one’s. So the neighbour is flat, its neighbours are flat, and flatness spreads round the loop and through every shared crease to the whole face.

The measurement agrees. Sampling the Miura face at 48 driven angles from −2.4 to 2.4 radians, two different surviving assignments at the same driven angle are never closer than 1.53 times that angle, measured over all twelve creases. Their separation grows in proportion to how far the face is folded, which is the signature of curves that leave a common point in different directions and never come back.

The number is worth reading. Two assignments that leave the flat face in different directions separate at a rate set by the angle between those directions, and 1.53 radians of separation per radian driven is a wide angle — the survivors are not nearly coincident curves that a small imperfection could join, but distinct motions from the first moment of folding. A manufacturing error small compared with the fold would not bring two of them together, which is the practical form of the statement that the junction is at flat.

So a loop adds no junction. It removes combinations — fifteen of sixteen on a generic face, twelve on a Miura face — and it leaves every survivor crossing the others only at the flat sheet, exactly as the chain’s did.

The plain fold the census leaves out #

One kind of configuration is deliberately outside the count, and it is the one place a region could move on its own.

A Miura vertex has a straight crease through it: two of its creases are collinear. The Miura folds two ways records the consequence. One of the vertex’s two branches is not a zigzag motion at all but a plain fold about the straight line, with the other two creases staying flat. On a Miura sheet every straight row line can be folded alone while everything else stays flat, and folding one straight line does not force its neighbours to fold.

That is local motion, and it is the only kind the Miura face has. A region of a Miura can bend along a straight line without the rest of the sheet going flat. But it is not a switch between the resting states the census counts: those have every crease folded, and reaching a plain fold from any of them means unfolding its zigzag creases to nought, which by the spreading argument above unfolds the whole face first. The straight lines are a separate family of motions attached to the flat sheet, not a bridge between the folded ones.

For a generic face there is nothing of the kind. Its vertices have no collinear creases, no plain-fold branch, and one combination; the face is a mechanism with one motion and one resting state on each side of flat.

What this makes of a meshed wing #

The chain gave a wing built as a fan of vertices a clean picture: a switch with many positions, each held firmly, changed only by driving the whole fan through flat. The loop keeps the second half of that picture and changes the first.

A meshed wing has fewer positions, not more. Every loop of vertices removes combinations the chain allowed, and a mesh of many loops, each requiring its product to close, keeps only the combinations every loop agrees on — which on a generic mesh is the single motion one crease decides the sheet found, where fixing any fold angle fixes every other. The Miura’s four survive because its repeated vertex makes its loops agree more often; only four creases decide a Miura counts the same ambiguity from the other end.

And its switches are still global. A wing with loops cannot flip a region by climbing a small local wall, because there is no configuration in which one region has switched and the rest has not. The wall is the flat sheet found the barrier for one vertex at the flat state and the chain found it adding over vertices; the loop keeps the junction at the same place, so the wall is still the whole mesh’s flat energy.

That gives the fan-of-vertices hypothesis for insect wings a definite limit. If a wing locks open and shut by its hinges alone and has loops of rigid panels, it must pass through its flat, fully open configuration to change state, and the energy it must be given is its whole flat energy. A wing that folds into nothing set out the packing a folded wing achieves; a wing that switched regions separately would need something the rigid-panel model does not contain — panels that bend, or vertices of more than four creases.

What a designer of a folding wing would take from it #

The loop result is negative about local switching and positive about something else. A meshed sheet with no repeated vertices is determined by its loops: every combination but one fails, so the sheet has one motion and no choice to make, and a single actuator anywhere drives the whole of it the same way every time. That is the property one crease decides the sheet found by propagation, now traced to the arithmetic that causes it — the loops, not the vertices, remove the choices.

A designer who wants a sheet with several stable positions therefore wants repeated vertices, as the Miura has, or wants the positions to come from springs rather than from branches. A designer who wants a sheet that cannot be confused wants every vertex different. The same choice of pattern sets how many states the sheet can be in and how reliably it can be told which one, which is the trade only four creases decide a Miura found from the actuator’s side and the loop explains from the sheet’s.

For a wing the reading is the same one the chain gave, with a sharper edge. A wing that needs to lock in two positions — open for flight and folded for rest — can get them from its hinges as a chain or a symmetric mesh, but it must pass through fully open to change between them, and a wing built as a generic mesh cannot lock in two folded positions at all. Two states and a flat wall between them is the most a rigid meshed wing offers, and anything richer needs bending panels or higher-degree vertices.

What the census assumes #

The vertices are degree four and flat-foldable, so each is geared and each branch is one number times a fixed vector. That is what makes the product condition exact and the spreading argument valid.

The panels are rigid. A panel that bends is not a loop constraint at all — it lets the fold angles round the loop disagree by bending — and a wing’s membrane panels bend readily.

Every crease is folded. Assignments leaving a crease at nought are excluded, which removes the plain folds of the Miura and nothing on the generic face.

The census counts at a driven crease’s own positive angles. Each combination is a curve through flat that continues to negative angles as its own mirror; the mirror of a combination is counted as the same combination, and a face’s resting states under springs would come in such mirrored pairs.

And the faces are two particular faces: a Miura with one lean, and a generic face solved from one seed. A different generic face is expected to keep one combination for the same reason, and a Miura with a different lean four, and neither has been counted.

What the census cannot show #

It counts combinations, not resting states under springs. The chain’s census placed springs on every crease and found a resting state per combination; the loop census counts the combinations the springs would have to choose among, and a sprung loop’s resting states have not been counted.

It does not follow a loop through the rest of a mesh. A face is one loop; a mesh of many faces imposes every face’s product at once, and a combination must pass all of them. On a generic mesh the single survivor of each face is the same motion, which is why the whole mesh has one; on a larger Miura the four survivors of each face must also agree between faces, and how many do on a sheet of many faces is the census only four creases decide a Miura takes from the actuator’s side, crease by crease.

It does not treat vertices of higher degree. A degree-six vertex has three freedoms, so a crease at nought does not force it flat, and the spreading argument fails at it. That is where a region could genuinely switch alone, and it is not measured here.

And it measures faces, not wings. Whether any insect’s folding wing has the rigid-panel loops the model assumes, or relies on the bending the model excludes, is a question of anatomy.

Still open: a vertex with room to stay flat in part #

The spreading argument has one hinge, and it is the property that makes degree four special: one crease at nought forces the vertex flat. A vertex with six creases, like the waterbomb’s, has three freedoms, and it can hold one crease at nought while its other five fold. A loop containing such a vertex could have a junction away from the flat sheet — two combinations sharing a configuration in which that one crease is unfolded — and a region beyond it could switch while the rest of the sheet stayed folded.

Whether that happens on a real pattern is a census of the same kind on a mesh with degree-six vertices, and the base that tiles provides one: the waterbomb tessellation, whose degree-six vertices alternate with degree-four ones. If a waterbomb sheet has junctions away from flat, its degree-six vertices are where a folding structure could switch piece by piece, and the degree of a pattern’s vertices would be the design parameter that decides whether a fold switches globally or locally.

The habit worth carrying is about constraints that close. Before expecting a loop to add behaviour, ask what it requires to come back round. A condition that has to be met on return can only remove possibilities, and whatever junctions the system has were already there before the loop was closed — so the question about new behaviour is always a question about the parts, not about the loop.