Origami & Folding
what one sheet can do
Concept

Angle deficit — where it appears

How far the angles round a point fall short of a full turn. It is zero everywhere on a flat sheet, and removing a wedge makes it positive, which forces the paper into a cone whose sharpness the deficit sets.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

a crease — concentrated at a pointdeficit 0.5236 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 0.5236 rad, none of it anywherecurved at every pointa 30° wedge removed, against Ω = 1 − 0.0400 r² · same total curvature, 0.5236

A crease carries no curvature

A fold looks like the sharpest curvature a sheet could have, and intrinsically it has none at all. Developability — the first of the four conditions this site's checker runs — is exactly the statement that folding an uncut sheet creates no curvature anywhere, including at the creases.

biology · Growth as metric 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 growth profilelargest |K| it carries∫K dAgrown by the same factor everywhereflat — it can be laid in a plane0.0000.0e+0 — nothinggrown more at the rimnot flat at any radius1.400-3.258grown more at the centrenot flat at any radius7.8426.767the growth of a spherical capnot flat at any radius0.3911.118the growth of a hyperbolic discnot flat at any radius0.391-1.360grown so that the curvature cancelsnot flat at any radius1.4004.8e-5 — nothing∫K dA = −2πR (ln Ω)′(R) — the total is decided at the rim, so the interior cancels out of it

The test measures the rim

Flattening the specimen is the right test and the measurement anybody actually makes on a flattened specimen is a boundary one — how far the margin overruns its chord. Total curvature is a boundary quantity too: it equals minus two pi R times the growth profile's slope at the rim, and nothing else about the interior survives into it. So a sheet can be curved everywhere and integrate to nothing, and the test reports it flat.

biology · Growth as metric 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.

ConeGaussian curvatureDevelopabilityDevelopable surfaceGrowthKirigamiMetricPleatBoundaryCurvature concentrationDifferential growthGore

All concepts