A crease carries no curvature

There is a natural way to think about a crease that is completely wrong, and it is worth dismantling because the correct version explains something this site has been asserting since its first phase without ever putting it this way.

The wrong version: a crease is a line where the surface turns sharply, so it is a line of enormous curvature, and a sheet with two hundred creases in it is a strongly curved object.

The right version: a fold changes no distance measured along the paper. Curvature is determined by distances measured along the surface. So a folded sheet has exactly the curvature it had before it was folded, which is zero, everywhere, including along every crease and at every vertex.

The theorem, used rather than proved #

Gauss’s Theorema Egregium says curvature is intrinsic: computable from the metric alone, and therefore unchanged by any deformation that does not change distances along the surface. That theorem belongs to cartographic-projection.com in this fleet, and this site holds a standing licence to use it and not re-derive it.

Folding is exactly such a deformation. The definition of a fold on this site is a motion that preserves every distance measured along the paper — no stretching, no compression, no cuts. Whatever else it does, it cannot change the curvature.

A flat sheet has curvature zero at every point. So does the same sheet folded into a crane, a Miura, or anything else, at every point of it. That is not an approximation and not a modelling choice; it follows from the definition of what folding is.

What developability is really saying #

The site has been checking this the whole time without describing it in these terms, which is the finding this essay exists to record.

The developability test takes an interior vertex, sums the sector angles between consecutive creases, and refuses the pattern if the total is not 360°. It has been described here as the condition that the vertex lies in a flat sheet — which is true, and is a statement about the drawing rather than about the geometry.

The geometric statement is sharper. The angle deficit at a vertex of a piecewise-flat surface is 2π minus the sum of the angles around it, and it is the curvature concentrated at that point. Zero deficit is zero curvature. So the first of the four conditions every pattern on this site has passed is the assertion that the fold has not created curvature anywhere.

Reading it that way changes what the condition is for. Described as “the vertex lies inside a flat sheet”, developability sounds like bookkeeping — a check that the drawing is a drawing of paper. Described as “the deficit is zero”, it is the reason the other three conditions are the only ones needed: Kawasaki, Maekawa and big-little-big are all statements about a vertex whose curvature is already known to be zero, and none of them would mean what they mean at a vertex with a deficit.

It also explains why developability is the condition that cannot be made to fail by a figure drawn on a flat sheet. The complexity phase found this the hard way, needing a witness for each of the four tests and being unable to construct one for the first: a vertex drawn inside an intact flat sheet has sectors summing to 360° by construction, because they are angles around a point in a plane. The check can only fail on a pattern that was built wrong, never on one that was drawn.

Then where does a folded object’s curvature come from? #

From the way the surface sits in space, and nowhere else. There are two kinds of curvature and only one of them is intrinsic.

A cylinder is curved in the ordinary sense — anybody can see it bends — and its Gauss curvature is zero at every point, which is why a flat sheet rolls into one. The bending is extrinsic: it is visible from outside and undetectable from inside. A creature living on the cylinder’s surface, measuring only distances along it, would find a perfectly flat world.

A folded crane is the same situation with the bending concentrated onto lines. Every point of it, including the points on the creases, has a neighbourhood that could be flattened without stretching anything — because the sheet was flat, and nothing was stretched. The dramatic appearance of a folded model is entirely a fact about where the sheet is, not about what it is.

What it takes to put curvature at a point #

Since folding cannot create curvature, something else has to, and there are exactly two options.

Remove material. Cut a wedge out of a flat disc and join the cut edges, and the angle around the apex is now less than a full turn. The deficit is the wedge angle, the curvature is positive and concentrated entirely at that one point, and the result is a cone. Everywhere except the apex the surface is still flat — a walker who circles the apex at any radius travels less than 2π times the radius, and a walker who never goes near it measures nothing unusual at all.

Insert material. Slit the disc and glue in an extra wedge, and the angle around the point exceeds a full turn. The deficit is negative, the curvature is negative, and the result is a saddle that cannot be flattened.

Both are cuts, and this site’s founding rule is one sheet with no cuts. That rule is normally presented as an arbitrary constraint that turns out to be generative, and it now has a sharper reading: the no-cuts rule is the reason nothing on this site has curvature. An origamist working from an uncut square is working entirely within a flat metric and always will be.

Growth is the third option, and plants have it #

Growth gets curvature without cutting anything, which is the whole reason this field exists.

A growing sheet does not remove or add material at a point; it changes the distances between the material it already has. The curvature that produces is computable from the growth field, it can be either sign depending on where the growth is concentrated, and it appears without any surgery.

So a plant has an option a folder does not. That is a genuine asymmetry between the two halves of this site’s subject, and it explains a fact that would otherwise be a coincidence: the shapes plants make routinely — domes, saddles, cups, frills — are exactly the shapes an uncut sheet cannot reach.

The sphere, which is the case everybody meets first #

The cleanest consequence is one most people have run into without recognising what it was.

A sphere has positive curvature at every point. A flat sheet has zero. Folding preserves curvature. Therefore no amount of folding an uncut flat sheet produces any patch of sphere, ever — not approximately in the sense of getting closer with more creases, but not at all in the sense that every point of the folded object still has a flat neighbourhood.

A folded approximation to a sphere is a polyhedron: flat faces meeting at edges and vertices, and at every vertex the angles still sum to 360° if the sheet was uncut. What makes it look spherical is that the faces are small and the fold angles are many. What makes it not a sphere is that every one of its points is intrinsically flat.

This is one of the four things about paper that are not true arriving from the geometric side, and it is the reason the three ways round the restriction all cost something. Gores pay in seams, which is a cut. Wet-folding pays in strain, which is a stretch. Curved creases pay nothing and reach only what a curved crease can reach.

Where a folder does get curvature, and what it costs #

There is one route by which a folder acquires genuine intrinsic curvature without cutting, and it is worth naming because it is exactly the mechanism a growing sheet uses.

Wet-folding changes the distances between points of the sheet. That is precisely what growth does, at a hundredth of the rate and by a completely different mechanism, and it produces the same kind of result: a metric that is no longer flat and a surface that will not lie down.

So the honest taxonomy has three entries rather than two. Folding preserves the metric. Cutting changes the topology. Stretching — whether by a damp thumb or by a year of growth — changes the metric, and only that third thing produces curvature in an intact sheet. A folder who wants a dome has to become, briefly and slightly, a grower.

How much sphere a strain buys #

The wet-folding figure reports about twenty degrees of sphere for a few per cent of strain, and the relation behind that number is one line and worth having, because it says how the two quantities trade.

Take a cap of a sphere of radius $R$, out to an angular radius $\theta$ from its pole. A geodesic circle at that radius has circumference $2\pi R \sin\theta$, and flattening the cap onto a plane demands a circle of circumference $2\pi R\theta$ instead. So the rim has to stretch by a factor of $\theta/\sin\theta$, which for a modest cap is $1 + \theta^2/6$.

Setting that excess equal to the available strain $\varepsilon$ gives

$$\theta = \sqrt{6\varepsilon}.$$

Two per cent of strain buys twenty degrees; three per cent buys twenty-four; five per cent buys thirty-one. The figure’s number falls straight out of it, and the derivation needs nothing about paper beyond a strain the material will take.

Which is a square root, and that is the problem #

The exponent is the useful half. The reachable cap grows as the square root of the strain, so doubling what the material will take buys only forty per cent more angle — and the return keeps shrinking.

Run it in the direction a maker would want. A hemisphere is ninety degrees, which is $\pi/2$ radians, and $\theta^2/6$ at that value is forty-one per cent. So a sheet would have to stretch by two fifths of its own length to be flattened out of a hemisphere, which is a fabric rather than a paper, and no amount of care with a damp brush gets near it.

That bounds the technique cleanly. Wet-folding is not a way of reaching spherical shapes with more patience; it is a way of reaching a cap whose angular size is fixed by the material and which is small for any material a folder uses. The dome on a wet-folded model is real, it is genuinely curved in the intrinsic sense, and it is a shallow cap of twenty or thirty degrees.

It also puts a number on the asymmetry with growth. A plant’s tissue accumulates strain over weeks and reaches arbitrary values, so the square root that caps a folder at thirty degrees is no constraint at all on a leaf — which is why the shapes plants make routinely are exactly the ones a folder can only approach.

Matching them, which is the experiment #

If both a cone and a growth field carry curvature, the natural question is whether they can be made to carry the same amount, and what happens when they do.

The total curvature of a surface is the integral of the Gauss curvature over its area. For a cone it is entirely at the apex and equals the wedge angle. For a grown disc it is spread out, and the generator finds the growth parameter that matches a given cone by bisection.

The two surfaces then agree on one number and on nothing else whatsoever.

The cone is flat at every point but one. It has a place where something happens and an infinite number of places where nothing does. Its curvature is a spike.

The grown disc is curved at every point and has no special place at all. The generator asserts this too: it refuses to draw if the grown disc has any point of zero curvature, because the comparison’s whole content is that the two distributions have nothing in common.

What the total does and does not tell a reader #

This is the transferable lesson and it applies well outside biology.

Total curvature is one number extracted from a whole function. Two surfaces with the same total can be as different as a spike and a smear, and a great deal of loose reasoning about curved things treats the total as though it characterised the shape.

Where the total is informative is when something forces it. There is a classical result relating the total curvature of a closed surface to a topological invariant — that result is cartographic-projection.com’s in this fleet, ruled on explicitly, and no figure here computes it. What is used above is nothing more than the divergence theorem applied to a Laplacian, which is why the module says so in its own docstring and why the argument stops where it does.

The reason for the care is not bureaucratic. An essay that reached for the topological identity here would be borrowing a much stronger statement to make a much weaker point, and the weaker point — a spike and a smear can share a total — needs no theorem at all.

What the picture cannot show #

The cone in the figures is drawn flat on the page, as a disc with a wedge taken out of it. That is the pattern, not the object: the object is what the pattern becomes when the two cut edges are brought together, and it is not drawn.

That is the standing caution of this site — a crease pattern flat on the page is not the folded object — arriving in a slightly different form, and it matters here because the flat drawing is where the deficit is legible. The wedge is visible as a wedge exactly while the cone is not yet a cone.

The grown disc is drawn as concentric rings, which is a diagram of a metric rather than a picture of a surface. The rings are where the sheet’s material rings are, and their spacing is not drawn to the grown metric’s scale. No embedding was solved for anywhere in this essay.

The idealisation, named #

Real paper does have a crease radius, and at the scale of that radius the surface genuinely is curved in the ordinary sense. That radius has a price and it is measurable.

It is still not intrinsic curvature. A crease with a radius is a cylinder of small radius joining two flat regions, and a cylinder has zero Gauss curvature. The sheet remains flattenable in principle at every point; what the radius costs is length, not curvature.

The one place a real fold does acquire intrinsic curvature is where the paper is damaged — crushed fibres at a vertex where many layers meet, or the deliberate stretching of a wet-folded surface. Those are departures from the model rather than consequences of it, and the model says so.

Where this ladder goes next #

This closes the growth-as-metric ladder’s first four rungs, and the field turns from metrics to patterns.

The next essays take folds that organisms actually use and ask what they achieve: a corrugation that packs a leaf, and the container that turns out to be doing the choosing. Those are ordinary crease patterns, verified the ordinary way, and they belong to the half of this field where folding rather than growing is the mechanism.

The connection worth carrying forward is the one this essay establishes. A leaf that corrugates and a leaf that ruffles are doing categorically different things — one is folding, with a flat metric throughout, and the other is growing, with a metric that forbids flatness. They can look similar from a distance and no amount of looking will separate them. Flattening the specimen will, and that is a better test than any picture.