The pile, not the panel

There are half a dozen ways to build a fold in a panel that has depth. The panels can be chamfered, offset, tapered, rolled, hollowed or joined by a membrane, and the useful way to arrange them is by what each one gives away rather than by what the cross-section looks like.

Every one of those cross-sections has two panels in it. That is not a simplification anybody chose; it is what a crease is — a line with a piece of sheet on each side — and a technique is a statement about a crease.

A folded model does not have two layers anywhere except at its last fold. Where a pattern actually closes, several sheets lie over one another and the crease at the edge of that pile has to take all of them round the corner at once. The number of layers is a property of the pattern, decided by its folded state, and no description of any technique mentions it.

What the pile is, counted rather than reasoned about #

The layer count is not estimated from the crease count or inferred from the shrink. The folded state of each pattern is built, its panels are placed where the folding puts them, and the footprint is sampled on a grid: at every point of it, how many panels lie over that point.

The deepest point of that map is the worst case a technique has to survive, and on the printed patterns it runs from seven to sixty.

Two things in that table are worth more than the maximum, and both of them are about the distribution rather than the peak. Every facet is a layer makes the same point about a crumpled sheet, where the count is not designed at all. The preliminary base is eight layers everywhere — its typical point and its worst point are the same, because a base collapses the whole sheet onto one footprint. And the fold-and-cut triangle is seven at its worst and 1.2 typically, because almost all of its sheet is untouched and the folding happens in a small region.

Those are different engineering problems with the same headline number, and a single figure for “the thickness of the model” describes neither.

The count is a conserved quantity #

The layer count is not free to be designed away, and the reason is an identity this collection has already proved: the folded footprint times the mean number of layers over it is the area of the sheet.

So a pattern that folds to a thirtieth of its area has thirty layers on average, exactly, and there is nothing a technique can do about it — the mean depth is the shrink factor, and the shrink factor is what the pattern was chosen for. A deployable that packs into a thirtieth of its deployed area has thirty layers of panel somewhere, whatever it is made of and however its hinges are built.

That identity is used here as a check rather than as a result: a depth map that mis-sampled would break it, and the layers counted over the footprint agree with the area of the paper to within 1.6 per cent on every pattern on the shelf.

What the pile costs, in millimetres #

A pile has to go round the corner, and the layers do not all travel the same distance.

The model is the plainest one available and is stated rather than assumed: the layers are concentric through the bend, layer j wraps a semicircle of radius j·t where t is the sheet’s thickness, and paper does not stretch. Layer j then spends π·j·t of its own length in the bend that the innermost layer does not spend, so the outermost of k layers falls short of the innermost by

creep(k, t) = π · (k − 1) · t

That is the arithmetic a bookbinder does under the name of page creep. There is no material property in it — no stiffness, no modulus, no energy — and every number below survives with the physics deleted.

On ordinary copier paper at a tenth of a millimetre, two layers give 0.31 mm and the Yoshimura’s sixty give 18.5 mm, which is eleven per cent of the 170-millimetre sheet the pattern is printed at. On cartridge paper it is 39 mm and a quarter of the sheet.

Why the technique cannot absorb it #

The techniques differ in what they give away, and it is worth being precise about which of them the pile actually reaches.

The offset panel moves the hinge axes off the mid-surface so that panels of finite thickness can close, and it costs the least interesting thing there is. Its offset is per crease, and the offset a crease needs is set by the thickness of what is folding at that crease — which is the pile, not the panel.

The tapered panel thins each panel toward the crease. The taper is bounded by the panel’s own depth, so a pile of sixteen needs sixteen tapers or one taper sixteen times as deep, and the second is not a panel any more.

A membrane hinge hands the problem to a different material and moves it rather than solving it: the membrane has to be as long as the creep, which is a length in millimetres and is the number above.

None of them is wrong and none of them is a fix, and the reason is arithmetic rather than mechanical: the creep is proportional to the layer count, the layer count is fixed by the pattern, and a technique acts at a crease.

The number a designer can read off before choosing anything #

The useful form of all of this is a single multiplier, available from the crease pattern alone.

Fold the pattern, count the deepest pile, subtract one. That is how many times the published two-layer allowance a real crease needs, whatever technique is eventually chosen and whatever the panels are made of — because the creep is linear in the layer count and the technique only sets the constant.

The last column is the multiplier on the published two-layer allowance:

A designer choosing between patterns for a deployable usually compares packing ratios. The multiplier is the same comparison seen from the manufacturing end, and by the conservation identity it is not an independent number — a pattern that packs into a sixtieth of its area is the pattern with a fifty-nine times allowance, and no cleverness at the hinge separates them.

How peaked a pattern’s depth can be #

The table carries two columns, the worst pile and the typical one, and the ratio between them is the quantity a designer actually wants — it says whether the allowance has to be paid everywhere or only at one place. That ratio is bounded, and the bound comes out of two facts already established rather than out of any new measurement.

The mean pile is the shrink factor, by conservation. The worst pile cannot exceed the number of panels, since a point can have no more paper over it than the sheet has pieces. So the ratio of worst to typical is at most the panel count divided by the shrink.

Run that against the shelf. The Yoshimura has sixty-five panels and shrinks by sixty, so its worst pile can be at most one and a tenth times its typical one — and the measurement gives exactly one. The preliminary base has eight panels and folds to an eighth of its area, so the bound is exactly one and the pattern has to be eight layers everywhere; there was never any other possibility. The waterbomb tessellation’s bound is 1.7 and it measures 1.03. The fold-and-cut triangle’s bound is 5.8 and it measures 5.8, which means its deepest point carries every panel the sheet has.

So two of the eight are pinned at their bound and two more are pinned by having nowhere to go.

Which removes the relief exactly where it would help #

The consequence is unwelcome and it is not a tendency, it is arithmetic.

A pattern packs hard by having its shrink approach its panel count — that is what packing hard is, since the mean pile is the shrink and the pile cannot exceed the panels. But the bound on peakiness is the panel count over the shrink, so as one approaches the other the bound approaches one. A pattern that packs hard is uniformly deep, necessarily.

There is therefore no pattern that folds to a sixtieth of its area and is sixty deep in one corner and shallow elsewhere. The conservation identity fixes the average at sixty, the panel count caps the worst at sixty-five, and between them there is no room for a distribution with a peak in it.

That closes off the one design escape the profile figure seems to offer. A shallow pattern like the fold-and-cut triangle really does concentrate its difficulty — 1.2 layers typically and seven at one point — so a technique for it has to survive one region and the rest of the sheet is nearly free. A deployable chosen for its packing ratio has no such region. Its allowance is paid at every crease, over the whole sheet, and the multiplier in the table is not a worst case to be designed around but the ordinary condition of every fold in the pattern.

The shelf’s own ordering makes the point. The four patterns with a peakiness above two are the four that shrink least, and the four that shrink most all sit within a few per cent of uniform. Relief exists exactly where it is not needed.

Which theorem was checked, and how #

The pile is counted from the folded state, not from the pattern: the panels are placed, their polygons are tested against a grid over the footprint, and the count at each cell is how many contain it.

The conservation identity is the cross-check. Layers summed over the footprint times the cell area must equal the total area of the panels, and it does — to within 1.6 per cent on every pattern, the residue being the sampling grid rather than the geometry.

The creep model is checked against its own limits. A pile of one layer must creep by nothing, so the model has no constant in it; and a pile twice as deep must cost twice as much, so no term in it is squared. Both are asserted rather than inspected.

A paper whose thickness has not been measured is refused. The four papers named carry measured thicknesses; a fifth name gets an error rather than a guess, which is the difference between a figure about paper and a figure about a plausible number.

Where the model stops #

A concentric bend is a model of a bend. Real layers slip against one another, and slipping is exactly how a book’s pages accommodate creep; the model says how much length has to come from somewhere and says nothing about where. What it will not do is disappear: the paper in the bend is the paper in the bend, whatever slides.

The pile is measured at a point, and a crease is a line. The number reported is the deepest point of the footprint, which may be a corner where a great many panels happen to meet rather than a crease with a pile along it. A technique has to survive the worst crease rather than the worst point, and the two coincide only when the deep region has a fold through it.

Nothing here computes a force. The crease has a radius and what that radius costs in material is a different measurement with a different model behind it. How hard the pile pushes back, whether the panels yield, what the hinge torque becomes — those belong to a subject about materials and structures, and no number here would change if the panels were made of anything else.

And a folded state is not the only thing a pattern has. A shrink measured along each axis separately is two numbers rather than one, and the pile is a third; none of the three determines the others.

And the shrink is a property of the folded state, not of the motion. Everything above is measured at the flat folded state; a deployable that never closes completely has a smaller pile and a proportionally smaller problem, which is a design decision available to anybody willing to give up some of the packing ratio.

What the picture cannot show #

A depth map is drawn as a profile — how much of the footprint is at least k layers deep — and that curve has no geometry in it. Where the deep region is matters enormously to a designer, and the profile is silent about it: sixty layers spread evenly across a Yoshimura and sixty layers concentrated at one corner produce the same curve.

The cross-sections have the opposite problem. They show exactly where the material goes and can only show two or three layers before the drawing becomes a grey band. There is no drawing that shows sixteen panels of finite thickness meeting at one crease and remains a picture rather than a smear, which is part of why the published accounts stop at two.

The idealisation, named #

The sheet has zero thickness, which is the idealisation this entire essay is about — so it is worth saying exactly where it is still being used. The layer count is computed on a zero-thickness folded state: panels lie on one another with nothing between them, and the state exists because the paper has no depth.

That is the right way round. Giving the panels depth would change the folded state, which would change the layer count, which would change the creep — a circular problem that the field solves by iteration and that this measurement deliberately does not attempt. What is reported is the layer count of the ideal folding, which is the input a technique is given.

The generalisation #

A property analysed at a component and a constraint imposed by an assembly are different things, and the second is usually the binding one. Every thick-panel technique is correct about what it does at a crease. The quantity that decides whether the mechanism can be built is a count nobody in that literature has a symbol for, and it comes from the pattern rather than from the joint.

The measurement to take away is a ratio rather than a length: the allowance a real crease needs is the published one multiplied by the pile depth minus one. Seven times for a preliminary base, fifteen for a Miura, thirty-one for a waterbomb tessellation, fifty-nine for a Yoshimura — and the multiplier is available before any technique has been chosen, because it is a property of the crease pattern and can be read off the folded state.

The surprising part, for anybody who has folded both, is that the Miura is not the hard case — the pattern behind every deployable the subject reaches sits in the middle of the shelf. It is the pattern deployables are built from, it packs well, and at sixteen layers it is in the middle of the shelf. The corrugations that pack best are the ones that stack deepest, exactly and by conservation, so the packing ratio a deployable is chosen for is the number that decides how hard it is to build.

Where the ladder goes next #

The obvious next rung is the one this measurement makes askable: where the deep region is. A pile of sixty at one corner is a local problem and a pile of thirty everywhere is a global one, and the map that distinguishes them exists — it is the depth map this rung only summarises.

Beyond that is the iteration the caveat names. Panels with depth fold to a different state from panels without, so the layer count that prices the technique is itself a function of the technique. Nobody here has solved that; the field solves it by building one and measuring.