Crossing the curve moves no start

Every tuck placement worked out for a gathered cap has been drawn under one rule that nobody chose. The paper reads strain placed the starts of straight tucks to keep the strain the paper takes as small as possible, and found them spreading outward from the centre rather than crowding toward the rim. Its broken line of hiding — how much length the tucks remove from each circle of the flat disc — was made to agree with the sphere’s own requirement at every start, and so it hid too much everywhere between them. Between two starts the paper could only stretch.

That essay named the restriction as the thing its picture could not show and predicted what lifting it would do: a line free to cross the sphere’s curve could split each stretch’s error evenly above and below, should roughly halve every figure, and might move the first start. The calculation turns out to be much cleaner than the prediction. The halving is exact, the starts do not move at all, and the reason is a single line of algebra that also says how to treat a paper that is stiffer one way than the other.

Hiding too little is compression #

The model is the one the earlier tuck essays used, going back to what a flat sheet can become. A flat disc is gathered into a spherical cap of unit radius, and a circle of flat radius $s$ has to lose length to become the shorter circle of the sphere. The length it must hide is

$$H(s) = 2\pi\,(s - \sin s),$$

and straight tucks started at a set of radii hide length along a broken line $L(s)$: straight between starts, bending upward at each start as new tucks begin.

Where $L$ is above $H$, the tucks remove more length than the sphere needs removed, the circle that remains is too short to reach round the sphere, and the paper there must stretch. Where $L$ is below $H$, the tucks remove too little, the remaining circle is too long, and the paper must take up the excess in compression. Either way the strain round the circle is the error divided by the circle’s length, and it has a sign:

$$\varepsilon(s) = \frac{L(s) - H(s)}{2\pi s}, \qquad \varepsilon > 0 \text{ stretch}, \quad \varepsilon < 0 \text{ compression}.$$

The earlier placements took the absolute value and found the line never went negative. That was not because negative strain is bad. It was because a broken line that meets a convex curve at its corners lies above it everywhere else, so the touching rule could only ever produce stretch. It asked paper to do the one thing it does least willingly and never asked it to do the other.

Lowering the line by a cone #

The whole argument rests on one observation about that formula. Take the broken line and subtract from it a quantity proportional to the radius, $2\pi\delta s$ — a straight line through the origin, which in tuck terms is the hiding a cone does, the same fraction $\delta$ of every circle. The signed strain becomes

$$\frac{L(s) - 2\pi\delta s - H(s)}{2\pi s} = \varepsilon(s) - \delta.$$

Every signed strain moves down by exactly $\delta$, everywhere, and nothing else changes. The starts are where they were, the slope of the hiding still bends at them, and on every stretch the tucks still hide at a constant rate — a rate lower by $2\pi\delta$ than before, which is the same reduction on every stretch.

So a crossing line is available from any touching line for the price of one number, and the best one is easy to name. The touching line’s signed strain runs from zero, at the starts, to its worst value $E$ somewhere in the stretches. Lower it by $E/2$ and it runs from $-E/2$ to $+E/2$: the paper takes half the old worst strain as stretch and half as compression, and the worst strain of either kind is exactly half what it was. On the hemisphere with four starts placed for strain, 1.81 per cent becomes 0.90 either way.

The picture of the lowered line shows what that means for the tucks. At each start the line now sits below the sphere’s curve, so every new ring of tucks begins hiding a little less than the sphere asks — the paper just outside a start is slightly compressed. Through the middle of each stretch it rises above the curve and the paper is stretched. At the next start it is below again. The error alternates in sign with equal peaks, compression at every start and stretch at the middle of every stretch, and that alternation is the signature of a best approximation.

Why nothing better exists #

A lowered touching line is one way to cross the curve. The claim that it is the best way needs an argument, and the argument is short.

Any continuous broken line from the centre, whatever its corners and values, can be written as a lowered line through a shifted curve: choose $\delta$, and ask the line to pass through $H(s) - 2\pi\delta s$ at its corners instead of through $H$. Shifting $H$ by a straight line through the origin changes neither its curvature nor the way a chord misses it, so the chord of the shifted curve between two starts sits above the shifted curve by exactly what the chord of $H$ sits above $H$. The crossing problem is the touching problem on a curve that bends the same way, and a paper that can stretch by $\varepsilon_s$ and compress by $\varepsilon_c$ is asking for a touching line whose stretch never exceeds $\varepsilon_s + \varepsilon_c$, lowered by $\varepsilon_c$.

That was checked the other way round rather than taken on trust. The crossing line was built independently — each stretch run as far as it could go with its corners on the lowered curve, and its largest stretch and largest compression both checked against their limits on every piece — and for two, four, eight and sixteen starts on a hemisphere it lands on the touching placement’s starts to within a fiftieth of a per cent of the radius, at exactly half the worst strain: 7.80 per cent becoming 3.90 at two starts, 1.81 becoming 0.90 at four, 0.43 becoming 0.22 at eight. A free search that moved every corner and every value of the crossing line at random, thirty thousand times from that starting point, found nothing better at one to eight starts.

So the earlier essay’s forecast was half right. Every figure halves, exactly rather than roughly. And the first start does not move, because nothing moves: the placement that was best for a line that had to touch is the placement that is best for a line that may cross.

Where the first start does move #

There is one reading of “the first start moves” that is true, and it is the reading a maker would care about. The earlier comparisons fixed the number of starts and asked how little strain they could leave. A maker holds the material fixed instead — this paper gives two per cent and no more — and asks how few starts it needs and where the first must go.

For tucks from the centre, the touching line’s worst strain sits at the apex of the cone they make, and it equals the excess where they stop, $f(s_1) = 1 - \sin s_1 / s_1$. Where a ring of divisions belongs derived the ring rule — a new ring of divisions each time the excess reaches another multiple of the material’s give — and the strain essay found that its first ring and the first strain-bounded start were the same radius by identity: both are where $f$ reaches $\varepsilon$.

Allowed to cross, the cone from the centre may hide a smaller fraction of each circle than the excess at its end, taking stretch at the apex and compression at its edge. With equal gives each way the cone can run until $f(s_1) = 2\varepsilon$. That is the ring rule’s second ring, exactly, at every give on the chart: 0.31 of the radius for a paper giving two per cent, against the touching line’s 0.22; 0.45 at four per cent against 0.31; 0.72 at ten per cent against 0.50. A coincidence that the strain essay explained as an identity has a second identity beside it, one ring further out.

The count falls with it. A hemisphere in a paper giving one per cent each way needs four starts where the touching line needs six, three where it needs four at two per cent, and two where it needs three at four to six per cent. Because the error in a stretch falls as the square of its length, doubling the allowance lengthens every stretch by about $\sqrt{2}$, so the count falls by a factor of about 0.7 before rounding — which on a shallow cap and a generous paper is the difference between a single cone and a cone with a ring of tucks beyond it: a 60° cap in a paper giving ten per cent each way needs one start allowed to cross and two touching.

Only the sum of the two gives #

The halving assumed that paper gives as readily in compression as in stretch, which is almost certainly not true of any real sheet, and it is not needed. A paper that stretches by $\varepsilon_s$ and compresses by $\varepsilon_c$ needs a touching line whose worst stretch is under the sum, lowered by $\varepsilon_c$.

For a paper that stretches one per cent, the hemisphere needs six starts if it takes nothing in compression, five if it takes a quarter or a half of a per cent, four at one or two per cent, and three at four. Each of those is the count a stretch-only paper would need if its give were the sum of the two, and the starts sit in the same places: the ratio between the gives decides only how far the line is lowered, which is to say only how the unavoidable error is shared out between the two kinds of strain.

That is a stronger statement than the halving and it is the useful one. It says the whole of the earlier placement work — the ring rule’s rings, the spreading starts, the density law $\sqrt{\sin s / s}$, the tables for caps of 60, 90 and 120 degrees — is right for a paper that can be compressed, provided the give it was computed with is read as the sum of the two gives rather than the stretch alone. Nothing needs recomputing. What changes is one sentence in every essay that says “a material giving five per cent”: it now means a material whose stretch and compression limits add to five.

A paper that resists compression more than stretch sits on the dial between the two cases drawn so far. At a ratio of one quarter, the four-start line is lowered by a fifth of 1.81 per cent, and the paper takes 1.44 per cent as stretch and 0.36 as compression. The worst strain has fallen by a fifth, not a half, and the stretch and compression always add to the touching line’s worst value, at every setting of the dial, which is the invariant the whole construction rests on.

The mechanism the theory came from #

The alternating error in the lowered line has a name, and the name belongs to someone who was thinking about machines. The theory of the best uniform approximation — the one function in a family whose largest error is smallest — was founded by Pafnuty Chebyshev in the 1850s, and his starting point was Watt’s parallel motion, the linkage that guides a steam engine’s piston rod along a line that is nearly but not quite straight. Chebyshev asked how to make the error of that nearly straight path as small as possible and found that the best answer always makes the error swing between equal extremes, alternately above and below, as many times as the family has parameters plus one.

A straight stretch of tucks is a straight line trying to follow a curve, and the lowered line on each stretch does exactly what Chebyshev’s theorem says the best line must: its error is largest, and equal, at three points, with the sign alternating — compression at each end, stretch in the middle. The paper in a gathered cap is solving the problem a steam engine’s linkage posed, one stretch at a time, and the only thing that stopped the earlier placements from reaching Chebyshev’s answer was that their line had to touch the curve at its ends, which forced both end errors to zero and doubled the one in the middle.

What makes the tuck problem easier than Chebyshev’s is that the stretches share their ends. Each wants its end errors at the same compression, so lowering every stretch by the same amount keeps them joined, and the best line for the whole cap is just the best line for each stretch, placed end to end. A linkage has no such luck: its links do not share a common offset.

What the picture cannot show #

Whether paper can take compression at all without changing shape. A thin sheet asked to shorten in its own plane does not usually compress; it wrinkles, because buckling out of the plane is cheaper than squeezing fibres together. The compression these figures ask for — under one per cent at every start — is small enough that a dense sheet may take it in the plane, and large enough that a thin one will pucker instead. Which of those happens is a property of the particular paper and its thickness, and nothing here decides it. The creases a sheet gives itself is about what a sheet does when it refuses a shortening outright.

Where the compression goes round the circle. The strain here is the average round each circle, the same simplification the strain essay named: real tucks are lines of three layers, and the paper between two tucks takes its strain unevenly. A compression averaged round a circle may be a small wrinkle beside each tuck rather than a uniform shortening, and a maker would see the wrinkle.

The cost of a start. Fewer starts are better, but nothing here prices one, so the comparison between four starts and six at a given paper is a count and not a verdict. Every start is a point where tucks begin and three creases meet in a way that cannot fold flat, and a straight tuck is a cone point is the account of what each one does to the sheet.

The idealisation, named #

The cap is a sphere of unit radius and the flat radius is measured along it, so every number is a share of the rim’s radius and the model has no size. Tucks are lines of no width that remove length without adding stiffness, as in every earlier account of gathered caps, and crowding outward costs almost nothing is where the cost of that assumption was first measured.

The paper has two numbers and no others: the strain it will take in stretch and the strain it will take in compression, the same in every direction and at every radius, free below the limit and forbidden above it. That is the model of paper that stretches on purpose with one more number, and it is still a model: a real sheet stiffens gradually, creeps under a held load, and gives differently along its grain and across it.

And the line is continuous. A tuck begins with zero width, so the hiding cannot jump at a start; only its rate can change. A pattern that allowed a jump — a small dart cut in at each start — would be a different problem, and it is not one this model can pose.

How the claims were checked #

The crossing line is built its own way and compared against the touching line, not derived from it. Its corners are required to lie on the curve lowered by the compression limit, each stretch is extended as far as both limits allow, and the resulting starts are required to match the touching placement at the summed give to within 0.02 per cent of the radius — at two, four, eight and sixteen starts, and at every ratio of compression to stretch on the chart.

The two limits must add to the touching line’s worst strain at every setting of the dial, which is the invariant the lowering argument predicts, and it is checked on the figure that moves.

The first start allowed to cross must fall on the ring rule’s second ring at each of five gives from two to ten per cent, as the first touching start falls on the first.

The lowered line must actually cross the curve — twice in every stretch past the first and once on the cone from the centre — so a lowering too small to change the picture would stop the figure rather than draw a line that only claims to cross.

Still open: what a start costs #

The lowering settles the question the strain essay left first and brings back the one it left second. A start is not free, and every figure here counts starts without pricing them. A pattern that honoured both the paper’s give and the cost of a start would choose the count as well as the places: fewer starts with more strain, or more starts with less, traded at whatever a start is worth to the person folding it.

Two measurements would make that trade concrete. The first is how many tucks a start can begin. Three answers, one count asked whether the divisions a rim needs fit round it at all, and the strain placement has now said where they must begin: a start at 0.72 of a hemisphere’s radius that has to double the rate of hiding needs a ring of tucks that fits in the circumference there, and whether it does is the width question at a particular place. The second is a cost for the compression half. If a sheet puckers rather than compresses, the compression allowance is not a limit but a price paid in wrinkles, and the best lowering is then not the one that balances the two peaks but the one that balances their costs.

Sideways from here, the lowering trick belongs to every problem in this subject where a broken line follows a convex curve and the error has a sign. A curve has no panels approximates a curved crease by straight segments and finds the kinks between them summing to a constant of the curve; whether the segments must meet the curve at their joints, or may cross it, is the same question asked of a fold rather than a tuck, and it decides how large each kink has to be.

The habit worth carrying is about restrictions that come with a drawing. Before optimising a quantity that can be negative, check whether the method allows it to be. The touching rule looked like a way of drawing a broken line and was actually a decision that the paper would only stretch, and it cost exactly half the strain on every pattern it touched.