Only one side can run out

Where the rulings run out found a curved fold’s own edge: the rulings leaving a curved crease are not parallel, so they cross, and past the first crossing two points of the paper have been sent to one point of space and there is no surface. The distance to that crossing is $\sin\beta$ times the crease’s own tightest radius of curvature, for every crease curve tried and at every ruling angle — one of the cleanest laws this subject has.

It ended by naming what it had not done. A curved fold has two surfaces, one each way from the crease, each with its own rulings, and everything measured was one of them. Whether the other one has a boundary, whether it is the same distance away, and whether one always runs out before the other were left as a measurement nobody had made.

The measurement turns out to be a sign, and the answer is sharper than the question allowed for.

The denominator, and what crossing the crease does to it #

The distance from the crease to where its rulings cross is

$$\frac{\sin\beta}{\beta' + \kappa}$$

with $\beta$ the angle the rulings leave at, $\beta'$ its rate of change along the crease, and $\kappa$ the crease’s curvature — both rates per unit of arc length, which is a detail that is invisible on a circle and matters everywhere else.

The second surface is reached by flipping the normal. That does two things at once and they are the whole argument. It negates the curvature, because a crease that bends toward one side bends away from the other. And it reverses the sense in which the ruling angle is measured, so it negates $\beta'$ as well. The second surface’s denominator is the first’s with a minus sign in front of it.

A negative denominator means the rulings diverge rather than converging, and there is no crossing to reach — the surface is unbounded in that direction, at that point. So at any one point of any curved crease, at most one of the two denominators is positive, and:

At most one of a curved fold’s two surfaces can run out there. Measured at 241 points along four crease curves at four ruling angles each, no point has both sides bounded, on any of the sixteen.

That is a stronger statement than one always runs out first, which is how the question was posed. There is no race. The two surfaces bound disjoint stretches of the same crease, and the stretches are decided pointwise by a sign.

Three creases with an unbounded surface #

Once the statement is pointwise, its consequence for a whole crease is arithmetic. If the sign never changes along the crease, one surface is bounded everywhere and the other is bounded nowhere.

For a constant ruling angle $\beta' = 0$, so the sign is the sign of the curvature — and a crease that never changes the way it bends has a curvature of one sign along its whole length. The circle, the ellipse and the parabola are all of that kind, and at every ruling angle tried the outer surface of each of them has no crossing anywhere. Not a distant one. None.

The wave is the exception in the set, and it is the exception for the reason that makes it a wave: it inflects. Its curvature changes sign twice over the range drawn, so each surface is bounded over the stretch where the crease turns toward it — exactly half of the sampled points each, on a wave symmetric enough to split them evenly.

The numbers on the bounded stretches are the same law as before. Wherever a surface is bounded, its reach is $\sin\beta$ times the tightest radius — 0.3894 at a ruling angle of 0.4 radians, 0.6442 at 0.7, 0.8415 at 1.0, 0.9636 at 1.3, on both sides of the wave and on the inner side of the other three, to four figures. The law is a law about a surface and it holds on whichever surface the sign selects.

The two surfaces of one fold #

It is worth setting out what the second surface is, because the subject’s own vocabulary makes it easy to lose.

A curved crease divides the flat sheet into two regions and each of them folds into its own developable surface. The two are not copies of one another: each is ruled by its own family of straight lines leaving the crease, and the two families leave at equal angles on opposite sides, which is what a fold means. The sculptors got there first is the history of people making these objects decades before anybody could compute the ruling lines that determine them, and the objects are two-sided in exactly this way — a curved-crease sculpture is a pair of surfaces joined along a curve, and the eye reads it as one thing.

A curve has no panels is the negative result that makes the pair irreducible. A rigid folding is a finite list of flat pieces joined along lines, and a curved crease admits no such list: refining one leaves a total kink that does not fall, because it is a constant of the curve. So the two surfaces cannot be approximated apart and then joined; each is a genuinely smooth object and each has its own boundary or does not.

What an unbounded surface is not #

It would be easy to read unbounded as unlimited, and the reading is wrong in two ways that matter to anybody drawing a design.

It is unbounded by this and by nothing else. The crossing of rulings is one way a developable surface can stop existing and it is not the only one. The paper has an edge. Adjacent creases have their own rulings, and the paper between two curved creases has to be reachable from both, which is a much tighter limit than either crease’s own reach and is what actually bounds a concentric pleat. A single crease with an unbounded outer surface in an infinite sheet is a statement about the mathematics of one crease, and every real design has neighbours.

And it is unbounded in the flat sheet, not in space. The rulings that never converge in the pattern are rulings whose images in the folded surface also never converge, so the surface exists arbitrarily far out — but it is a surface that keeps curving away, and how much of it is useful is a different question with a different answer. One curve and one number settled what a curved fold adds: the curvature a crease has within the surface is unchanged by folding, and everything the fold produces is curvature out of the surface, controlled by one number. An unbounded outer surface is one whose out-of-surface curvature keeps going; the paper is there and the shape is not necessarily wanted.

What the shares add up to #

The last column of the census is the share of the crease each side bounds, and its arithmetic is the argument in one line. On the circle, the ellipse and the parabola it reads a hundred per cent and nothing: one surface bounded along the whole crease, the other along none of it. On the wave it reads fifty and fifty.

Those pairs sum to one, every time, and that is not a coincidence about these four curves — it is the sign statement counted. Every point of every crease is bounded on exactly one side unless the denominator is exactly zero, and a denominator of exactly zero is a crease that is momentarily straight, which happens at isolated points and contributes nothing to a share.

The wave’s even split is a different kind of fact and is about the wave rather than about folds. A sine has as much crease bending one way as the other, so its two surfaces get half each. A crease that inflected once near one end would split its crease ninety to ten, and the surface with the short share would be bounded only over a stub — which is the shape of design a reader should expect to be hard to reason about, because its two halves obey different rules.

Why this was not found earlier #

The measurement was always available. The closed form was already computed with a guard: a non-positive denominator returned infinity, with the reasoning written beside it that a negative root is a crossing behind the crease and is on the other sheet. That sentence is correct and it was the whole of the gap. The other sheet is the fold’s second surface, the crossing behind the crease is that surface’s envelope, and calling it somebody else’s problem left half of every fold unmeasured.

It is worth noticing what kind of blind spot that is. Nothing was wrong: every number reported about the reach was right about the surface it was about. Nothing was unchecked: the reach law was put against a second, independent computation that intersects neighbouring rulings and knows nothing of the formula. What was missing was a question, and the guard had answered it in a comment rather than leaving it open. A quantity returned as infinity because the sign was inconvenient is a quantity nobody will think to ask about again.

The number that is the same on both sides #

One quantity survives the whole of this unchanged, and it is the one established before any of it. Wherever a surface is bounded, its reach is the sine of the ruling angle times the crease’s tightest radius of curvature — and that holds on the wave’s outer stretch exactly as it holds on the circle’s inner one, to four figures at all four angles measured.

So the law is not about a side. It is about a surface that is bounded, and it says the same thing wherever the sign lets it apply. That is worth stating because the obvious guess would have been the other way round: two surfaces of one fold, curving away from each other, might reasonably have had different reaches, and a designer could be forgiven for expecting the concave side to run out sooner than the convex side by some factor depending on the fold.

There is no factor. There is a sign, and on whichever side it selects, the same sine.

What the measurement rests on #

Four crease curves and a constant ruling angle. The circle, the ellipse, the parabola and a wave, at four angles each. The constant angle is what makes the sign the sign of the curvature; a design in which $\beta$ varies has $\beta'$ in the denominator too, and a large enough negative $\beta'$ would bound the outer surface of a crease that never inflects. Nothing here tries one.

The crease is a plane curve. A curved crease in a real fold is a space curve once the paper leaves the table, and the quantity in the denominator is its curvature in the flat sheet, which is what the rulings are drawn in. That is the right quantity for this question and it is not the crease’s curvature in space.

Sampling at 241 points is not a proof. That the two regions never overlap follows from the denominators being negatives of one another, which is algebra; the sampling checks the algebra rather than establishing the claim. A sign that changed between two samples would be missed, which is why the inflection count is read off the curve rather than off the samples.

And the surface is a developable of zero thickness. Real paper has a crease with a radius, and what that radius does to a curved fold is a different subject from what the rulings do. Nothing here is a statement about paper.

What a designer gets from it #

Two things, and the second is the useful one.

The first is a warning about the bound. A reach computed for a curved crease is the reach of one of its surfaces, and quoting it as the fold’s reach is quoting one of two numbers. If the design’s material lies on the other side, the number does not apply to it — and on a crease that does not inflect, no number of this kind applies to it at all.

The second is a design rule with a sign in it. Put the part of the design that needs to extend a long way from its crease on the side the crease bends away from. That side is the one whose rulings diverge, and it is unbounded by crossings however far the design runs. The side the crease bends toward is where the $\sin\beta$ bound bites, and it is where a design should keep its material close.

That rule inverts on an inflecting crease, once per inflection, which is a real constraint on a design that wants a long reach along a wavy fold: the far side alternates. Where curved creases meet found the subject’s hard conditions living at the finitely many points where creases cross; the inflections are a second finite set of special points, of a different kind, and nothing had marked them as special before.

Still open: the angle that varies #

The whole of this rests on $\beta' = 0$, and that is the one assumption a real fold does not satisfy.

A curved fold’s ruling angle is set by the crease’s curvature and by the fold angle, and neither is constant along a designed crease. So $\beta'$ is a function, it sits beside $\kappa$ in the denominator, and it can carry the sign on its own. A crease that never inflects can still have both surfaces bounded, in different places, if the ruling angle turns fast enough against the curvature — and where the sign changes is then a property of the fold rather than of the crease’s shape, which is a much more useful thing for a designer to know.

Computing it needs one thing this file does not have: the relation between the fold angle and the ruling angle, which is a standard result and has not been implemented here. With it, the sign becomes a function of two things a designer chooses, and the question which side of my crease is unbounded becomes answerable at every point of a design instead of at every point of a curve.

Sideways from here, the same trick applies to the bound between two creases. That bound is computed from the reach of the inner crease of a pair, which is the reach on one side; if the pair’s creases bend the same way, the second surface of each is unbounded and the binding constraint is the one already computed, but a pleat whose creases alternate in curvature has a different accounting, and concentric pleating is the case where they do not.

There is also a cheap thing to measure that would sharpen the design rule. The unbounded side is unbounded by crossings, and what actually limits it is the sheet — so the useful quantity is not a reach but how far the rulings on that side have travelled by the time they leave the paper, which is a property of the crease’s position in the sheet rather than of its shape. The fold a machine can make is the essay about what can be specified rather than drawn, and a curved crease is a function where a straight one is a pair of coordinates; a bound that depends on where the crease sits in its sheet is a bound a specification has to carry with it.

The habit worth carrying is about guards that answer questions. A branch that returns a default for an inconvenient sign has decided something, and the decision is invisible because the computation runs and the numbers are right. The way to find them is to read every early return and ask what the caller would have learned if the computation had continued.