Panels with somewhere to go

The zero-thickness sheet is the assumption the whole subject runs on, and thickness is the assumption’s bill. It arrives the moment anything is manufactured, and every response to it is a way of paying — which is why naming the idealisations is the first thing this subject does.

What is being paid for is specific. A crease in an ideal sheet is a line, and the paper on either side rotates about it. A crease in a thick panel is a hinge somewhere in a slab, and the two slabs meeting there occupy space that the ideal sheet did not — so they collide, at a fold angle well short of flat, and the mechanism jams.

Why symmetric thickening fails #

Take the ideal surface and grow each panel to thickness $t$, half above the surface and half below. It is the obvious construction and it is the one everybody tries first.

Fold it. At a valley crease the two panels’ lower halves rotate toward one another and meet before the fold is complete; at a mountain the upper halves do. The maximum fold angle is set by the thickness and the panel length, and it is nowhere near $180°$.

That is not a limit anybody can design around by choosing better proportions. It is the same argument as the layer count: a stack of $n$ layers is $nt$ thick, the paper has to get round the outside of it, and the getting-round consumes length that the flat pattern did not budget for.

The figure measures the collision rather than describing it: the panels’ overlap is computed by sampling one slab against the other and reporting the fraction of area shared. At the fold shown it is a few per cent, which is enough to jam a mechanism entirely.

The three answers, and what each costs #

There are broadly three families of technique, and they are best understood by what they give up.

Trimming. Cut material away near each crease so the panels have room to close. The kinematics changes: a trimmed panel’s effective hinge line is not quite where the ideal one was, so the folded shape drifts from the pattern’s prediction, and the drift accumulates across a large array.

Shifted hinges. Move the hinge axis off the ideal crease line, to the top surface at a mountain and the bottom at a valley. This closes fully and it changes the linkage: the arc lengths in each vertex’s spherical quadrilateral are no longer the pattern’s sector angles, so the vertex’s motion is a different motion. For a single vertex it can be compensated; for a tessellation the compensations conflict, which is the difficulty at a corner in its sharpest form.

Offset panels. Leave every hinge axis exactly on the ideal crease line, and move the panel off the ideal surface instead — each one entirely to whichever side leaves room. The axes are unmoved, so the linkage is unchanged, so the motion is exactly the ideal one. What is given up is that the panels no longer form a surface: they are slabs at different offsets, and the thing being folded is no longer a thickened sheet but an assembly.

The third is Tomohiro Tachi’s, and its claim is the strongest available: the kinematics is preserved exactly, not approximately. A rigid-foldable pattern remains rigid-foldable with the same one degree of freedom and the same path.

Why leaving the axes alone preserves everything #

The argument is short and worth having, because “preserves the kinematics exactly” is the kind of claim that invites suspicion.

A rigid folding is determined entirely by the hinge axes and the constraint that panels are rigid bodies. The panels’ shapes enter only through collisions; they do not enter the kinematics at all. Two assemblies with the same hinge axes and the same connectivity have the same configuration space, whatever the panels look like.

The offset-panel construction changes only panel shapes and positions, and leaves every axis where it was. So it changes only the collisions, which is exactly the thing it was introduced to fix.

That reasoning also says where the method must fail: it fails when a panel offset far enough to clear one of its neighbours collides with a different one. The figure’s generator finds this if the fold is tightened — at a shallow enough panel angle the offset arrangement overlaps too, by a few per cent, and the assertion inside the figure is only that the offset arrangement overlaps less than the symmetric one, which holds throughout.

The offset at a tighter fold #

The technique is not unconditional, and the figure can be pushed until it says so.

The limit is geometric and can be stated: the offset a panel needs is the sheet thickness, and the room available to move it into is set by how far its neighbours have swung. Below a certain fold angle there is not enough room, and the arrangement that cleared at $100°$ does not clear at $60°$.

That does not undermine the method, and it is worth being clear why. A deployable’s working range is from flat to some packed state, and the packed state is chosen by the designer. If the required packing needs a fold tighter than the offsets allow, the answer is thinner panels or a shallower pattern, not a different technique — every technique meets the same wall, and this one meets it later than the alternatives.

Which theorem was checked, and how #

The overlap is measured, not stated. Each arrangement is built as a set of quadrilateral slabs in cross-section from the ideal centre-line, and the shared area of each neighbouring pair is computed by sampling.

The figure refuses to draw a case in which the offset arrangement overlaps more than the symmetric one, which would mean the picture argued the opposite of its caption. That check is not decorative: the first version of the sampling reported zero overlap everywhere, because it tested points against a fixed winding and one of the two slab families is built on the other side of its centre-line and therefore has the other winding. The figure looked entirely convincing and the numbers under it were nonsense.

What is not checked here is that the offset assembly folds all the way without any collision, at any fold state, for a whole pattern. That is a three-dimensional interference problem across a whole array, and it is what the simulators built for this do.

Closing round a vertex #

The two-dimensional condition deserves a statement even though it cannot be drawn in cross-section, because it is where the technique either works or does not.

Round a degree-four vertex there are four panels and four creases. Each panel is assigned an offset — a signed distance from the ideal surface — and each crease imposes a relation between the offsets of the two panels it joins: they must differ by the sheet thickness, with a sign set by whether that crease is a mountain or a valley.

Going round the vertex, four such relations must be satisfied simultaneously, and the composition must return to the offset it started from. With four creases and three of one letter and one of the other — which is what Maekawa forces — the four increments are $+t, +t, +t, -3t$ or a rotation of it, and the sum is zero. The loop closes.

That is a pleasant result and it is why the technique works on a Miura at all: Maekawa’s theorem, proved for an entirely different reason, is exactly the condition that makes the offsets consistent. A vertex where the letters were two and two would give increments summing to zero as well; a vertex that violated Maekawa could not fold flat in the first place.

What does not follow automatically is the geometry. The offsets closing means the assignment is consistent; it does not mean the offset panels avoid each other, which depends on lengths and angles and is checked numerically.

The steps are three thicknesses, not one #

The closure argument carries a number the essay states and does not read, and the number is the design specification the last section asks for.

Going round a degree-four vertex the four increments are $+t, +t, +t, -3t$. Three of them are the sheet thickness and the fourth is three times it, because Maekawa forces a three-to-one split and three unit steps up against one unit step down would leave $+2t$ rather than closing.

Follow the offsets themselves rather than the increments. Starting at zero they run $0, t, 2t, 3t$ and then back to zero in one jump. The four panels around a vertex sit at four different levels spanning three sheet thicknesses.

Which is the stepped surface, measured #

That is the step the essay names as a specification, and it now has a size: on a Miura built this way, the top face is not stepped by one panel thickness but by three.

For an array meant to present a flat face, three thicknesses is what has to be shimmed, filled or tolerated. For a five-millimetre panel that is fifteen millimetres of relief across every vertex, which is not a finishing detail.

The count generalises. At a vertex of degree $d$ the majority letter appears $d/2 + 1$ times, so the staircase climbs that many unit steps before the single large one brings it back, and the span is

$$\big(\tfrac{d}{2} + 1\big)\,t$$

Three thicknesses at degree four, four at degree six, five at degree eight. A higher-degree vertex is a deeper assembly, in exact proportion, which is a cost of degree that nothing in the zero-thickness world hints at.

And it prices the collision limit #

It also explains why the offset arrangement runs out of room at a tighter fold, which the figure demonstrates and does not account for.

A panel three thicknesses off the surface has to clear neighbours it was never adjacent to. The clearance it needs grows with its offset, and the offsets are fixed by the letters rather than chosen — so the panel at $3t$ is always the one that collides first, and it collides with a panel two creases away rather than with either of its own neighbours.

So the failure at 60° is a failure of the largest step, and the way to postpone it is to reduce the span rather than the thickness: a vertex arrangement needing a smaller majority would be shallower. Maekawa forbids one, so three thicknesses at degree four is a floor rather than a starting point.

What the picture cannot show #

The figure is a cross-section of an accordion, which is the one-dimensional case. Real thick-panel origami is two-dimensional and its difficulty is at the vertices, where several panels meet and the offsets have to be consistent all the way round.

That consistency is the substance of the technique and it cannot be drawn in cross-section. Round a degree-four vertex, the four panels must be assigned offsets such that each adjacent pair clears, and the assignment has to close — go round the vertex and come back to the offset one started with. Whether it can is a condition on the pattern, and for some patterns it fails.

The picture also shows panels of uniform thickness. Tapered panels are a standard refinement, and they blur the line between offsetting and trimming: a taper is a trim that has been chosen so as not to move the axis.

Assembly, and the thing that is actually being built #

Once the panels stop forming a surface, the object has changed category, and it is worth saying what it has changed into.

A thickened sheet is a sheet. An offset-panel assembly is a set of plates held in a fixed relative arrangement by hinges — closer to a linkage with plates on it than to a folded material. Nothing in the physics minds, and two practical things follow.

The first is that the assembly has a front and a back that the pattern did not. Panels sit at different offsets, so the top surface is stepped, and anything mounted on it — a solar cell, a reflector, an antenna element — is stepped too. For an array whose job is to present a flat face, that step is a specification to be met rather than an artefact to be tolerated.

The second is that the assembly can be made of parts that were never one piece. A folded sheet has to be cut from one sheet; an assembly of plates and hinges does not, and that is why nearly every deployed structure that is described as origami is in fact bolted together from separate panels. The pattern survives the translation because it was never about the paper — it was about the hinge axes, which is the same reason the offsets are free to move.

The idealisation underneath #

The hinge is still a line. Real hinges are pins, films or living joints, all of which have width, and a hinge of width $w$ behaves like an axis whose position is uncertain by something of order $w$.

For a membrane hinge — a flexible film bonded across the gap between two panels, which is how most deployables are actually built — the “axis” is not a line at all. The film bends over a region, the effective axis migrates as the fold closes, and the exact-kinematics claim becomes an approximation whose error depends on the film’s stiffness. That is a real limitation and it does not undermine the method: an approximation whose error is a film thickness is very much better than one whose error is a panel thickness.

The second idealisation is that panels are rigid. They are not, and a large array’s panels bend under their own weight, which changes the geometry by more than the offsets do.

What the technique means for the pattern #

There is a consequence for design that is easy to miss and that reframes the whole area.

If the kinematics is preserved exactly, then the thick-panel question and the pattern question separate. A designer can work entirely in the zero-thickness world — choose a slant, check the folding, measure the packing — and only afterwards ask whether the offsets close. That is a much better position than the alternatives put a designer in, where thickness changes the motion and the pattern has to be redesigned around the hardware.

Separation of concerns is a modest-sounding virtue and it is the reason this technique has been adopted rather than admired. It lets the mathematics of rigid folding stay clean while the engineering happens somewhere else, and the two only have to meet once.

The surprising connection #

The offset-panel idea has a close relative in a subject with no folding in it: printed circuit boards.

A multilayer board is an assembly of conducting layers at different offsets from a nominal surface, connected by vias, and the routing problem is to assign each conductor a layer such that no two that must not touch are ever in the same place. The layer assignment is a discrete choice; the geometry is otherwise fixed.

Assigning offsets round a vertex is the same problem. Each panel gets a discrete offset, adjacency imposes constraints, and the question is whether a consistent assignment exists. Where the analogy pays is that the board people have long known such problems are graph colourings and that some graphs need more layers than others — which suggests, correctly, that some crease patterns need more distinct offsets than a single sheet thickness provides.

Who found it, and when #

Tomohiro Tachi introduced the offset-panel technique in 2011, in work on rigid-foldable thick-panel structures; the earlier and more restricted tapered-panel method is his too. The problem itself is old and was worked on by the deployable-structures community for decades before origami vocabulary arrived: Koryo Miura’s own solar array work in the 1980s addressed it with membrane hinges and generous clearances.

The shifted-axis approach has a long engineering pedigree under other names, and the systematic comparison of the families — trimming, shifting, offsetting, tapering — was set out by Lang, Tolman, Crampton, Magleby and Howell in a 2018 review that remains the map of the area.

The ladder from here #

This rung solves the one-dimensional case exactly and states what the two-dimensional case needs. The rung above it is that assignment problem round a vertex: when a consistent set of offsets exists, what a pattern must satisfy for it to, and what to do when it does not.

Below it, getting thickness round a corner sets out the problem at a vertex without solving it, and the sheet has a thickness establishes the accounting that makes all of it necessary.