One curve and one number

A curved crease produces a shape nobody creased into it. Fold along a circle drawn on a flat sheet and the paper on both sides is forced into a surface it did not have to be told about — and the interesting question is how much of that surface is decided by the curve, and how much is left to choose.

The answer is that the curve decides almost everything and one number decides the rest. That number is the fold angle, and following it through is what turns curved folding from a surprising phenomenon into a construction.

Curvature, split in two #

A curve drawn on a surface bends in two independent ways, and separating them is the whole of the argument.

Geodesic curvature, $\kappa_g$, is how much the curve bends within the surface — what an ant walking along it would report, having no access to the third dimension. A great circle on a sphere has $\kappa_g = 0$; a line of latitude does not.

Normal curvature, $\kappa_n$, is how much the curve bends because the surface bends, in the direction of the surface’s normal. The ant cannot detect it at all.

The curve’s actual curvature in space combines them:

$$\kappa^2 = \kappa_g^2 + \kappa_n^2.$$

That is not specific to paper; it is the standard decomposition of a curve on a surface, and the two parts are orthogonal, which is why the relation is Pythagorean.

What an isometry cannot change #

Folding paper is an isometry: distances measured along the surface are preserved, because the sheet neither stretches nor tears. That single constraint is where the whole subject comes from.

Geodesic curvature is an intrinsic quantity — it is defined entirely in terms of distances along the surface. So an isometry cannot change it.

The consequence is immediate and it is the point of this essay. The geodesic curvature of a folded crease equals the plain curvature of the crease as drawn on the flat sheet, whatever the fold angle, forever. A crease drawn as a circle of radius $r$ has $\kappa_g = 1/r$ before folding and $\kappa_g = 1/r$ after, at every stage in between, at every fold angle anybody chooses.

Two corollaries follow without further work.

A crease drawn straight on the flat sheet has $\kappa_g = 0$, so it has $\kappa_g = 0$ folded, so it is a straight line in space. Straight creases stay straight — a fact everyone knows and nobody usually derives.

A crease drawn curved has $\kappa_g \neq 0$, so its total curvature $\kappa \geq \kappa_g > 0$ regardless of folding, and the moment it is folded at all, $\kappa_n$ becomes nonzero and the crease leaves the plane. A curved crease cannot be folded flat. That is why a curved fold makes a three-dimensional object rather than a flat one, and it is a two-line argument rather than an observation.

The case that can be computed exactly #

Concentric circles are the case where all three quantities are available in closed form, which makes them the right case to check a general claim against.

Fold along a circle of radius $r$ so that the paper on each side becomes a cone of half-angle $\beta$. In the folded state the circle is a circle of Euclidean radius $r \sin\beta$ — a cone of half-angle $\beta$ maps a slant distance $r$ to a radius $r\sin\beta$. So

$$\kappa = \frac{1}{r\sin\beta}, \qquad \kappa_g = \frac{1}{r}, \qquad \kappa_n = \frac{\cot\beta}{r}.$$

The middle one is fixed by the isometry, as promised. The first grows without bound as $\beta \to 0$ and the fold tightens. The third is the difference, and it is entirely at the disposal of the fold angle.

Substituting confirms $\kappa^2 = \kappa_g^2 + \kappa_n^2$, since $\csc^2 = 1 + \cot^2$. The figure evaluates all three across sixty fold angles and asserts the identity at each one rather than at the ends.

The fold angle is the angle in the triangle #

The three curvatures form a right triangle, and the closed form says which angle of it the fold angle is.

Divide through. $\kappa_g / \kappa = \sin\beta$ and $\kappa_n / \kappa = \cos\beta$, exactly, at every radius and every angle.

So $\beta$ is not merely a parameter that the three quantities happen to depend on — it is the angle of the Pythagorean triangle itself, and $\cos\beta$ is the fraction of the crease’s total bending that has left the surface. A sheet at $\beta = 90°$ is flat and none of it has left; at $\beta = 60°$ exactly half has; as $\beta \to 0$ all of it has, and the intrinsic part becomes a vanishing share of a diverging total.

That is the sharpest available statement of what a folder is adjusting. Working the crease harder does not add curvature to the surface and subtract it from the drawing. It rotates a fixed intrinsic quantity out of the surface, and the total in space grows because a leg of fixed length is being held against a shrinking angle.

Which puts a hard limit on how tight a curved fold can be #

The same closed form, met with one material fact, forbids something the flat-fold theory has no way to forbid.

Paper cannot be bent past a minimum radius $R$ — the crease has a radius and that radius is a property of the sheet rather than of the folder. The surface’s bending along the crease is $\kappa_n = \cot\beta / r$, so the fold can only be tightened while

$$\frac{\cot\beta}{r} \leq \frac{1}{R}, \qquad\text{that is,}\qquad \tan\beta \geq \frac{R}{r}.$$

The tightest achievable fold depends only on the ratio of the drawn radius to the paper’s own. A circle drawn at $r = 100R$ can be closed to within about half a degree of flat. One drawn at $r = 2R$ stops at 27°, and one at $r = R$ at 45° — a fold that is barely a fold.

That is a prohibition with no analogue in the straight-crease theory, where a crease of any length folds to any angle and the sheet’s minimum radius costs a fixed gap rather than a limit. Here it removes configurations outright, and it removes them fastest on the smallest curves.

It also explains a choice every curved-crease sculptor makes and few state: the curves are drawn large. A tight curve is not harder to fold well — it cannot be folded far at all, and the ceiling is set before the paper is touched.

The rulings, and what fixes them #

A developable surface is ruled: through every point runs a straight line lying entirely in the surface. Those rulings are what the paper is bent along, and they are the physical thing a folder is manipulating.

For the concentric-circle fold the rulings are radial, by symmetry. In general they are not, and their directions are the remaining content of the theory.

Huffman’s relation ties them to the crease. At a point of a curved crease, the rulings leaving into the two sheets make angles $\beta_1$ and $\beta_2$ with the crease’s tangent, and those angles are related to the crease’s geodesic and normal curvature — which is to say, to the flat drawing and the fold angle. Given the crease curve and the fold angle at one point, the rulings on both sides follow, and with them the whole surface.

So the counting is: a curved crease has one function’s worth of freedom (the curve, drawn on the flat sheet) plus one number (the fold angle at a single point). Everything else is determined. That is a very small amount of input for a surface, and it is why curved-crease sculpture looks inevitable — because it is.

Folding the same curve two ways #

The cleanest demonstration that the fold angle is the only freedom is to fold one crease at two angles and compare.

Two things change and one does not. The overall scale changes, because every curvature is proportional to $1/r$. The proportion of the total that is normal curvature changes with the fold angle, and that is what a folder is adjusting when the piece is worked. What does not change is the relation among the three, which holds at every radius and every angle.

That is the sense in which the crease curve and the fold angle are separate controls. The curve sets the scale and the shape of the pattern of curvature along the crease; the angle sets how much of it goes out of the surface. A sculptor working curved folds is manipulating exactly these two and nothing else.

Why the rulings have to be straight #

There is a step above that is easy to accept without noticing it is a theorem, and it is worth pausing on.

Paper away from a crease has zero Gaussian curvature. A surface with zero Gaussian curvature that is smooth enough is developable, and a developable surface is ruled: through every point there is a straight line lying in the surface, and the surface’s tangent plane is constant along it.

That is why a curved-crease model has visible straight lines running across it, and why running a finger along one finds it flat. The rulings are not a stylistic feature of the drawing; they are where the paper is not bending, and they carry the whole surface between them.

The rulings also cannot be arbitrary. Where two rulings from the same side of a crease would cross, the surface folds back on itself and the model creases somewhere nobody creased it — which is the standard failure mode of an over-ambitious curved design, and which is predicted entirely by the crease curve.

Which theorem was checked, and how #

The Pythagorean relation is asserted at every one of sixty-one sample angles, to a part in $10^{12}$, before anything is drawn. Asserting it only at the ends would pass for any number of wrong formulas.

The claim that the geodesic curvature is constant is not asserted so much as constructed: it is computed as $1/r$ and drawn as a horizontal line, and the honest description of the figure is that it exhibits the relation on a case where all three quantities are known rather than measuring them off a surface.

That distinction matters, so the surrounding machinery is what carries the weight. The cone construction elsewhere on this site checks the isometry directly, on the first fundamental form, on the actual folded surface, to $5 \times 10^{-7}$ — and the constancy of $\kappa_g$ is a consequence of that isometry rather than an independent assumption.

What the picture cannot show #

The chart shows curvature against fold angle for one crease radius. It cannot show the surface, and the surface is what a reader wants.

More significantly, the chart is of the circular case, and generalising from it is exactly where the intuition fails. For a general curve the fold angle is not constant along the crease — it varies, and it must, because the geometry forces a relation between the crease’s curvature and torsion and the fold angle at each point. The single number in this essay’s title is one number for the concentric-circle fold and one number plus a compatibility condition in general.

The chart also stops at $\beta = 90°$, which is the unfolded sheet, and does not continue past it. Beyond that the two cones would be on the same side, which is a fold in the opposite sense rather than a new regime.

The idealisation underneath #

Zero thickness and perfect inextensibility, and here the second one is doing something specific.

A real curved crease is not a crease at all in the sense the rest of this site uses. It is a region of the paper that has been bent through a large radius, and the bending is distributed rather than concentrated — the crease has a radius, and for a curved fold that radius is deliberately large. The mathematical crease is the limit of a narrow band of high curvature, and the surface either side is genuinely developable only outside that band.

There is a second point that this essay’s own logic forces. If $\kappa_g$ cannot change, then a fold that appears to change it must be doing something other than folding — and wet-folding is exactly that. Damp paper stretches, the map ceases to be an isometry, and geodesic curvature becomes free. That is why a wet-folded surface can be doubly curved and a dry-folded one cannot.

What this rules out #

A relation that fixes so much is most useful for what it forbids, and three prohibitions follow immediately.

No curved crease folds flat. Established above in two lines, and worth restating because it separates this whole field from the rest of the site: every theorem about flat-foldability is about straight creases meeting at points, and none of it applies here. There is no Kawasaki condition for a curved crease, because there is no flat state to have a condition about.

No sphere, however many curved creases are used. A curved fold changes the shape of a developable surface and does not make it non-developable, so the Gaussian curvature stays zero everywhere away from the creases. Approximating a dome with curved folds is approximating it with a faceted developable, exactly as gores do with straight ones.

No independent choice of ruling. A designer cannot draw a crease and then decide how the paper leaves it. The relation fixes the rulings from the curve and the angle, and a model whose rulings run somewhere else is a model whose paper has stretched.

The surprising connection #

Every statement above is a statement about a surface of zero Gaussian curvature, and the reason it applies to paper is the Theorema Egregium: Gaussian curvature is intrinsic, a flat sheet has it zero everywhere, and no isometry can change it. That theorem belongs to a neighbouring subject and is used here rather than re-derived.

The surprise is where else the same relation turns up. A sheet of steel rolled into a cone, a fabric draped over a form, a wooden board bent into a boat’s hull — all are the same constraint with different failure modes, and all use the same split of curvature into a part they cannot change and a part they can.

The boatbuilders got there first and have the best vocabulary for it. A plank that will not take a shape is said to require edge-set, meaning it would have to change its geodesic curvature, which timber will not do; the traditional response is to cut a differently-shaped plank rather than to force it. That is the same reasoning as choosing a different crease curve instead of a different fold angle, arrived at by people fitting planks to hulls several centuries before anybody wrote $\kappa^2 = \kappa_g^2 + \kappa_n^2$.

What a folder is actually doing #

The theory says the surface is determined; the practice is that a curved fold has to be coaxed, and the two are not in conflict.

Determined does not mean easy to reach. The paper’s equilibrium shape is the one that minimises bending energy subject to the crease, and it has to be got there — which means working along the crease, easing the rulings into place, and letting each part of the sheet find the position the geometry has already chosen for it. A curved fold made carelessly buckles instead, adding creases nobody drew.

That is a difference in kind from straight-crease folding, where a fold either is or is not made. Here the crease is made first, lightly, and the shape arrives afterwards as the sheet relaxes. Folders describe it as the paper doing the work, which is a reasonable description of a surface settling into a determined configuration.

The one genuine choice left is the fold angle, and it is chosen by how hard the crease is worked. So the single number in this essay’s title is, in the hand, a matter of pressure.

Who found it, and when #

David Huffman — better known for the coding algorithm — spent decades on curved-crease geometry from the 1970s onward, and the relation between a crease’s curvature and its rulings is his. He published very little of it; most survives in models and in notes, and it has been reconstructed and extended since his death in 1999.

The mathematical treatment was given proper form by Dmitry Fuchs and Serge Tabachnikov in 1999, in a paper on the differential geometry of paper folding that derives the relation between the fold angle, the crease’s curvature and its torsion. Erik Demaine, Martin Demaine and Duks Koschitz have since worked on Huffman’s models and on the reconstruction of his method.

The engineering-side history is older and separate: the geometry of developable surfaces was worked out for sheet metal and for shipbuilding long before anybody applied it to paper, and the folding literature has been slowly rediscovering results that naval architects had by 1900.

The ladder from here #

This rung establishes what is fixed and what is free for a single curved crease. The rung above it takes several: two curved creases on one sheet must agree about the rulings between them, and the compatibility condition is restrictive enough that most pairs of curves cannot both be creased on the same sheet.

Below it, a crease that curves establishes the phenomenon, and the sculptors got there first records that the practice preceded the theory by about forty years.