An angle that turns faster than the crease

Only one side can run out established that a curved fold’s two surfaces have envelope denominators that are exact negatives, so at most one of them can have its rulings converge at any point of the crease. It then found that a crease which never changes the way it bends has one surface bounded everywhere and the other bounded nowhere — the circle, the ellipse and the parabola at every ruling angle tried.

That finding rested on an assumption stated at the time and not relaxed: the ruling angle was constant. The denominator has two terms in it, the crease’s curvature and the rate the angle turns at, and a constant angle sets the second to zero. Everything about which surface runs out was then the sign of the curvature, which is why a crease of one curvature gave one answer along its whole length.

A real fold does not have a constant ruling angle. The angle is set by the crease’s own curvature and by how far the fold is closed, and neither is constant along a designed crease. Letting it turn is the one experiment the earlier finding named and did not run.

The second term, given something to do #

The distance from the crease to where its rulings cross is $\sin\beta$ over a denominator, and the denominator is

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

where $\kappa$ is the crease’s curvature in the flat sheet, $\beta'$ the derivative of the ruling angle along the parameter, and $v$ the speed — both terms rates per unit of arc length, which is the detail that makes them comparable at all.

Crossing to the fold’s other surface negates the whole expression, so the sign of that sum decides which surface is bounded at each point. With $\beta$ constant the sum is $\kappa$, and a crease whose $\kappa$ never changes sign therefore never changes which surface it bounds. With $\beta$ turning, the first term can outweigh the second, and it does so exactly when

$$\left|\frac{\beta'}{v}\right| > |\kappa|$$

which in words is: the ruling angle turns faster than the crease bends.

The threshold, measured #

Running a circular crease with the angle set to $1.4 + a\sin t$ and sweeping $a$ puts the threshold where the algebra says and nowhere else.

At $a = 0$ the outer surface is unbounded along the whole crease, which is the earlier finding. At $a = 0.25$, $0.5$, $0.75$, $0.9$ and $1.00$ it is still unbounded along the whole crease — five rates approaching the threshold from below with nothing happening.

At $a = 1.10$ the outer surface is bounded over 14 per cent of the crease. At $1.25$, twenty per cent. At $1.40$, twenty-five per cent. The change is at one, and one is where the angle’s turning rate equals the crease’s curvature, both measured per unit of arc.

That is a threshold with no fitting in it: the parameterisation was chosen so that the rate and the curvature are the same number, and the sweep confirms that the crossing happens at the value the algebra names rather than near it.

Why the threshold is at one and not near it #

A threshold that lands on a round number usually means the parameterisation was chosen to make it land there, and that is exactly what happened — which is worth saying, because it is what makes the measurement a check rather than a discovery.

The crease is a circle of radius $r$, so its curvature is $1/r$ and its speed is $r$. With the angle running $\beta_0 + a\sin t$, the derivative with respect to $t$ is $a\cos t$ and the rate per unit arc is $a\cos t / r$. The denominator is then

$$\frac{a\cos t}{r} + \frac{1}{r} = \frac{1 + a\cos t}{r}$$

which is negative somewhere exactly when $|a| > 1$, whatever the radius is. So $a$ is the ratio of the two rates by construction, and the threshold being at one is the statement that the two terms are comparable and cross where they are equal.

What the sweep adds is that the geometry agrees with the algebra: the rulings drawn at each rate, intersected with their neighbours, converge or diverge exactly where the sign says. That is the same separation this subject keeps: the closed form is one instrument and intersecting two neighbouring rulings is another, and neither knows the other exists.

What the turning costs before it changes anything #

The column beside the shares is the one a designer would read first, and it moves the whole way.

The bounded surface’s reach falls monotonically as the rate rises: 0.2956 at a rate of zero, 0.2350 at a quarter, 0.1909 at a half, 0.1484 at three quarters, 0.1199 at nine tenths, 0.0993 at one, and 0.0000 at 1.40. Turning the angle costs reach long before it costs the sign, and by the time the sign flips the surface that was bounded has lost two thirds of the paper it had.

That is the same arithmetic read forwards. A larger denominator is a nearer crossing, and turning the angle in the direction that adds to the curvature is a larger denominator everywhere it acts. The sign change is what happens when the added term is large enough to take the sum through zero on the far side of the crease, and by then the near side is very close indeed.

So the two effects are not alternatives. A design that turns its ruling angle appreciably is buying a boundary on the surface that had none and paying for it with the reach of the surface that had one, and the exchange rate is poor: at a rate of 1.4 the outer surface is bounded over a quarter of the crease at a reach of 0.725, while the inner surface’s reach has gone to nothing.

What a designed ruling field would look like #

There is a reading of the turning rate that makes it less abstract, and it is worth putting beside the numbers.

The rulings leaving a crease are the straight lines the surface is flat along. A constant ruling angle means they leave at the same angle everywhere, so they fan out at a rate set entirely by the crease’s own bending — the picture where the rulings run out draws, where the crossing distance is $\sin\beta$ times the tightest radius and nothing else about the curve matters.

A turning angle means the fan is being opened or closed along the crease independently of how the crease bends. Turning it with the bending crowds the rulings on the inner side and the crossing comes in; turning it against the bending spreads them, and past the threshold they stop converging on that side altogether and start converging on the other. The threshold is the rate at which the turning exactly undoes the bending, so at $a = 1$ there is a point of the crease where the rulings are locally parallel and the surface has no boundary in either direction there.

That point is worth naming because it is what moves as the rate passes one: below the threshold it does not exist, at the threshold it is a single point, and above it there are two of them with a bounded stretch of the far surface between.

Where a real fold’s angle comes from #

The rate is a parameter here and in a fold it is not, which is the gap between this measurement and a design.

The ruling angle on a curved fold is determined: given the crease’s curvature in the sheet and the fold angle at each point, the angle the rulings leave at follows. So $\beta'$ is not free — it is what the crease’s shape and the fold’s closure between them produce, and a designer chooses those rather than choosing $\beta$.

What this measurement establishes is that the threshold exists and where it is in the quantity that matters. What it does not establish is which designs reach it. A crease whose curvature varies slowly and whose fold angle is nearly constant will have a small $\beta'$ and behave like the constant-angle case; a crease with a tight region, or one folded much further at one end than the other, will not.

That is a question about the relation between the fold angle and the ruling angle, which is standard and is not implemented here — the same gap the earlier finding named, now narrowed to a single number to compute rather than a phenomenon to look for.

The quantity that decides it is worth seeing held still before it is allowed to move. On a circle with a constant ruling angle the sign is the curvature’s, and the curvature of a circle is one number: the sum never approaches zero, never crosses it, and is the same distance from it at every point of the crease. There is nothing in that picture for a turning angle to compete with until the turning is put in.

And the crossing itself is worth looking at closely rather than across the whole range, because a threshold read off nine widely spaced rates is a threshold located to within the spacing. Taken at six rates from 0.90 to 1.20, nothing happens at 0.95 and nothing at 1.00, and at 1.05 the outer surface is bounded — so the change sits in the interval the algebra names and not merely near it.

The reach at the threshold, and the number it is not #

One coincidence in the table is worth disarming, because it invites a reading it does not support.

At a rate of exactly one the bounded surface’s reach is 0.0993, and at a rate of zero it is 0.2956 — very nearly three times as much. A reader could take that for a law: the reach falls to a third at the threshold. It does not. The ratio depends on the constant the angle is centred on, because the numerator of the reach is $\sin\beta$ and the angle at the point where the denominator is smallest is $\beta_0$ plus or minus the rate. Shift the centre and both numbers move, and not together.

What does not depend on the centre is where the threshold is, because the numerator plays no part in the sign. That is the division worth keeping: the denominator decides which surface is bounded and the numerator decides how far it reaches, and only the first is a clean statement.

The same division explains the last row. At a rate of 1.40 the inner reach is 0.0000, which is not the threshold arriving — the threshold was four rows earlier — but the numerator and denominator conspiring at one point of the crease, where the angle has come round to near zero and the sine with it.

What the sweep assumes #

One crease and one shape of variation. A circle, with the angle running a constant plus a rate times a sine. The sine is periodic, which a closed crease requires, and it is otherwise arbitrary: a different profile with the same maximum rate would put the threshold in the same place and distribute the shares differently.

The angle stays a ruling angle. The constant is chosen so that $\beta$ never leaves the interval where a ruling exists; at a rate of 1.4 it runs from 0 to 2.8 radians, which is close to the limit. A larger rate would take it past, and the numbers there would be about the parameterisation rather than about folds.

The reach is the distance to the first crossing, as it has been throughout, and it is measured on whichever surface the sign selects. A reach of 0.0000 at the highest rate is the crossing arriving at the crease itself, which is a surface of no width rather than a failure of the computation.

And it is a plane curve with a curvature in the flat sheet. That is the quantity the rulings are drawn against and it is not the crease’s curvature in space, which is a different number and the one a photograph shows. The sculptors got there first is the reminder that these objects were made for decades before the ruling lines could be computed at all, and what was being made was the second curvature rather than the first.

What the threshold does not decide #

It does not say a turning angle is bad. A boundary on the surface that had none may be exactly what a design wants — a fold with a defined edge on both sides is a fold that can be cut to shape — and what this gives is the price rather than a verdict.

It does not cover a crease that inflects as well. Both terms can change sign along one crease, and the sum can then cross zero more than twice. The sharing would be finer than the two stretches seen here and nothing computes it — the wave in only one side can run out is the inflecting case with the second term set to zero, and the two effects together have not been put on one crease.

It says nothing about the surface between two creases. The gap between two curves is the bound that actually limits a pleated design, and it is computed from reaches on one side. With both sides bounded and both reaches varying, that bound would have to be recomputed — and a concentric pleat, where every crease bends the same way, is exactly the case where a turning angle would change which crease of a pair binds.

And it assumes a surface at all. A curve has no panels is the negative result that a curved crease admits no finite list of flat pieces, so everything here is about a smooth object that a machine cannot specify by coordinates. A turning ruling angle makes that worse rather than better: it is a second function along the crease.

And it is a bound on the surface, not on the fold. One curve and one number established that folding cannot change a crease’s curvature within the surface and that everything a curved fold adds is controlled by one number; nothing here contradicts it, and nothing here says the folded shape is any different past the threshold — only that the flat sheet’s ruled region is.

Still open: the rate a design actually has #

The threshold is a statement about a quantity nobody has computed for a real design, and computing it is one implementation away.

Every curved-crease design has a turning rate at every point, fixed by its crease curvature and its fold angle, and the question the sweep makes askable is simply whether that rate ever exceeds the curvature. It is a comparison between two functions along a curve, both of which a design determines. A design that never exceeds it has one unbounded surface and behaves exactly as the constant-angle account says; one that exceeds it somewhere has a boundary on both sides, at places the designer did not put there.

The second thing the sweep suggests is a design variable nobody is using. If the rate is what decides the boundary, and the rate follows from the fold angle, then folding a crease unevenly along its length is a way of deciding where each surface stops. That is a control with no name in this subject, and whether it is usable depends entirely on how much of the rate a design can vary without changing the shape it is trying to make.

Sideways from here, the shape of this result is worth carrying to any formula with a sum in its denominator. Two terms added together, one of which every measurement has set to zero, is an assumption disguised as an equation — and the way to find it is to ask which term the experiments have varied.

The habit worth carrying is narrower and more useful. When a closed form’s denominator has two rates in it, they are comparable, and the ratio of them is a dimensionless number with a threshold at one. That threshold is worth looking for before any sweep is run, because it says what to sweep and where the interesting part will be.