Cone — where it appears
Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.
What one cut buys
A fold moves paper about and cannot change how much of it surrounds a point. A cut can, and that one difference is the whole of what this site's founding rule is worth. Take a wedge out and the sheet closes into a cone; let one in and it has more paper than the plane will accept.
A cut that removes no paper
Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.
The corrugation that curves
A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.
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.
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.
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.
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.
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.
Growth that only turns
Grow a sheet by the same two stretches at every point — 1.4 along one direction, 1 across it — and let only the direction change from place to place. No rate varies anywhere, and the sheet still cannot lie flat. A cut reads two whole turns' worth of geometry: the number of times the direction turns round it, less the circulation of a field built from how fast the direction turns moving along itself and moving across itself. Directions radiating from a point put all the curvature at that point, a cone of 2π(1 − 1⁄1.4); directions running round it make an anticone; a spiral pitched forty degrees from the radius holds nothing at its centre; and a direction that turns steadily across the sheet curves it in stripes of alternating sign.
Named alongside it
The objects these essays reach for when they reach for this one.
Gaussian curvatureAngle deficitDevelopable surfacePleatDevelopabilityConstraintKirigamiTrade-offAnisotropyBoundaryCorrugationCurvature concentration