Origami & Folding
what one sheet can do
Concept

Cone — where it appears

The surface a sheet takes when the angles round a point do not sum to a full turn. It is developable everywhere except at its apex, which carries all of the curvature, and it is what a single cut at a point produces.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

the sheeta wedge of 60° marked for removal60°56.4°what it closes intoa cone of half-angle 56.44°83.3% of the turn is left, and the sine of the half-angle is that same fractionthe circles of latitude are the disc's own, arriving shorter than a flat sheet would needno fold can do this: folding moves paper about and never alters how much of it surrounds a point

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.

material · Kirigami a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

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.

material · Kirigami parallelstraight creases spread 0.0°flat, they spread 0.0°fan 5.2°straight creases spread 25.5°flat, they spread 30.9°fan 9.2°straight creases spread 46.6°flat, they spread 55.0°fan 13.8°straight creases spread 72.4°flat, they spread 82.5°

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.

tessellation · Miura the bar is the worst gap between the hiding straight tucks do and the hiding a sphere needsa cap of 90°, as a share of what the rim hides — every tuck straight, started at evenly spaced radiifrom 1 radius36.9%straight from the centrefrom 2 radii12.3%2.99 times smaller than 1from 4 radii3.3%3.73 times smaller than 2from 8 radii0.8%3.93 times smaller than 4from 16 radii0.2%3.98 times smaller than 8a broken line through a smooth curve is out by the curvature times the square of the spacing

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.

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

Named alongside it

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

Angle deficitDevelopabilityDevelopable surfaceGaussian curvatureKirigamiPleatBoundaryCorrugationCurvature concentrationDeploymentFold angleGore

All concepts