Getting thickness round a corner

Every crease pattern describes a surface of no thickness. Everything anybody builds has some, and the gap between the two is not a tolerance to be swallowed — it is the central engineering problem of the subject, and it has a literature.

What that literature contains is a set of techniques. What it mostly lacks is a way of comparing them, because the obvious comparison — which one looks most like the ideal fold — is the least useful one available.

The control case, which does not move #

Start with the arrangement nobody uses, because it explains why the others exist.

Put the hinge on the mid-plane of a solid panel, where the zero-thickness pattern says the crease is. Now rotate. The material on either side of the axis occupies the same place the instant the rotation begins, and the two panels interfere immediately.

The travel is not small. It is zero. A solid panel hinged on its own mid-plane cannot rotate at all, and the contact test reports exactly that.

That is a stronger statement than “thickness makes folding harder”, and it is the right starting point. The problem is not that a thick fold is approximate. It is that the obvious construction does not work by any amount, so something has to be given up before there is a mechanism at all.

Four things to give up #

The techniques divide by what they sacrifice, and once sorted that way they stop looking like variations on a theme.

Give up material. Taper the panel to a wedge at the crease. The two wedge faces then have room to swing until they meet, and the travel is a straight function of the wedge angle: a thirty-five degree taper gives a hundred and ten degrees of fold, a fifty degree taper gives eighty, and reaching a full flat fold would need a knife edge. The cost is that the panel is thinnest exactly where it is worked hardest.

Give up symmetry. Put the hinge on one face rather than in the middle. The panels then swing away from the material behind them and the fold closes fully — in one direction only. The cost is that the mountain-valley assignment is built into the hardware: a hinge on the top face folds one way and cannot fold the other, so the pattern’s assignment has to be decided before the thing is manufactured.

Give up contact. Separate the panels and join them with a flexible membrane, as an airbag or a stent effectively does. Travel is nearly complete and the panels never touch. The cost is a gap, and with it stiffness, a seal, and a folded pattern that is no longer geometrically the one on paper.

Give up coplanarity. Offset alternate panels out of the reference plane so the hinge axes all lie on one surface. The flat state is then a step rather than a plane, which for a deployable that has to stow flat is a real loss.

Those are four; the review literature lists several more, including rolling contacts, doubled hinges and deliberately strained joints. The taxonomy is the point rather than the completeness.

Which theorem was checked, and how #

The travel figures are measured, and the measurement is worth describing because it is the sort of number that usually gets asserted.

Each technique is represented as a cross-sectional outline with a hinge point. The generator rotates the two outlines toward one another about that point, and tests at each angle whether the two convex polygons overlap, using a separating-axis test — for every edge of either polygon, project both onto its normal and check whether the projections are disjoint. A separating direction means no contact.

The travel is then found by bisection on the first angle at which no separating direction exists. Sixty bisection steps put it well inside a hundredth of a degree.

Two consequences of doing it that way rather than by formula. The chamfer’s travel comes out as exactly a hundred and eighty degrees minus twice the taper angle, which is a clean relation nobody put in — it was discovered by running the test at several taper angles and noticing. And the surface-hinge case revealed a bug: the first version rotated every technique the same way and reported that a surface hinge cannot move, which is the opposite of what it is for. A hinge on one face folds away from the material behind it, and the direction is part of the technique.

What the test cannot show is anything about strength, fatigue, sealing or cost, which are what actually decide between these in practice.

Where the chamfer’s relation comes from #

The relation found by running the test is worth deriving, because two lines settle it and the derivation carries a consequence the measurement does not.

Each panel is tapered so that its face at the crease makes an angle $a$ with the mid-plane. Folding rotates each panel toward the other, and the two tapered faces meet when the total rotation has taken up the angle each of them started away from the closed position. That is $a$ apiece, so

$$\theta = 180° - 2a$$

exactly, with no thickness in it anywhere. The travel a chamfer buys depends only on its angle, which is why the relation came out clean and why it holds at every panel thickness.

And why the last twenty degrees are the expensive ones #

The same geometry prices the sacrifice, and the price is not linear.

A taper of angle $a$ on a panel of thickness $t$ removes material over a length $(t/2)/\tan a$ from the crease. At the thirty-five degrees that buys 110° of travel, that weakened strip is $0.71t$ long. At the five degrees needed for 170°, it is $5.7t$.

So travel is linear in the taper angle and the weakened region goes as its reciprocal. Going from 110° to 170° of fold is a further sixty degrees of travel bought with an eight-fold longer stretch of thinned panel — and since bending stiffness goes as the cube of thickness, the strip that is doing the least work structurally is also the strip carrying the fold.

Which explains why the chamfer is used where it is used. A pattern needing modest fold angles gets them almost free: forty-five degrees of travel costs a taper of 67.5° and a weakened strip a fifth of a thickness long. A pattern needing to close flat cannot have a chamfer at any price, and that is the same statement as a knife edge is not a panel — arrived at as a limit rather than as an objection.

The trade is not optional #

It is tempting to look for the technique with no cost, and the reason there is not one is structural rather than accidental.

A zero-thickness fold has two properties that a thick fold cannot both keep: the panels meet along a line, and they are free to rotate through a straight angle about it. Meeting along a line means the material is present right up to the axis, and material at the axis is exactly what blocks the rotation.

So every technique breaks one of the two. Tapering removes the material near the axis. A surface hinge moves the axis out of the material. A membrane separates the panels so there is no material at the axis to begin with. An offset panel moves the panels rather than the axis.

There is no fifth option because there are only two properties to break. That is why the taxonomy by sacrifice is the right one: it is not a convenient way to organise a list, it is the structure of the problem.

What happens at a vertex #

Everything above is one fold in cross-section, and a crease pattern is not made of isolated folds.

At a vertex, four or more thick creases meet, and the techniques have to be compatible with one another there. A surface hinge folding one way meets a surface hinge folding the other, and the two hinges are on opposite faces of the panel between them, so that panel has to be thick enough to carry both.

That is where the three-to-one sign split stops being an abstract fact about assignments and becomes a manufacturing constraint. Three creases on one face and one on the other means three of the four hinges are coplanar and the fourth is not, and the panel geometry has to absorb the difference.

Tachi’s offset-panel method solves it systematically: panels are shifted perpendicular to the reference surface by amounts computed from the assignment, so that every hinge axis lands where the ideal pattern says. The shifts accumulate across a pattern, which is why a thick-panel Miura is a stepped surface rather than a flat one.

The numbers, side by side #

Putting the measured travels together makes the shape of the trade visible in a way the individual figures do not.

A solid panel hinged on its mid-plane reaches zero degrees. A panel tapered at thirty-five degrees reaches a hundred and ten. A membrane fold with a gap of thirty percent of the panel thickness reaches a hundred and seven. A surface hinge reaches a hundred and eighty, which is a complete fold.

Read down that list and the surface hinge looks like the answer, which is why it is the technique most often reached for. What the number does not carry is that it is a hundred and eighty degrees in one direction and zero in the other, so a pattern needing both assignments needs hinges on both faces and a panel thick enough to carry them.

The taper’s number is the one with the cleanest structure: a hundred and eighty degrees minus twice the taper angle, exactly. So a taper is a dial, and a designer can buy any travel short of complete by giving up a proportional amount of material. Nothing else on the list is adjustable in that way.

Which is the useful summary. One technique gives everything at the cost of a permanent choice; one gives a continuously adjustable amount at the cost of strength; one gives almost everything at the cost of contact; and the ideal gives nothing at all.

Where the thickness comes from #

It is worth asking why the panels are thick in the first place, because the answer determines which technique is available.

Sometimes thickness is structural: the panel has to carry a load, and stiffness scales with the cube of depth, so it is thick because it must be. A solar array’s substrate is in this category, and so is anything architectural. Tapering such a panel removes material exactly where the moment is highest, which is the worst possible place, so the taper is usually unavailable.

Sometimes thickness is functional: the panel contains something. A photovoltaic laminate, a battery, an electronics layer, an insulating core. Here the depth cannot be reduced at the crease because the contents are there, and the hinge has to be moved instead.

Sometimes it is manufacturing: the sheet is thick because that is the stock available or because thinner cannot be handled. This is the friendliest case and the one where a membrane fold is usually right.

So the taxonomy of techniques maps onto a taxonomy of reasons, and a technique chosen without asking which reason applies is a technique chosen by appearance. The deployables that get built are almost all in the first two categories, which is why the offset and hinge-shift methods dominate the engineering literature and the taper appears mostly in demonstrations.

Where the model stops #

Two dimensions. A cross-section is a slice through one crease. It cannot show what happens where creases meet, which is where the real difficulty is.

Convex outlines. The contact test assumes convex cross-sections, which is true of these four and is not true of a rolling-contact hinge.

Rigid bodies. Panels are treated as undeformable. A real panel of any span flexes, and the flex is sometimes what makes an interference tolerable.

No hinge hardware. The hinge is a point in these figures. A real one is a pin, a living hinge, a flexure or a laminate, each with its own thickness, its own travel and its own failure mode.

Static contact. The test asks whether the panels overlap at a given angle. It does not ask whether they can reach that angle without colliding on the way, which for a multi-panel assembly is a different and harder question.

Symmetric folding only. Both panels are rotated by the same amount about the shared axis. A real fold is often driven from one side, and the asymmetric case has a different contact angle.

Zero thickness is still the design language. Every pattern in the field is drawn as if the sheet had none, including the ones that get manufactured, and the thick realisation is produced afterwards from the ideal one.

Nothing about the crease pattern. A technique is chosen per crease, and a crease belongs to a pattern. Whether the per-crease choices are globally consistent is the vertex question, and it is not visible in a cross-section.

The surprise: the thickness changes what folds #

The instinct is that thickness is an implementation detail — the pattern is the design, and thickness is what the manufacturing engineer worries about afterwards.

It is not. Thickness changes which patterns are foldable at all.

A pattern that is rigid-foldable at zero thickness may have no thick realisation, because the accommodation techniques impose their own constraints and those can be inconsistent across a pattern. Conversely — and this is the surprising direction — a pattern that is not rigid-foldable at zero thickness can sometimes be built thick, because the gaps the techniques introduce give the panels room the ideal pattern denied them.

So the ideal theory is neither an upper nor a lower bound on the buildable set. It is a different set, overlapping it, and the relationship between the two is not understood in general.

That is an uncomfortable position for a field whose theory is entirely about zero-thickness sheets, and it is the honest reason the engineering literature on thickness is as large as it is.

Who found it, and when #

The problem is as old as trying to build one of these and the systematic treatment is recent.

Hoberman’s patents from the early 1990s contain the offset-crease idea, arrived at while designing deployable structures rather than while studying folding. Tomohiro Tachi’s offset-panel method appeared in 2011 and is the first general construction: given a rigid-foldable pattern, it produces thick panels that fold the same way.

The taxonomy itself is due to Robert Lang, Kyler Tolman, Erica Crampton, Spencer Magleby and Larry Howell, whose 2018 review collected the techniques and compared them systematically. Before it, the techniques existed as separate papers with no common frame.

It is worth putting beside the other direction the subject has taken. Self-folding sheets attack the same problem from the material end — make the sheet thin enough and flexible enough that thickness stops mattering — and they inherit a different difficulty in exchange. Neither approach removes the thickness; they choose which of its consequences to live with.

That review is the reason this essay can be organised by sacrifice. Somebody had to put the list in one place before the pattern in it became visible.

The pattern is drawn thin and built thick #

It is worth ending on the working relationship between the two descriptions, because it is not the one the vocabulary suggests.

A design starts as a zero-thickness crease pattern. It is checked as one — the vertex conditions, rigid-foldability, the kinematics — and everything the theory has to say is said about that object. Only then is a thickness assigned and a technique chosen.

That order is not laziness. The ideal pattern is where the mathematics is, and there is no theory of thick patterns to design in directly. The thick realisation is a construction applied to a finished design rather than a design method of its own.

The cost is that the two can disagree, and the disagreement is discovered late. A pattern that checks out perfectly can turn out to have no consistent thick realisation, and the discovery happens at the point where somebody tries to build it. That is the ordinary shape of a field whose theory and whose practice describe different objects.

The ladder from here #

Later rungs against this anchor: Tachi’s offset-panel construction, worked through on a Miura. Rolling contacts and doubled hinges, which the contact test here cannot handle. The vertex case, where the techniques have to agree. Thick-panel patterns with no zero-thickness counterpart. Membrane materials and what they cost in stiffness. And the open question of characterising which rigid-foldable patterns admit a thick realisation, which nobody has answered.

The ideal fold has no thickness and every real one does, and the whole of this field is the arithmetic of that difference.