A mechanism that closes on itself

Rigid folding treats the panels as rigid plates and the creases as hinges, and asks whether the object can move from flat-open to flat-closed without any panel bending. It is the model everything manufactured uses, because manufactured panels are metal or plastic and do not bend.

Its central quantity is the degree of freedom: at a degree-four vertex, four fold angles related by three equations, leaving one, so the vertex moves in a one-parameter family. Propagate that across a pattern and the whole sheet has one degree of freedom, which is what makes the Miura what it is.

All of that is local. Close the sheet and a condition arrives that is not.

The loop closure #

A mechanism’s configuration is described by its fold angles. Going round a closed path of panels, each hinge contributes a rotation, and the composition of those rotations has to bring the walker back to where they started — because the panels are one connected body and the path returned to the panel it left.

On a sheet with a boundary every closed path bounds a region, and the closure is implied by the conditions inside it. The composition is automatically the identity, and there is no separate constraint.

On a closed sheet there is a path that does not bound, the composition round it is not implied by anything, and requiring it to be the identity is a genuine equation — at every stage of the motion, not merely at the flat state.

Why it is not the flat-folding parity #

The two conditions look alike and they are about different things.

The flat-folding parity is a condition on the flat folded state: a loop crosses an even number of creases, or the paper comes back the other way up. It is discrete, it is decided by counting, and it is checked before anything moves.

The loop closure is a condition on every configuration of the mechanism. It is continuous, it involves the fold angles, and it has to hold throughout the motion rather than at one end of it.

So a tube can pass the parity and still have no rigid motion, or have a rigid motion and no flat state, and neither implies the other.

What the closure equation says #

Writing it out makes the difference from the flat-folding condition concrete.

Each crease is a hinge with a fold angle. Rotating about a hinge by its fold angle is a rigid motion of three-dimensional space, and walking from a panel to a neighbour applies one such rotation.

A closed walk applies a sequence of them, and for the mechanism to be a connected rigid body, the composition has to be the identity — the walker’s panel has to come back to where it is.

On a disc every closed walk bounds, and the composition round it is the product of the compositions round the vertices inside, each of which is the identity by the vertex kinematics. So nothing new is required.

On a closed sheet a walk round the loop composes rotations whose product is a rigid motion of space, and the equation that product is the identity is six scalar equations — three for the rotation, three for the translation — holding at every value of the fold parameter.

Six equations at every configuration is a strong condition, and the surprise is that any pattern satisfies it. The Miura does, which is why Miura tubes deploy.

Two models, two conditions, one shape #

The parallel with the flat-folding case is exact and worth laying out.

Flat folding, disc. The vertex conditions are necessary; the two-colouring is implied.

Flat folding, closed sheet. The vertex conditions are unchanged; the two-colouring becomes an independent parity condition on each non-shrinkable loop.

Rigid folding, disc. The vertex kinematics is necessary; the loop closure is implied.

Rigid folding, closed sheet. The vertex kinematics is unchanged; the loop closure becomes an independent equation on each non-shrinkable loop, at every configuration.

Same structure both times: a global condition that is a consequence on a disc and an independent requirement elsewhere, arriving in whichever model happens to be in use.

That is a reasonable reason to expect a third instance in whatever model comes next, and it is a reason to look for the condition before building the object rather than after.

Rigid and flat, which do not imply each other #

Worth restating, since a reader may expect one condition to cover both.

A pattern can fold flat without being rigid-foldable: the paper bends on the way even though both end states are flat, which is the condition that is not flat-foldability.

A pattern can be rigid-foldable without folding flat: the panels move perfectly well and never reach a flat state.

Closing the sheet adds a condition to each, and the conditions are unrelated. A tube can pass the parity — an even number of creases round it, so a flat state exists — and have no rigid motion. Or it can have a rigid motion and an odd crease count, in which case it moves and never flattens.

So a designer of a folded tube has two independent things to check and the collection can currently check one of them.

What would have to be built #

For the record, since the essay records a gap.

The collection’s rigid-folding machinery solves for every panel’s position at once, from a set of fold angles and the pattern’s connectivity, on a patch.

Extending it means: identify panels across the glued edges, add the closure equation for each independent loop, and solve the enlarged system. The equations are ordinary rigid-body equations and the solver already handles systems of them.

What is not obvious is how the degree of freedom count comes out. Adding six equations per loop might remove all the freedom, leaving a rigid object that does not move at all; or the equations might be dependent on the existing ones, leaving the count unchanged. Which of those happens is the interesting question and it is exactly the question the computation would answer.

That the Miura tube deploys says the answer is unchanged for at least one pattern. Whether that is general or particular is not known here.

What it does to the degrees of freedom #

A closed loop adds equations, and adding equations removes freedom.

A rigid mechanism’s degree count is the number of angles minus the number of independent constraints. Closing a sheet adds a closure constraint per independent loop, so a cylinder loses some freedom and a torus loses more.

For the Miura specifically the answer is known to engineers and it is a good one: a Miura tube retains a one-degree-of-freedom motion, which is why the pattern is used for deployable booms. The closure is satisfiable and satisfied.

That it is satisfiable is a fact about the Miura rather than about closed sheets, and it is not obvious in advance.

The condition is not a vertex condition #

Worth saying plainly, since rigid-foldability is normally decided vertex by vertex.

Every vertex of a glued sheet is an ordinary vertex with an ordinary degree-four kinematics. Its fold angles satisfy the same relations, its motion has the same one parameter, and nothing about the identification touches it.

So the vertex analysis is complete and unchanged, and the closure is an extra condition on top of it — which is exactly the shape the flat-folding conditions take on the same sheets, one model further along.

What has been computed here #

Not the mechanism. It is worth being explicit.

This collection’s glued-sheet work is combinatorial: which letterings are consistent, how many free letters there are, what a search costs, whether a flat folded state exists. Every one of those is about the flat state and none is about the motion.

The rigid-folding machinery in this collection solves the panel positions for a patch, and it has not been given a glued sheet. Doing so means adding the closure equation, which is a well-posed thing to do and has not been done.

Recorded as owed.

Counting freedoms, carefully #

The degree-of-freedom arithmetic is worth doing once, because it is the quantity the essay is about and it is easy to state loosely.

A mechanism of $p$ rigid panels in space has $6p$ coordinates. A hinge between two panels removes five of the six relative freedoms, leaving one — the fold angle. So a connected pattern with $c$ creases has $6p - 5c$ freedoms, of which six are the rigid motions of the whole object, leaving $6p - 5c - 6$ internal ones.

For a patch of Miura that count comes out at one, after the vertex relations are accounted for, and it is the pattern’s famous single freedom.

Gluing identifies panels and creases, so both $p$ and $c$ fall — and they do not fall in the same proportion, which is why the count can move either way. On the Miura’s two-period cell, panels go from fifteen to eight and creases from twenty-two to sixteen; the naive arithmetic then gives a different answer from the patch’s.

That arithmetic is not reliable, because it assumes the constraints are independent and in a folding mechanism they are famously not — which is exactly why a naive count says a Miura has no freedom at all and it has one.

So the honest position is that the count has to be computed rather than derived, and it has not been.

A tube in the hand #

The condition is easier to believe after making one, and a Miura tube takes ten minutes.

Fold a Miura from a rectangle — say eight columns by six courses — and collapse it a few times so the creases are willing. Then bring the two short ends together and tape them so that the corrugation runs round the tube.

Push the two open ends towards each other. The tube collapses along its axis, smoothly, in one motion, and springs back. That is the one degree of freedom, surviving the closure.

Now try to move it any other way — twist the two ends relative to each other, squash it sideways. It resists, and it resists because the closure has used up whatever freedom a strip of Miura would have had for those motions.

An open strip of Miura can be twisted slightly along its length, because the two ends are unconstrained relative to each other. Joining them removes that, which is the closure equation felt directly: a motion that was available becomes a motion that would break the seam.

So the closure is not an abstraction. It is the difference between an object with free ends and an object whose ends are the same end, and any strip of anything exhibits it.

The Kresling case, which is different #

One pattern deserves mention because its closure behaves unusually.

A Kresling tube is a cylinder of triangulated panels that twists as it collapses: shortening it rotates one end relative to the other. It is a standard deployable form, it is bistable, and it is used where a snap-through is wanted.

The twisting is exactly a closure phenomenon. On an open strip the two ends can rotate freely relative to each other; joining them into a tube ties the rotation to the extension, and the coupling is what produces the twist.

So the loop closure is not only a restriction. On some patterns it couples motions that were independent, and the coupling is the useful behaviour rather than a cost.

That is worth knowing before treating the closure as a constraint to be satisfied. On the Miura it is satisfied and invisible; on the Kresling it is satisfied and is the whole point of the object.

Three tubes, three behaviours #

A short catalogue, since the closure produces different things on different patterns and the variety is the interesting part.

The Miura tube. One degree of freedom, retained through the closure. Collapses along its axis, springs back, and is the standard deployable boom. The closure is satisfied and invisible.

The Kresling tube. The closure couples extension to rotation, so collapsing twists it. Bistable, snaps between two stable states, used where a definite click is wanted.

The waterbomb tube. Collapses radially rather than axially, and is used for stents and for grippers. Its closure produces a coupling between the tube’s diameter and its length.

Three patterns, three closures, three qualitatively different motions — and in each case the closure is what turns a strip’s behaviour into a tube’s.

That variety is the argument for computing the closure rather than treating it as a constraint to be checked. It is not merely something that can fail; on these three it is the thing that makes each object what it is.

Why the mechanism question is the one that matters #

For everything this collection has measured on glued sheets, the flat-state question has been the one asked, and it is worth saying why the mechanism question is more important for the objects involved.

A deployable structure is not used flat. It is used moving: packed, then unpacked, then packed again, and what it is judged on is whether the motion is smooth, whether it needs one actuator or many, and whether it goes where it is meant to.

The flat state is the packed configuration and it matters, but the existence of a flat folded state is a very weak statement about a mechanism. It says the endpoint exists; it says nothing about whether anything can get there.

That is the standing distinction in this collection between a theorem and a motion: a folded state existing and a path to it existing are different claims, and the second is the engineering one.

So the phase has measured the weaker of the two properties on the objects where the stronger one is what people care about. That is honest and it is a limitation, and it is the reason this essay is a statement of what would have to be built rather than a report of what was.

Where the rigid model stops #

A caution, since the essay is about a rigid mechanism and real folded tubes are not quite rigid.

The panels of a manufactured tube bend a little, the hinges have thickness and play, and the material stores energy — which is where bistability comes from and is not in the rigid model at all.

So the closure condition as stated is about an idealised mechanism. A real tube that fails it slightly will still move, by bending its panels a little, and a designer’s question is how much bending the closure error demands rather than whether it is zero.

That is the same relationship the rigid model has to everything else: the mathematics decides whether an exactly rigid motion exists, and the engineering decides whether an approximately rigid one is good enough. Both are needed and they answer different questions.

Why engineers already know the answer #

For the patterns that matter, the closure is satisfied and the knowledge is empirical.

Miura tubes are made and they deploy. Kresling tubes are made and they twist as they collapse. Waterbomb tubes are made and they have a characteristic bistable snap. Each of those is a closed sheet with a rigid or near-rigid motion, and each was arrived at by construction and testing rather than by solving a closure equation.

That is the ordinary relationship between the engineering and the mathematics here: the objects exist, they work, and the condition explaining why they work is available afterwards.

The sentence to add #

Rigid-foldability is decided at the vertices on a sheet with a boundary.

On a closed sheet there is a further condition — the composition of the hinge rotations round every non-shrinkable loop is the identity, at every configuration — and it is continuous where the flat-folding conditions on the same sheets are discrete.

Both are the same phenomenon: a global condition that is implied on a disc and independent everywhere else, arriving in whichever model is being used.