Paper that stretches on purpose

Every theorem here assumes the sheet does not stretch. That assumption is what makes folding an isometry, what makes a sphere unreachable, and what makes the packing condition true.

Wet-folding breaks it on purpose. The interesting question is not whether that is cheating but how much it buys, and the answer is a number.

What wet-folding is #

The technique is simple to describe and hard to do.

Dampen the paper — a spray, a sponge, sometimes a cloth — until it is pliable but not sodden. Fold while it is damp. The paper accepts curves it would crease, holds shapes it would spring out of, and can be pushed and moulded rather than only folded. Then let it dry, and it sets.

The setting is the point. Paper is a mat of cellulose fibres held together partly by hydrogen bonding between them. Water disrupts those bonds; drying re-forms them in whatever configuration the fibres have been left in. A wet-folded model dries into its shape rather than merely being held there.

Many papers are also sized — treated with a starch or a synthetic that stiffens them — and the sizing softens when wet and re-hardens on drying, which adds to the effect. A wet-folded model of heavy sized paper is a rigid object.

The theorem is not repealed #

The obvious question is whether wet-folding escapes the developability constraint, and it does not.

Gaussian curvature is an intrinsic property: it is determined by distances measured within the surface. A map that preserves those distances preserves the curvature, so a flat sheet mapped isometrically stays flat in that sense. That is Gauss’s theorema egregium, and it belongs to cartographic projection, where it is used to close the map question permanently.

Wet-folding does not repeal it because wet-folding does not preserve distances. The fibres move relative to one another; the sheet genuinely stretches and shrinks; the map is no longer an isometry, so the theorem does not apply.

What is bought, therefore, is exactly as much curvature as the strain allows. The theorem says a sheet with no stretch reaches nothing but developables; it says nothing about a sheet with one percent of stretch, and the answer for that sheet is a calculation rather than a prohibition.

The calculation #

Here is the quantity, and it is short.

To make a spherical cap of angular radius α from a flat disc, the rim has to change length. On the sphere the rim is a circle of circumference proportional to sin α; on the flat sheet it is proportional to α. So the material at the rim must shrink by a factor of sin α / α, which is a strain of 1 − sin α / α.

Everything inside the rim shrinks by less, so the rim is the binding case.

Now put numbers in. For a cap of fourteen degrees the strain is one percent. For twenty-four degrees it is three percent. For thirty-five degrees it is six percent. For a full hemisphere it is thirty-six percent, and nothing made of cellulose supplies that.

So the honest statement about wet-folding is: it moves the reachable set from zero degrees of sphere to about twenty or thirty degrees of sphere, and the boundary beyond that is as hard as it ever was.

One formula, and the reach goes as the square root #

The strain function has a small-angle form that is worth extracting, because it is accurate over the whole range that matters and it makes the answer arithmetic rather than a lookup.

Expanding, $1 - \sin\alpha/\alpha \approx \alpha^2/6$. At fourteen degrees that gives 0.995% against the exact 1.0%; at twenty-four, 2.93% against 2.91%; at thirty-five, 6.22% against 6.14%. The approximation is good to better than two per cent of itself right out to the edge of what any paper supplies.

Inverted, the reach is

$$\alpha \approx \sqrt{6s},$$

which returns 14.0°, 24.3° and 34.4° for one, three and six per cent — the figure’s own annotations, from one square root.

The square root is the discouraging part. Tripling a sheet’s strain buys only $\sqrt 3$ of cap angle, so the gap between dry paper and the best damp paper is a factor of 1.7 in reach and no more.

Which prices wet-folding in a single sentence #

The same expansion gives the number a reader actually wants, because a cap of angular radius $\alpha$ covers a fraction $(1 - \cos\alpha)/2 \approx \alpha^2/4$ of a sphere. Dividing by $s \approx \alpha^2/6$, the $\alpha$ cancels:

$$\frac{\text{sphere covered}}{\text{strain}} \approx \frac{6}{4} = 1.5.$$

A sheet that stretches by a fraction $s$ can cover about $1.5s$ of a sphere — one and a half per cent of a sphere for dry paper’s one per cent, seven and a half for damp paper’s five. The relation is linear, with no threshold and no diminishing return, and it holds across the whole usable range: the ratio measured at the figure’s three annotations is 1.49, 1.49 and 1.48.

That is the honest headline for the technique, and it is smaller than the technique’s reputation. What wet-folding buys geometrically is a fifteenth of a sphere. What it buys in retention and compliance is most of why anybody does it, and neither of those appears anywhere in this arithmetic.

Which theorem was checked, and how #

The curve in the figure is the strain function evaluated directly, and the annotations are found by bisection on it.

For each stated strain limit the generator solves 1 − sin α / α = ε for α, by bisecting on a function it has already established is monotone — and it checks the monotonicity rather than assuming it, throwing if a larger strain allowance ever produced a smaller cap.

That check is not decorative. Bisection on a non-monotone function converges to something, and the something is wrong; asserting the property the method depends on is the difference between a computed number and a plausible one.

What the figure cannot show is how much strain a given paper actually provides. One percent for dry paper and five or six for damp is a range taken from the materials literature and from what folders report, not something measured here. The shape of the curve is exact; the horizontal lines drawn across it are estimates, and they are labelled as the estimates they are.

What the strain buys in practice #

Twenty degrees of sphere sounds like very little and is worth more than it sounds.

The shapes wet-folding is used for are not caps. They are the gentle double curvatures of an animal’s back, the swell of a chest, the roundness of a beak — surfaces whose departure from developable is small but whose absence is immediately visible. A dry-folded animal reads as faceted; a wet-folded one reads as modelled, and the difference is a few percent of strain distributed over a large area.

The second thing it buys is retention. A dry sheet folded into a curve springs back, because the fibres were never released and the elastic energy is still there. A wet-folded curve dries into place with no stored energy, so it stays. That is the property that matters for sculpture and it is not about the geometry at all.

The third is compliance during folding. Damp paper tolerates being pushed through positions it would tear through dry, which lets a folder shape a region rather than crease it. That is a process advantage rather than a geometric one and it may be the largest of the three.

The alternative: take the seam instead #

There is a second way round the theorem, and comparing the two is the most useful thing this essay can do.

Cut the sphere into gores — long tapering strips, each narrow enough that flattening it costs almost nothing — and join them. The strain left in each strip falls with the square of the number of gores: two gores need something like forty percent, four need ten, eight need two and a half, and twelve need about one. That is why a paper globe has twelve of them and why nobody makes one out of two.

The estimate is a leading-order one and its derivation is short: a surface of curvature K, developed along a central line, is out by about K u² / 6 at transverse distance u, and a gore of an n-gore sphere is at most π/n wide either side of its meridian.

So the two routes trade different things. Wet-folding pays in strain and keeps the sheet whole. Gores pay in seams and keep the sheet unstretched. And a seam is a cut, which is the one thing the rest of this site forbids.

That is why the third route — curved creases — is the one this subject actually took. A curved fold reaches doubly-curved appearance from a genuinely developable surface, with no strain and no cut, by putting the curvature into the fold rather than into the material.

How much strain a sheet really has #

The horizontal lines on the figure are estimates, and it is worth saying where such numbers come from and how soft they are.

Paper’s tensile behaviour is measured routinely by the industry that makes it, and the quantity reported is usually strain at break: how far a strip stretches before it tears. For machine-made printing paper that is typically one and a half to three percent along the grain and two to five percent across it, and the anisotropy is not a small effect — the fibres are aligned by the manufacturing process and the sheet is genuinely two different materials in two directions.

Wetting changes it substantially. Wet paper is weaker and much more extensible, with strain at break often doubling or better, because water disrupts the hydrogen bonds between fibres and lets them slide.

What a folder can actually use is less than strain at break, because a sheet strained to its limit is a sheet about to tear at the first stress concentration. Working values are perhaps half of it, and they vary by paper more than by technique.

So the numbers here — one percent dry, three damp, six wet-folded — are the right order and should not be read to two figures. The shape of the curve is exact and the annotations are placed on it by bisection; where they sit horizontally is the soft part.

The other three idealisations #

Four assumptions underlie everything on this site, and it is worth comparing how each behaves under scrutiny, because they are not equally good.

No stretch is the one this essay measures, and it comes out well. The available strain is one to six percent, the reachable departure from developable is correspondingly small, and treating the sheet as inextensible is accurate for nearly every question.

No thickness is much worse. A sheet is a few hundredths of a millimetre thick and a complex model has dozens of layers, so the accumulated depth at a flap’s base is measured in millimetres against a model measured in centimetres. That is a ten-percent effect, not a one-percent one, and it is the reason engineering versions of this subject are mostly about thickness.

Creases are lines is worse still for curved work. A crease has a radius set by the material, and on a tightly curved fold that radius is a substantial fraction of the band it borders.

Perfect memory is the quietest and the most annoying in practice. Paper relaxes; a model opens slightly the moment it is put down; and wet-folding exists partly to defeat exactly this.

Ranked by how much error each introduces, the no-stretch assumption is the best of the four — which is not the ranking most accounts imply, since it is the one with a famous theorem attached and therefore the one that gets discussed.

Where the model stops #

Strain is not a single number. Paper is anisotropic: it stretches more across the grain than along it, and the difference is a factor of two or more. The figure treats it as one number, which is a simplification a papermaker would object to.

Elastic and plastic are not distinguished. Some of the deformation in wet-folding is recoverable and some is permanent, and only the permanent part is retained on drying. The strain limits quoted mix the two.

The cap is the easy case. A spherical cap is the most symmetric double curvature there is. A real wet-folded surface has curvature varying from place to place and the strain field is correspondingly non-uniform, with concentrations wherever the curvature changes fastest — which is where paper actually tears.

Nothing about drying. The shape a model dries into is not exactly the shape it was held in; paper shrinks as it dries, unevenly, and experienced folders compensate. That is craft knowledge and it is not in any model.

The gore estimate is asymptotic. It is the leading term of a series and it is good for narrow strips. For two gores, where the strips are not narrow, it is an over-estimate of the accuracy rather than a reliable number.

The surprise: the idealisation is the useful part #

There is a temptation to treat the no-stretch assumption as a regrettable simplification that a more careful theory would drop. The wet-folding numbers argue the opposite.

Paper’s strain capacity is one percent dry and a few percent damp. Fourteen degrees of sphere against zero is a difference, and against the ninety degrees a hemisphere needs it is a rounding error. For nearly every question anybody asks about a folded sheet, treating the strain as zero is not an approximation — it is accurate to within the precision of the question.

Which means the idealisation is not a weakness of the theory. It is a good model, in the technical sense: it discards a quantity that is genuinely small and it says so. The other three idealisations are worse in this respect — thickness in particular is not small, and pretending otherwise breaks anything that gets built.

So the interesting thing about wet-folding is not that it escapes the constraint. It is that measuring how far it escapes shows how tight the constraint was, and how little was being given up by ignoring it.

What a folder is actually doing #

The account so far is geometric, and it leaves out the part that occupies the hands.

Wet-folding is not folding with water added. The technique is closer to modelling: the sheet is coaxed into curves with the fingers and with tools, held while it dries, and adjusted as it stiffens. Creases still exist, but a great deal of the shaping happens between them, in regions the crease pattern says nothing about.

That is why a wet-folded model cannot be recorded as a crease pattern. The pattern captures where the sheet bends sharply; it has no notation for a region that has been rounded, and none for how much. A photograph of the finished piece and a written note are what the tradition actually passes on.

Which is an odd position for a subject whose central artefact is a diagram. The pattern is the object for straight-crease work, exactly and completely. For wet-folded work it is a partial score, and the performance carries the rest.

Who found it, and when #

The technique has a single well-attested origin, which is rare in this subject.

Akira Yoshizawa developed wet-folding in the 1950s and it is the innovation his reputation rests on more than any individual model. Before him, folded paper was faceted by necessity; his work is full of curved, rounded, animal-shaped forms that nobody had made from paper before, and the technique is why.

Yoshizawa also devised the mountain-valley notation this whole field uses, which makes him responsible both for the way patterns are written down and for the technique that most conspicuously escapes what they can express. A wet-folded model’s shaping cannot be recorded in the diagram language he invented.

The quantitative account came from elsewhere and much later. The mechanics of paper — its strain limits, its anisotropy, the role of hydrogen bonding — belongs to the pulp and paper literature, which has been measuring these properties for industrial reasons since well before Yoshizawa and has never had much to do with folding.

The ladder from here #

Later rungs against this anchor: paper’s stress-strain curve, measured, wet and dry. Anisotropy and grain direction, which every folder knows about and no model here includes. The mechanics of drying, and why models change shape as they dry. Sizing, and what it does. Strain concentration at curvature changes, which is where tearing happens. Backcoating and paper laminates, which are the modern way of getting the same result. And the question of how much of a wet-folded model’s final shape is strain and how much is curved creases the folder made without noticing, which nobody appears to have separated.

The assumption is false, the falseness is measurable, and measuring it turns out to be the best argument for keeping the assumption.