Where curved creases meet

A curved crease forces the paper either side of it into a shape nobody creased, and the flat-folding theorems say nothing about it — they are statements about straight creases meeting at a point.

That is true of a crease along its length. It is not true where two of them cross.

Zoom in and the curves go away #

The whole argument is one observation, made properly.

Take a point where several curved creases meet and look at a smaller and smaller neighbourhood of it. Each crease is a smooth curve, so in a small enough neighbourhood it is indistinguishable from its tangent ray. In the limit, the picture is a set of straight rays from a point — a straight-crease vertex.

Everything the subject knows about straight-crease vertices therefore applies to the tangents. The sectors between consecutive tangent directions are the sectors of that vertex; the conditions on those sectors are the ordinary conditions; and whether the vertex can behave the way a design wants it to is decided there.

The curvature does not appear. How sharply each crease bends after it leaves the vertex is a fact about the crease and not about the junction, and the figure above says so by drawing the same vertex three times: straight creases, gently curved creases, sharply curved creases, and identical sectors in all three.

What the sectors have to do #

Two conditions and they are not equally interesting.

The sectors between the tangent directions sum to a full turn. That is automatic — they are the angles around a point in a flat sheet — and it is the same observation as developability being the statement that folding creates no curvature: a vertex drawn inside an intact sheet has sectors summing to 360° by construction, and there is nothing to check.

The condition with content is Kawasaki’s: the alternating sums of the sectors must each come to a straight angle. In the vertex drawn here the sectors are 60°, 120°, 120° and 60°, and the alternating sums are 180° and 180°.

The refusal #

The clean way to state a condition is to break it.

Turn one tangent by seven degrees and the sectors become 53°, 120°, 120° and 67°. The alternating sums are now 173° and 187°, out by fourteen degrees, and the vertex has stopped satisfying the condition.

The important part is what happens when the curvatures are then varied. Nothing. All three panels of the refusal above have the same broken sums, because the curvature was never in the calculation. A designer who has a vertex that does not work cannot fix it by bending the creases differently. The only available repair is to move a tangent, which means changing where a crease leaves the vertex, which changes the design.

The generator asserts both halves of that. Given tangents satisfying the condition it refuses to draw if the alternating sum is anything but zero; given a disturbed tangent it refuses to draw if the condition still holds, since a disturbance that changed nothing would mean the check was measuring something else.

The discrete skeleton #

The consequence for design is a change of view rather than a technique.

A curved-crease pattern presents itself as a smooth object: continuous curves, continuous surfaces, a continuum of choices. Its combinatorial content is finite. The creases meet at finitely many points, each of those is a vertex with a degree and a sector list, and the conditions live there and nowhere else.

Between the vertices the curves are free in a way a straight crease never is — free to be any curve at all, subject to the smooth relations that govern a single curved fold along its length. That is where the design’s character comes from, and it is also where none of the conditions in this essay apply.

So a curved-crease design has two entirely separate difficulties. Getting the junctions right is a discrete problem with the ordinary vertex conditions in it, and it is finite. Getting the curves right is a smooth problem governed by the relations between a crease’s curvature in the sheet and its curvature in space, and it is where the surfaces actually come from.

The first is the one that can be checked by adding up angles. The second cannot.

That division has a practical shape. A designer sketching a curved-crease pattern is making two quite different kinds of decision without usually noticing the difference: where the creases meet, which is a finite set of choices with hard constraints on it, and how they travel between meetings, which is a continuum with soft constraints. The first can be got wrong in a way that is provably fatal and cheap to check. The second can only be got wrong in ways that show up as a surface nobody wanted.

The advice that follows is unglamorous and worth stating: fix the junctions first. Choose the vertices, choose the tangent directions at each of them, check the sums, and only then decide what the curves do in between. Doing it the other way round — drawing beautiful curves and finding out at the crossings — is how a curved-crease pattern ends up with a junction that no amount of redrawing will save.

The reduction does not depend on how much of each curve is drawn, which is easiest to see by drawing more of it.

Why the count of creases still matters #

One consequence of the tangent reduction is worth drawing out, because it is the sort of thing that looks like an accident and is not.

A vertex where curved creases meet has a degree: the number of creases arriving there. And every parity argument that applies to a straight-crease vertex applies to it, because the argument is about the tangents.

So a curved-crease vertex with an odd number of creases is in the same position as a straight one: Kawasaki’s alternating sum is not even defined for an odd count, and the panels around it cannot be two-coloured. Neither fact has anything to do with curvature, and both survive the zoom.

That is a strong practical constraint on curved-crease design and it is easy to violate by accident. Curves are drawn freehand, they meet where they happen to meet, and there is nothing in the drawing to prevent three curves from crossing at a point. In a straight-line pattern that mistake is conspicuous — a Y of three creases looks wrong to anybody who knows the subject. In a spray of smooth arcs it looks like part of the design.

The other things the vertex decides #

The tangents settle more than one question and it is worth separating them, since only one of them is about angles.

The sector angles are the tangents’ business, and Kawasaki’s condition is a statement about them.

The assignment — which creases fold toward the reader and which away — is a separate choice constrained by the same reduction: Maekawa’s difference of two applies to the tangent vertex, so the counts of the two kinds around a curved-crease vertex differ by two, exactly as they would if the creases were straight.

And the smallest sector carries the big-little-big condition, so a curved crease bounding a strictly smallest tangent sector has to differ in assignment from the crease on the other side of it.

All three transfer for the same reason and none of them mentions curvature. The vertex is a straight-crease vertex; the curves are what happens next.

There is one thing that does not transfer, and it is the important exception. A straight-crease vertex that satisfies all four conditions has a flat folded state. A curved-crease vertex that satisfies all four does not, because the creases leaving it are curved and a curved crease has no flat folded state at all. The conditions come across; the conclusion they usually support does not.

Which theorem was checked, and how #

What this essay computes is the tangent geometry, and the check is that it fails when it should.

Sector angles are computed from the tangent directions by sorting them around the vertex and differencing — a computation that knows nothing about the curves, which is the point. The sum is checked against a full turn and the alternating sums against each other.

The refusal is checked in both directions. A set of tangents that does not satisfy the condition is refused before anything is drawn, which is the ordinary discipline: a figure claiming a foldable vertex must not be able to draw an unfoldable one. And a disturbed vertex that still satisfied the condition would also be refused, which is the less obvious half — it is the check that the disturbance is doing something.

What is not checked is the folded state, because this repository does not compute one for a curved-crease vertex. Everything above is a necessary condition on the tangents, and a necessary condition is a filter and not a certificate.

And the arithmetic is worth watching on a vertex chosen to have no symmetry in it, because a symmetrical vertex can satisfy the condition for reasons that are about the symmetry rather than about the sums.

How small the neighbourhood actually is #

The reduction is exact in the limit and the drawing is not a limit, so it is worth putting a number on the region in which the tangent picture is honest.

A curve of curvature $\kappa$ has turned through an angle of about $\kappa s$ after arc length $s$. That turning is exactly the error in treating the crease as its tangent, so keeping the sector angles right to within $\varepsilon$ requires

$$s \lesssim \varepsilon / \kappa.$$

Put a designer’s numbers in. A crease drawn at a radius of a fifth of the sheet has $\kappa = 5$ per sheet-width, and holding the sectors to one degree — $0.0175$ radians — needs $s \leq 0.0035$ of the sheet. On a 150 mm square that is half a millimetre, which is thinner than the line the crease is drawn with.

Which is why the condition cannot be checked by eye #

That is the practical content of “exact in the limit”. The region where the picture shows the vertex the condition is about is smaller than the ink, so a designer looking at a junction is never looking at the object being tested.

A straight-crease vertex can be checked by inspection: the sectors on the page are the sectors in the condition. A curved-crease vertex cannot, and the failure mode is specific — a junction whose curves leave at plausible angles and whose tangents do not satisfy the sums looks correct at every magnification a page can carry.

So the check has to be arithmetic on the tangent directions, computed from the curves rather than measured off them. That is the same conclusion the vertex-count constraint reached from the other side: a Y of three straight creases is conspicuous and a Y of three arcs is not, and in both cases what has gone wrong is invisible precisely because curvature hides it.

Where the argument stops #

Three limits, and the first is the one most likely to be over-read.

The condition is necessary and nothing more. A vertex whose tangents satisfy it may still be undesignable for reasons no vertex condition sees, exactly as a crease pattern that passes every local condition may still not fold. Satisfying the tangents is where a curved-crease design starts, not where it finishes.

The result is local. It says what happens in a neighbourhood of the vertex small enough for the curves to look straight, and the arithmetic above says how small that neighbourhood has to be. A sharply curved crease departs from its tangent quickly, and the region in which the argument is a good approximation to the drawing shrinks in proportion to its curvature. The condition is exact in the limit and approximate in the picture.

The folded state near a curved vertex is not flat. Away from the vertex, a curved crease forces the paper into a curved surface, so there is no flat folded state to be talking about at all. What the tangent conditions describe is the vertex’s own local structure, and the surfaces that leave it are governed by the relation between a crease’s curvature and the fold angle.

Everything above assumes a crease is a curve rather than a band, which is one of the standing idealisations — and the one a curved fold strains hardest, because the paper either side of a curved crease has to bend along its length rather than merely turn about it.

Who established it, and when #

The modern characterisation of curved creases and their rulings is recent — the twenty-first century — and it is the work that turned curved-crease folding from a practice into a subject with theorems in it. The tangent condition at a vertex belongs to that literature, where it appears as a necessary condition among several rather than as a headline.

The practice is a great deal older than the theory, and this is one of the clearest cases of it on this site. Curved-crease models were being made in the 1920s and 1930s, and the best of them predate by decades any ability to compute the surfaces they consist of. Designers working by hand knew perfectly well that a junction either worked or did not, and had no way to say why.

What the mathematics added was not the ability to make better objects. It was the ability to say, before folding anything, that a proposed junction cannot work — which is a smaller contribution than it sounds and a genuinely useful one, because the alternative was to find out with paper.

It is worth noticing how the two halves of the subject arrived in the opposite order from the straight-crease case. There, the theorems came first in the sense that matters: Kawasaki and Maekawa were written down and then used, and the design methods that followed were built on them. With curved creases the objects came first by half a century, and the theory has been catching up ever since — which is a pattern this site has recorded more than once.

There is a reason for the asymmetry and it is not that anybody was slow. A straight-crease vertex is a finite object and its conditions are arithmetic; a curved fold is a surface, and describing one requires the differential geometry of ruled surfaces plus a way of computing them, which is a much later toolkit. The tangent condition in this essay is the piece of the curved theory that could have been stated at any time, because it is the piece that reduces to the straight case.

That is probably why it was not stated. A result that says this hard problem behaves like the easy one, in one particular place is easy to regard as not worth writing down, and its value only appears when somebody has a design that fails and no idea why.

There is a material cost to any of this that the geometry does not reach: a real crease is a band rather than a curve, and a curved crease is a band whose width varies along its length. Nothing above models it, and nothing above needs to — the tangent condition is a statement about directions at a point, and a point has no width.

What a checker for this would look like #

It is worth describing what would have to be built to check a curved-crease pattern properly, because the gap between that and what exists here is the honest measure of the field’s state.

The vertex half is easy and this repository does it: take the tangent directions, sort them, difference them, add up the alternating sums. It is the same arithmetic the straight-crease checker performs and it does not care that the creases curve.

The rest is not easy. A full checker would have to compute the folded surface — which means solving for the ruling directions along every crease, propagating them across each panel, and detecting where rulings converge or cross, since a surface whose rulings meet has a singular edge the paper cannot reach. It would then have to check the panels against each other for interpenetration, which is the curved analogue of a layer-ordering problem and is no easier for being smooth.

None of that exists here, and this site’s crease patterns are straight for that reason rather than by preference. The verification is what makes a pattern publishable on this site, and the verification for curved creases is a substantially larger program than the one for straight ones.

So the material field’s curved-crease essays are all about single creases and single vertices, where exact statements are available. Where a curved-crease design becomes a design — several creases, several junctions, a surface — this repository can state the necessary conditions and cannot certify anything, and it says so rather than drawing a pattern it has not checked.

Where the ladder goes next #

Upward, into the smooth half. The relation between a curved crease’s curvature in the sheet and its curvature in space is one number and governs everything between the vertices, which is most of a curved-crease design.

And sideways, into the ordinary vertex conditions this essay has been borrowing. Everything applied here to tangents is what applies to straight creases at a point, and the borrowing works because a vertex does not know whether its creases are about to curve.