Origami & Folding
what one sheet can do
Concept

Gore — where it appears

A tapered panel used to approximate a curved surface out of flat pieces. It is the standard workaround for developability and it costs a seam, which is what a folded sheet is trying to avoid.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

cylinderreachablecurved one way onlyconereachablecurved one way, from a pointsphereunreachablecurved two ways — impossiblesaddleunreachablecurved two ways — impossible

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.

material · Developability 5%10%15%15°30°45°60°75°90°strain the rim must takehow much of a sphere the cap coversdry paper1% of stretchreaches 14°damp paper3% of stretchreaches 24°wet-folded6% of stretchreaches 35°a hemisphere needs 36% and nothing made of cellulose is going to supply it

Paper that stretches on purpose

Wet-folding breaks the assumption every theorem of flat folding rests on, deliberately. It does not repeal the geometry — it buys a few percent of strain, and a few percent of strain is worth about twenty degrees of sphere.

material · Idealisation curved tucksthe disc is the flat sheet; the dark lines fold each tuck under, the light line folds it in half8 curved tucks, a cap of 90°hidden at the rim: 36.3%rim thickness on average: 1.571 sheetsthe tuck widens as the cube of the radius

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.

material · Developability the bar is the worst shortfall in hidden length, as a share of what the rim hidesa cap of 90°, straight tucks started at evenly spaced radii and at the best radii for the same count2 starts, evenly spaced12.3%2 starts, best spaced8.5%31% smaller3 starts, evenly spaced5.8%3 starts, best spaced3.7%36% smaller4 starts, evenly spaced3.3%4 starts, best spaced2.0%38% smaller8 starts, evenly spaced0.8%8 starts, best spaced0.5%40% smaller16 starts, evenly spaced0.2%16 starts, best spaced0.1%41% smallerthe best radii give every stretch between starts the same worst error, which crowds them toward the rim

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.

material · Developability the same arithmetic three waysm divisions leave a residual of f ⁄ m inside each piece, whatever the divisions are made ofthe material givesdivisions neededas goresas tucksas curved creases2.0%1919 cuts, 59.7 of seam19 tucks, 3 sheets deep19 creases, no cut and no pile5.0%88 cuts, 25.1 of seam8 tucks, 3 sheets deep8 creases, no cut and no pile10.0%44 cuts, 12.6 of seam4 tucks, 3 sheets deep4 creases, no cut and no pile20.0%22 cuts, 6.28 of seam2 tucks, 3 sheets deep2 creases, no cut and no pilecap of 90°, rim excess 36.3% · the count is ⌈f ⁄ ε⌉ in every column; only the cost of a division changes

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.

material · Developability

Named alongside it

The objects these essays reach for when they reach for this one.

Developable surfaceGaussian curvaturePleatIsometryAngle deficitConeConservationLayer countShape retentionSpherical capStrainThe Theorema Egregium

All concepts