The paper reads strain

Two ways of placing features on a gathered cap have been worked out from criteria that have nothing in common, and they put their first feature in almost the same place. Crowd the tucks toward the rim placed tuck starts to make the worst shortfall in hidden length as small as possible, and put the first one past the centre at 0.36 of the radius. Where a ring of divisions belongs put a ring of new divisions wherever the excess, shared equally among the divisions already running, would pass what a five per cent material takes, and put the first at 0.35. The second essay called the agreement a fact about the sphere and left it there, with a suggestion: two proxies agreeing may be two readings of a third quantity.

There is a third quantity, and it is the one the paper actually feels. A sheet does not register a shortfall in length. It registers a shortfall in length per unit of length — a strain. A millimetre that a circle of ten millimetres cannot find is a ten per cent stretch; the same millimetre missing from a circle of three hundred is a third of a per cent. The length criterion weighed those two millimetres the same, and the ring rule never looked at how a straight tuck’s hiding grows along it. Judged by strain, one of the two coincidences becomes an identity and the other placement turns out to have been pointing the wrong way.

Strain is a shortfall over a circumference #

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

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

on a sphere of unit radius, which grows as $\pi s^3/3$ near the centre and reaches $\pi^2 - 2\pi$ at a hemisphere’s rim. Straight tucks started at a set of radii hide length in a broken line through that curve: between two starts the hiding grows at a constant rate, and at each start the rate steps up as new tucks begin. The broken line agrees with $H$ at every start and misses it in between.

What the paper has to do about the miss is stretch. Where the tucks hide too much, the circle that remains is shorter than the sphere’s, and the paper has to take up the difference as strain round the circle; the circle it takes it up over has length $2\pi s$. So the strain at radius $s$ is

$$\varepsilon(s) = \frac{\lvert L(s) - H(s)\rvert}{2\pi s}$$

where $L$ is the broken line. That is the quantity a material limit is a limit on, and it is the quantity the ring rule was reaching for when it compared a residual against a material’s give.

The picture shows the thing the length criterion could not see. Every one of the three curves is at its highest at the very centre of the sheet, where a millimetre of shortfall is spread round almost nothing. The length placement — the one that crowded its starts toward the rim — has the longest first stretch and so the tallest peak there, and on this measure it is the worst of the three. Even spacing, which the earlier essay beat by 38 per cent on length, beats it by a factor of two on strain.

The first start is fixed, and it is the first ring #

Look at the first stretch alone. Tucks that run from the centre to a radius $s_1$ with straight edges hide length at the constant rate $H(s_1)/s_1$, which is to say they hide the same fraction of every circle: the fraction $f(s_1)$, where

$$f(s) = 1 - \frac{\sin s}{s}$$

is the excess the earlier essays have used throughout. That is the definition of a cone, and a straight tuck is a cone point says exactly this about tucks from the centre: they make a cone, not a sphere.

A cone hides the same fraction everywhere and the sphere hides nothing near its pole. So at the centre the tucks are hiding $f(s_1)$ of each circle while the sphere needs none hidden, and the paper there must stretch by the full $f(s_1)$. Further out the sphere needs more hidden and the gap closes, reaching zero at $s_1$. The strain of tucks from the centre is largest at the centre and equals the excess where the tucks stop.

That turns the first start into a calculation rather than a search. If the paper can take $\varepsilon$ and no more, the first stretch can run exactly as far as $f(s_1) = \varepsilon$ and no further. And that equation is the ring rule’s first ring, character for character: the radius where the excess first reaches the material’s give. The two placements agreed on their first feature because, for the first feature, they are the same equation.

There is a second route to the same radius that makes the point without tucks at all. Leave the middle of the disc plain and gather nothing until $s_1$. A plain disc gathered onto a sphere has to stretch by the whole excess, $f(s)$ at radius $s$, which is the ring rule’s own account of the region before its first ring. Its worst strain is $f(s_1)$, reached at the edge of the plain region. A cone from the centre carries its worst strain at its point; a plain disc carries the same strain at its rim; either way the first feature cannot sit further out than the radius where $f$ reaches the material’s give. Whatever pattern is chosen, that radius is fixed by the paper.

Past the first stretch, the starts spread out #

Once the first start is fixed, the question is how long each later stretch may run. The answer comes from how a broken line misses a smooth curve. On a stretch of length $h$ the chord misses the curve by about an eighth of the curvature times $h^2$, and the curvature of the hidden length is

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

so the miss in length is about $\tfrac{\pi}{4} h^2 \sin s$. Crowd the tucks toward the rim equalised that miss across the stretches, which meant making $h$ fall as $1/\sqrt{\sin s}$ — shorter stretches where the curve bends harder, which is toward the rim.

Divide the same miss by the circle it is spread over and the strain on a stretch at radius $s$ is about

$$\varepsilon \approx \frac{h^2}{8}\,\frac{\sin s}{s}.$$

Equalising that means making $h$ grow as $\sqrt{s/\sin s}$. On a hemisphere, $s/\sin s$ runs from one at the centre to $\pi/2$ at the rim, so the stretches should lengthen toward the rim, by a factor of $\sqrt{\pi/2}$, about 1.25, from the first to the last. The length criterion had the direction of the effect wrong, not only its size. The sphere’s hiding does bend hardest at the rim, and the rim is also where the circles are longest, and a strain divides by the second faster than it grows with the first.

The placement found by searching agrees with the argument. With four starting radii on a hemisphere the strain placement puts them at 0, 0.21, 0.46 and 0.72 of the radius, so that the stretches are 0.21, 0.25, 0.26 and 0.28 of it: the first is short because the cone’s point is the most expensive place on the sheet, and every later one is a little longer than the last. At every stretch the worst strain is the same 1.81 per cent, and it equals the excess at 0.21, as the cone argument requires. The length placement’s first stretch is 0.36, and the excess at 0.36 of a hemisphere’s radius is 5.34 per cent, which is the whole of its bad score.

How much the proxy cost #

It is worth putting the three placements side by side at several counts, because the gap between them is not a fixed penalty. It grows.

At two starting radii the length placement leaves 13.8 per cent strain, even spacing 10.0 and the strain placement 7.8. At eight they are 2.09, 0.64 and 0.43. The length placement falls further behind as the count rises, 1.8 times the strain placement’s figure at two radii and 4.9 times at eight, and even spacing sits between the two throughout, at 1.3 to 1.5 times the best.

The reason it falls behind is the first stretch again. The length criterion does not care how long the first stretch is, because near the centre the hidden length is tiny in absolute terms and a long first stretch costs it almost nothing; so as the count grows it spends its extra starts at the rim, where its own measure is expensive, and leaves the first stretch comparatively long. The strain at the centre is the excess at the end of the first stretch, which falls only as the square of that stretch. The strain placement shortens the first stretch in step with every other, and its advantage compounds.

None of this was wrong in crowd the tucks toward the rim. It minimised what it said it minimised, and on that measure the crowded placement beats even spacing by the 38 per cent it reported. What that essay could not show is that the measure was the wrong one for the paper, and that the placement it recommended is, for the thing a maker has to worry about, worse than doing nothing clever at all.

Where each rule spends its divisions #

The three rules can be compared without choosing a count, by asking where each would put its features per unit of radius when it has many to place.

The ring rule’s density is the slope of the excess, $f'(s)$, which starts at nothing and rises steeply: it is the rule that most wants to crowd. The length rule’s is $\sqrt{\sin s}$, which rises from nothing too but more gently. The strain rule’s is $\sqrt{\sin s / s}$, which is one at the centre and falls to $\sqrt{2/\pi}$, about 0.80, at a hemisphere’s rim. Of the three, only the strain rule is not zero at the centre, and that is the picture of the cone’s point: the one place on the sheet where a length criterion sees nothing to do is the place the paper is most exposed.

The law is not only an approximation for many features. Placed by search at twenty-four starts on a hemisphere, the stretches past the first multiplied by the predicted density agree to within 0.8 per cent. So the placement at any useful count is the density curve read off at that count, with one correction: the first stretch is set by the cone and not by the curvature, and it sits a little shorter than the curve would give it.

Two questions the ring rule answered as one #

The ring rule and the strain rule agree about the first feature and disagree about everything after it, and the disagreement is instructive because it is about counting.

At five per cent of give, the ring rule puts seven rings on a hemisphere, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the radius. Straight tucks bounded by the same strain need a cone from the centre to 0.35, a start there and one more at 0.78, and they reach the rim. At two per cent the rings number eighteen and the tuck starts three past the centre, at 0.22, 0.48 and 0.76. At one per cent, thirty-six rings and five starts.

Those counts are not like for like, and the difference is the useful part. A ring in the ring rule is one more division, and the rule’s residual, $f(s)/m$, assumes that each of $m$ divisions takes an equal share of whatever excess is present. A start in the tuck model is a radius where the rate of hiding changes, and a single start can begin as many tucks as its stretch needs; the model says nothing about how many, only about where the slope of the broken line bends. So the ring rule was answering two questions at once — how many divisions are needed at this radius, and where must something new begin — and it answered both with the same list. The count of divisions crowds toward the rim because the excess grows; the places where something must begin do not, because a straight tuck already running keeps hiding more as it goes.

That is the sense in which the rings crowd for a reason that belongs to counting and not to the sphere. Seven divisions at the rim is a true requirement of a five per cent material. That they need seven separate radii to begin at is not.

What the strain law says about other caps #

The spreading is not a property of the hemisphere. The density $\sqrt{\sin s/s}$ thins toward the rim on any cap, and the deeper the cap the more it thins.

On a 120° cap with four starting radii, the strain placement’s stretches are 0.20, 0.24, 0.26 and 0.31 — the last half as long again as the first — and it leaves 2.87 per cent against the length placement’s 8.57 and even spacing’s 4.51. On a shallow cap of 60° the stretches are 0.22, 0.25, 0.26 and 0.27, very nearly even, because on a shallow cap $s/\sin s$ barely leaves one. So the rule of thumb a maker could carry is short: on a shallow dish space the tuck starts evenly; on a deep one let them spread, and never crowd them; and in every case pull the first start in to where the excess meets the material’s give.

What this picture cannot show #

The broken line is required to meet the sphere’s curve at every start. That is how the tuck model has always been drawn, and it is a restriction rather than a property of paper. A line allowed to cross the curve — hiding a little less than the sphere asks just after a start and a little more before the next — could split each stretch’s miss into equal parts above and below and roughly halve every figure here. The ranking of the three placements would survive that, since it comes from how the miss is divided by the circle, but none of the percentages would.

The strain is taken as uniform round each circle. A real tuck is three layers of paper at a line, not a smooth loss of length spread round the circle, so the paper between two tucks takes its strain unevenly: most in the middle of the panel, least beside the fold. The strain here is the average round the circle, which is what a material limit on a thin sheet averaged over a panel would see, and not what the paper next to a tuck sees.

The number of tucks at each start is left free. A start is a radius where the slope of the hiding bends; how many tucks carry the new slope, and whether they fit side by side, is not decided. A tuck keeps what a gore cuts found single tucks running into one another at a cap of 130.6°, and nothing here moves that limit.

And the strain is taken up by stretching. The other thing a sheet can do with a shortfall is refuse it and take a different shape: a cone at the middle instead of a dome, which is exactly what tucks from the centre make. A stiff card would do that. The comparison is between placements for a material that stretches a little and then stops — the model of paper that stretches on purpose — and for a card that does not stretch at all, the cone point at the centre is a shape error of the same size rather than a strain.

The idealisation underneath #

The sphere is a unit sphere and the flat radius is measured along it, so a circle at flat radius $s$ becomes a circle of length $2\pi \sin s$. Tucks are lines of zero width that remove length without adding any stiffness. The material has one number, the strain it can take everywhere and in every direction, and it takes any strain under that number at no cost. Crowding outward costs almost nothing priced the thickness a start adds and found it nearly constant, three sheets of its own on a gathering of about one and a half, and none of that enters here. Every one of these can be relaxed, and the first to relax would be the equal share round the circle, because that is where a real tuck differs most from a smooth loss of length.

How the claims were checked #

The cone identity is checked on three stopping radii: tucks from the centre to a quarter, 0.45 and 0.65 of a hemisphere’s radius leave, at the centre, a strain equal to the excess where they stop, and nowhere more than that.

The strain placement is found by halving on the worst strain, the same way the length placement was found on the worst shortfall: each stretch is extended as far as a trial strain allows, and the trial is halved until the stretches just reach the rim with the count given. Its worst strain is then required to equal the excess at its first start, and its stretches past the first to lengthen outward.

The ordering of the three placements is required at every count drawn, strain below even and even below length, so a count at which the crowded placement came out ahead would stop the figure rather than print a table that said otherwise.

The density law is checked against the placement it predicts: at twenty-four starts, each stretch past the first times $\sqrt{\sin s / s}$ at its midpoint varies by 0.8 per cent on both a hemisphere and a 120° cap.

And on every setting of the material’s give on the dial, the first strain-bounded start is required to fall on the ring rule’s first ring, which is the coincidence the earlier essay reported and could not explain, now checked as an identity.

Still open: a line that is allowed to cross #

The obvious next calculation is the restriction named above. A broken line that follows the sphere’s hidden length without having to touch it at the starts is the true best approximation, and for a convex curve its miss on each stretch splits evenly above and below. Computing the strain placement under that freedom would say how much of every figure here is the model’s and how much the paper’s — and whether the first start, which the cone argument fixes when the line must pass through the centre, moves once the line may begin below the curve.

The second is to give a start a cost. Every start past the centre is three creases meeting at a point that cannot fold flat, and a maker would rather have two than five; the strain placement tells how few starts a material allows, and the ring rule how many divisions the rim needs, and a pattern that honoured both would begin its tucks at the strain placement’s radii and add tucks at each start until the rim’s count is met. Whether those tucks fit — whether a start at 0.78 of a hemisphere’s radius can begin five tucks in the circumference it has — is the width question three answers, one count left open, now with a place to ask it.

The habit worth carrying is about units. Before optimising a shortfall, divide it by whatever it is a shortfall of. A length missing from a short circle and the same length missing from a long one are not the same fault, and a criterion that counts them alike will put its effort where the number is largest, which is not always where the damage is.