The tube that gets built

Folding that gets built surveyed the engineered applications: solar arrays that pack for launch and open in orbit, airbags folded into a steering column, stents threaded through an artery, starshades, radiators, energy absorbers.

Every one of them is a closed sheet. A boom is a tube. A stent is a tube. An airbag is a bag. A bellows is a tube. A packed antenna is a shell.

The mathematics has been done on flat rectangles.

What a rectangle is a rectangle of #

The flat pattern is not wrong; it is the manufacturing artefact.

A folded tube is made by cutting a flat sheet, scoring it, folding it, and joining two edges. The flat sheet is what the machine handles, the pattern is what gets printed on it, and the joining is the last step.

So the flat rectangle is real and it is the thing that exists in the factory. What it is not is the object that works, and the properties anybody cares about — how far it packs, how it deploys, whether it flattens — are properties of the joined thing.

What changes when the edges meet #

Everything global, and the counts are exact.

On a two-period Miura cell: twenty-two free letters and fifteen panels as a rectangle; eighteen letters and ten panels as a tube. A fifth of the pattern’s freedom is a property of the rectangle’s edges and does not survive the join.

And a condition arrives. A path running once round the tube cannot be shrunk to a point, so the number of creases it crosses has a parity that nothing local forces, and an odd count means no flat folded state at all.

The applications, one by one #

It is worth going through them, because the claim that every engineered folded object is a closed sheet is a strong one and deserves checking.

A deployable boom. A tube of Miura or Kresling, collapsed along its axis for launch and extended in orbit. Closed in one direction; two open ends. A cylinder.

A stent. A tube of a folded lattice, compressed to pass through a catheter and expanded in place. A cylinder.

An airbag. A closed bag rather than a tube — a sheet joined to itself all round — folded into a small volume and inflated. Closed in a stronger sense than a cylinder, and the least studied of these mathematically.

A starshade. A large petalled shell, packed round a hub. Closed in one direction round the hub.

A packed antenna or reflector. A dish or a shell, wrapped. Closed in one direction.

An energy absorber. A crushable tube, usually a corrugation, closed round.

A folded bellows. A tube, closed round, with the corrugation along it.

Seven applications and every one is closed in at least one direction. The only flat folded objects that get manufactured are decorative or packaging inserts, which is a real category and not the one the deployable literature is about.

That is a striking ratio and it is worth pausing on: the subject’s mathematics is about discs and none of its applications is one.

The airbag, which is the hard case #

Of the list, one is closed in a way this collection cannot describe, and it is worth naming.

A bag is a sheet joined to itself all round — a sphere, topologically, with no boundary at all and no loops that cannot be shrunk. That is a different sheet again from a cylinder or a torus, and it is not obtainable by identifying a rectangle’s opposite edges.

A sphere has Euler number two and no non-shrinkable loops, so it has no parity condition of the kind this essay is about. What it has instead is a different problem: a sphere is not flat, so a sheet folded into one is not developable, and the material has to stretch or gather.

Which is why airbag folding is a different subject from crease-pattern folding: the sheet is not a developable surface and the whole apparatus of isometries does not apply.

So the list has six objects this collection’s machinery can now describe and one it cannot, and the one it cannot is the one that is not a folding problem in the usual sense.

What the packing ratio is a property of #

One quantity deserves a careful look, since it is the number deployable work is judged on.

The packing ratio is how far a structure collapses: extended length over packed length, or extended volume over packed volume.

It is computed from the fold angles and the panel geometry, vertex by vertex, and it therefore transfers from the flat pattern to the tube unchanged. A Miura strip that collapses to a tenth of its length makes a Miura tube that collapses to a tenth of its length.

Where the join matters is at the ends. A tube’s packed form has its two open ends, and whatever is attached there — a hub, a plate — does not collapse, so the achievable ratio is the pattern’s ratio degraded by the fittings.

That is an engineering matter rather than a folding one, and it is the usual reason a real structure underperforms its pattern. Worth separating from anything in this essay, which is about the pattern’s own properties.

How much of the literature is affected #

An honest estimate, since the essay makes a claim about a field.

The kinematics literature — degrees of freedom, motion paths, Poisson’s ratio, bistability — is local and transfers. That is most of it.

The flat-foldability literature about specific patterns is about flat sheets and its results are about flat sheets. Applying them to a tube requires the parity check and nothing else, and the check is rarely stated.

The counting literature — how many configurations a cell admits, how large a tunable family is — is measured on cut cells and overcounts, by the free letters the rim supplies.

So the exposure is: kinematics unaffected, flat-foldability needs one extra check, counting is biased. That is a narrow correction to a large body of work, which is the usual size for a hypothesis that was true of everything anybody looked at.

The rule engineers already have #

The parity is known in practice as a rule of thumb: a folded tube needs an even number of facets round it.

It is arrived at by trying. A tube with an odd count does not collapse cleanly; pushing it produces a buckle, and the buckle is an unplanned crease appearing where the arithmetic requires one. Anybody who has made a few tubes has learnt to use even counts.

What the mathematics adds is that the rule is exact rather than approximate, that it is the same rule that refuses a loop of paper with three creases, and that it is checkable by counting rather than by testing.

For a designer that is worth having: an addition performed on the drawing rather than a prototype that buckles.

Why the flat analysis is nearly right anyway #

The essay would be alarmist if it stopped there, and most of the flat analysis transfers.

The kinematics transfers completely. A Miura’s single degree of freedom comes from a degree-four vertex, the vertices are unchanged by the join, and the tube moves the way the strip does. That is most of what deployable analysis is about.

The packing ratio transfers. How far the structure collapses is a geometric fact about the fold angles, computed vertex by vertex.

Poisson’s ratio and stiffness transfer, being local.

What does not transfer is anything counting configurations, anything about whether a flat state exists, and the loop closure of the mechanism, which is a condition on the motion that a strip does not have.

The numbers, for one pattern #

Since the essay claims a fifth of a pattern’s freedom is a property of the rectangle, the Miura’s numbers deserve to be quoted directly.

One period. Two interior vertices. Seven free letters and six panels as a rectangle; five and three as a tube glued across; six and four glued along; four and two glued both ways.

Two periods. Eight vertices. Twenty-two letters and fifteen panels as a rectangle; eighteen and ten across; twenty and twelve along; sixteen and eight both ways.

Three periods. Eighteen vertices. Forty-five and twenty-eight; thirty-nine and twenty-one; forty-two and twenty-four; thirty-six and eighteen.

Read the vertex column and nothing moves, which is the control: the tube’s vertices are the rectangle’s vertices, at the same angles, satisfying the same conditions.

Read the letter column and about a fifth goes at each size, in the direction the tube is closed. That is the freedom the rectangle’s edges were supplying and the tube does not have.

And on the odd sizes, glued across, the tube has no flat folded state at all, while the rectangle folds perfectly well.

What this collection can now say about a tube #

A short inventory, since the point of building the machinery was to be able to say something.

Whether it has a flat folded state. By parity, in one addition, exactly. Available.

How many free letters its pattern has. By counting the creases the seam joins. Available and exact.

How many panels it has, and its Euler number. Available, and the second is the check that the description is sound.

Whether a lettering exists and what finding one costs. Available, on cells small enough to search.

Whether the mechanism’s motion survives the closure. Not available; the computation has not been built.

Whether the layers collide. Not available; the collision test compares pieces rather than panels.

Four things that could not be said at all a short while ago and two that still cannot. That is the honest state, and the two missing ones are the two an engineer would ask first — which is a fair description of how far a piece of mathematics usually is from the application it is about.

Making one, to see the condition #

The parity is worth meeting physically, and a tube takes five minutes.

Fold a corrugation from a strip — a simple zigzag will do, six columns across — and join the two short ends into a tube. Collapse it: it goes, into a flat ring.

Now make the same thing with five columns. The corrugation folds exactly as before while the strip is flat, and the tube does not collapse. Pushing produces a buckle, and the buckle is a sixth crease appearing where the count needs one.

Two objects, the same pattern, one column apart, and one of them has no flat state.

The thing to notice is that the second tube’s failure is not local. Every vertex of it is an ordinary corrugation vertex, folding exactly as its counterparts do on the strip. The refusal is a property of going all the way round, and no amount of inspecting the pattern near the buckle explains it.

That is the whole of what a closed sheet adds, in the hand, and it takes one strip of paper more than most demonstrations in this subject.

Which of the four sheets a designer is on #

A short guide, since the four have been used throughout and only two are objects.

The rectangle. What is drawn, cut, scored and handled. Every crease pattern in the literature. Four edges.

The tube. The product. Two edges. Half the free letters, a parity condition, and the same vertices.

The other tube. Joining the other pair of edges, which for a corrugation means closing it along its courses rather than across them. A real object — a ring of bellows — and a different sheet from the first with different counts.

The torus. Not manufacturable, included because the comparison needs an end of the scale.

A designer works on the first and ships the second. Everything in this collection until recently was about the first, and the properties being reported were the first’s.

Why the mathematics stayed flat #

The gap between what is studied and what is built has an explanation and it is not oversight.

Paper is flat. The subject’s method is folding paper, and paper arrives in flat sheets. A mathematician working on folding works on a flat sheet because that is what is in front of them.

The formats are flat. No crease-pattern format has a field for an identification, so a closed sheet cannot be recorded, exchanged or checked by any of the standard tools.

The theorems are flat. Every result in the subject is proved about a disc, correctly, and there was no reason to state the hypothesis while no counter-example existed.

The manufacturing joins last. The flat pattern is the working object all the way through design and production, and the tube exists only at the end.

Four independent reasons pointing the same way, which is why the gap survived. None of them is a mistake and together they made the closed sheet invisible.

What it would take to close the gap #

Three things, in increasing order of difficulty.

Record the identification. A pair of boundary edges with a direction, alongside the crease pattern. Trivial to specify, absent from every format.

Check the parity. An addition, per loop, before anything is cut. Available now.

Compute the mechanism’s loop closure. Whether the rigid motion survives the join, which is a real computation that has not been done and is the one engineers most need.

The first two are free. The third is the interesting one and it is where the mathematics could tell an engineer something they do not already know by building the thing.

A note on what engineers know #

The essay has been careful to say that the parity rule is already known in practice, and that deserves emphasis rather than being buried.

People who make folded tubes know that facet counts should be even. They know that Miura tubes deploy in one motion and Kresling tubes twist. They know where to put a seam and how much a fitting costs in packing ratio.

None of that came from the mathematics of crease patterns. It came from building things, and it is correct.

What the mathematics can offer is the reason, which converts a rule of thumb into a check — and a check can be applied to a pattern nobody has built yet, which a rule of thumb cannot. That is a modest contribution and it is the honest size of what this phase produced for the applications.

The order the checks should go in #

For somebody designing a folded tube, the cheap checks first.

Count the creases running round the tube, including the seam if it will fold. Even, or there is no flat state. An addition.

Check the vertices. Ordinary rigid-folding analysis, unchanged by the join.

Check the loop closure of the mechanism. Whether the motion survives being closed — satisfied by the Miura, exploited by the Kresling, and not computed in this collection.

Check the thickness. Layers accumulate and a closed sheet’s layers accumulate round a loop with nowhere to go.

Only the first is new and only the first is free.

Where the seam sits in all this #

A last practical note, because the seam is the one thing the mathematics treats as nothing and the manufacturing treats as everything.

Mathematically a seam is an identification with no thickness, no width and no stiffness. Every count here treats it that way.

In a real tube it is a weld, an adhesive line, a fold-over or a stitched joint. It is thicker, stiffer, and it is where the object fails. Where to put it is a manufacturing decision made late, and it interacts with the mathematics in exactly one place: whether the seam becomes a fold, which flips the parity.

That interaction is worth flagging to anybody in the loop, because the person deciding where the seam goes is usually not the person who checked the parity.

The sentence #

The subject’s objects are flat sheets and its applications are closed ones, and the gap has been invisible because a closed sheet was not something the mathematics could describe.

It can now, the description is a rectangle plus an identification, and the first consequence of having it is a condition that half the crease counts fail.