Developability — the series
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What a flat sheet can become
A sheet that cannot stretch cannot become a sphere. That much belongs to differential geometry; what belongs to folding is the three ways round it — seams, curved creases, and a few percent of stretch — and what each one costs.
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A tuck keeps what a gore cuts
A flat disc gathered into a spherical cap has more circumference than the cap, and a gore removes the excess while wet-folding stretches it away. A tuck folds it under, which keeps the sheet whole and turns the excess into thickness. At the rim of a gathered cap the paper is α ⁄ sin α sheets thick on average — π⁄2 for a hemisphere — and a simple tuck is three, so single tucks reach a cap of 130.6° before they run into one another. And because a sphere's circles fall short of a plane's as the cube of the radius, a tuck that follows the sphere widens as the cube too: its edges are curves.
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A straight tuck is a cone point
A tuck with straight edges hides length in proportion to how far past its start it has gone, which is a cone's law and not a sphere's. Started at the centre, straight tucks make a cone. Started at several radii, they hide length in a broken line that follows a sphere's cubic, and the worst shortfall falls as the square of the number of starting radii: 36.9 per cent of the rim's hiding from one start, 12.3 from two, 3.3 from four, 0.8 from eight. Every start is three creases at a point, which is a vertex that cannot fold flat — and it is exactly where the gathered sheet's curvature goes.
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Crowd the tucks toward the rim
Straight tucks started at several radii follow a sphere's hidden length in a broken line, and evenly spaced starts leave a worst shortfall that falls as the square of their number. Evenly spaced is not the best spacing. A sphere's hiding bends hardest near the rim, so the best starts crowd outward — on a hemisphere, four of them at 0.36, 0.60 and 0.80 of the radius — and leave 38 per cent less error than four evenly spaced. As the count grows the saving closes on 42 per cent, a limit set by the square root of how the sphere's hiding bends; on a shallow dish it approaches five ninths. To follow a hemisphere within one per cent takes six rings of tucks instead of eight, and within a tenth of a per cent eighteen instead of twenty-four.
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Crowding outward costs almost nothing
Placing the tuck starts for equal error crowds them toward the rim, where the gathered paper is already at its thickest, and the obvious worry is that the accuracy is bought with depth. Measured, it is not: on a hemisphere the crowded placement's deepest start sits in 3.37 sheets against the even placement's 3.32, because a start is three sheets of its own and the gathering beneath it is only one and a half.
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Three answers, one count
Seams, curved creases and a few per cent of stretch are the three ways round the sphere, and a tuck is a fourth. All four dispose of one quantity — the excess circumference a flat disc has over the sphere's circle — and all four dispose of it by dividing the circle. So the number of divisions needed is the same whichever answer is chosen: nineteen for a hemisphere in a material that gives two per cent, eight at five, four at ten. What differs is what a division costs, and one of the four runs out.
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Where a ring of divisions belongs
A pattern that divides the circle everywhere as finely as its rim requires is over-divided for most of its radius, because the excess grows from nothing. Putting a ring of new divisions in wherever the residual would otherwise pass what the material takes gives seven rings on a hemisphere at five per cent of stretch, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the way out — and the first of those sits where a completely different criterion put its first tuck start.
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The paper reads strain
Two placements of a gathered cap's tuck starts agreed on their first radius and nobody knew why. Both were proxies. What the paper takes is the shortfall at a circle divided by the circle's length, and judged by that the first start must sit exactly where the excess reaches the material's give — and past it the starts spread outward, not in. Four radii placed for length leave a hemisphere 5.34 per cent strain; the same four placed for strain leave 1.81.
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Crossing the curve moves no start
The broken line of hiding that straight tucks make on a gathered cap has always been drawn touching the sphere's curve at every start, so the paper between starts can only stretch. Let it cross, and the best line turns out to be the old one lowered by a cone: the starts stay exactly where they were at every count, the worst strain halves exactly — 1.81 per cent to 0.90 on a hemisphere at four starts — and a paper that stretches one amount and compresses another places its tucks exactly as a stretch-only paper giving their sum would.
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A start is worth an inverse square
Straight tucks gathering a flat disc into a cap leave a worst strain that falls as the starts multiply, and every essay on them counted starts without saying what one buys. It buys an inverse square. A chord of the hiding curve misses it by its width squared times the curve's bend over eight, so starts placed for strain leave a worst strain of I² ⁄ 8m², with I one integral of the cap's shape — 1.464 for a hemisphere. A paper that gives ε therefore needs about I ⁄ √(8ε) starts, a tenth of the give costs three times the starts, and letting the hiding cross the curve saves exactly a factor of √2. The same argument says why the starts spread outward, and by how much.