Developable surface — where it appears
Named by 16 essays across 2 fields — each of them below, with the objects they name alongside it.
A crease that curves
Bend a crease and the paper either side is forced into a shape nobody creased. The flat-folding theorems say nothing about it, because they are statements about straight creases meeting at a point.
The sculptors got there first
Curved-crease folding produced its best objects decades before anybody could compute one. The surfaces were made by hand, the ruling lines that determine them were not calculated until much later, and the mathematics has been catching up ever since.
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.
One curve and one number
Folding cannot change how curved a crease is within the surface — that is what an isometry means. Everything a curved fold produces is the curvature it adds out of the surface, and one number controls all of it.
A curve has no panels
A rigid folding is a finite list of flat pieces joined along lines. A curved crease has no such list, and refining one does not help: the kink at each joint falls as one over the segment count, and the total of the kinks does not fall at all, because it is a constant of the curve.
Where the rulings run out
A curved fold's surface is made of straight lines leaving the crease, and the lines are not parallel, so they cross. Past the first crossing there is no surface: two points of the paper have been sent to one point of space. The boundary is a curve nobody drew, no crease pattern shows it, and it sits at the sine of the ruling angle times the crease's own tightest radius — on every curve tried.
The gap between two curves
The rulings leaving a curved crease are not parallel, so they cross, and the surface exists only as far as the first crossing. That bound is usually read as a limit on how far a design extends outward. It is not: the paper between two curved creases has to be reachable from both, so the bound bites hardest where the circles are smallest, and a concentric pleat has a hole in the middle that no sheet size removes.
Found by people not folding paper
The shortest strip that makes a Möbius band has a literature, and it is in differential geometry rather than in origami. The two subjects have the same number, they reached it by completely different routes, and neither of them cites the other — which is the fourth time this collection has found that shape.
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.
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.
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.
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.
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.
Only one side can run out
A curved fold has two surfaces and every reach ever computed here has been one of them. The closed form's denominator is the crease's curvature plus the rate the ruling angle turns at, and crossing to the other surface negates both — so at any point of any crease at most one of the two surfaces can have its rulings converge. A crease that never changes the way it bends therefore has a surface with no such boundary at all, anywhere along it.
An angle that turns faster than the crease
Which of a curved fold's two surfaces runs out is decided by a sum of two rates — how fast the crease bends and how fast the ruling angle turns — and every measurement so far has set the second to zero. Let it turn and it carries the sign on its own: past a rate of exactly one, a crease of unchanging curvature bounds both of its surfaces, which no constant angle on that crease can do. Below that rate the turning costs reach without changing anything else.
Named alongside it
The objects these essays reach for when they reach for this one.
Gaussian curvaturePleatCurved creaseIsometryEnvelopeGoreRulingRuling lineCurvatureIdealisationLayer countAngle deficit