Origami & Folding
what one sheet can do
Concept

Developability — where it appears

The property of having zero Gauss curvature everywhere. It is what a surface must have to be reached from a flat sheet by bending alone, and it is the restriction every uncut folding works under.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

VMMM60°90°120°90°Kawasaki60° + 120° = 180°90° + 90° = 180°both 180° — satisfiedMaekawa3 mountains, 1 valleysdifference 2exactly 2 — satisfiedangles sum to 360°which is what a flat sheet requiresmountainvalley

Two conditions at a point

Whether a single vertex folds flat is decided completely by two tests — one on the angles, one on the assignment. They are independent, they are easy to check, and together they settle the case entirely.

flat-folding · Flat-foldability a = 0.6K(0) = -2.40opens — a rufflearea ×1.72a = 0.25K(0) = -1.00opens — a rufflearea ×1.27a = 0K(0) = 0.00stays flatarea ×1.00a = -0.25K(0) = 1.00closes — a domearea ×0.77a = -0.4K(0) = 1.60closes — a domearea ×0.65the growth factor is the same function throughout — only the sign of one number changesΩ = 1 + a r² · rim-heavy growth opens the sheet, centre-heavy growth closes it

Which way the disc curves

A growing disc either domes or ruffles, and which one it does is not a matter of how much it grew. It is decided by where the growth was — more at the rim opens the sheet, more at the middle closes it — and one number in one formula takes it through both.

biology · Growth as metric 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 straight creasescurvature 1.4curvature 2.6the tangents are the same in every panelsectors 60.0°, 120.0°, 120.0°, 60.0°they sum to 360.0°, and alternately to 180.0° and 180.0°which is Kawasaki, on tangents rather than on lines

Where curved creases meet

A curved-crease design looks like a smooth object and its constraints are not smooth. They live at the finitely many points where creases cross, and at each of those the conditions are about the creases' tangent directions — the curvature does not appear in them at all.

material · Curved creases 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 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 mismatch 0.066 radiansmismatch 8.5e-14 radiansevery vertex of both is developable and Kawasaki-exact to the last bit a double holds

Solving every face at once

A quadrilateral mesh that folds rigidly has to close round every one of its faces, and the rung that built the general mesh could close one. Four of them at once resisted a descent that drove each free length to its own root, because closing a loop is a condition on several lengths together — and solving them jointly finds a sheet with no two vertices alike that folds, and a surface of them sixteen dimensions wide.

rigid · Rigid folding the edge of the paperMVM40°60°20°60°the same sectors, in a lineMVM40°60°20°60°this lettering folds4 of 8 letterings foldVMV MMV VVM MVMno vertex theorem applies here at all— the sectors do not close, and there is no cycle to alternate round

The vertices nobody checks

The four conditions of flat folding are stated at an interior vertex, and the word interior carries the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, none of the four conditions says anything about them, and the condition that does decide them — a one-dimensional crease pattern, folded or refused exactly — was in hand all along.

flat-folding · Boundary 0204060801001200510152025foldssurface heldpeak at 64 foldsbox of side 1 · sheet thickness 0.01 · most surface at 64 folds · 128 folds fills the box with sheet alone

Nothing grown has a seam

A gut is a tube and a leaf is a disc, and neither was made by joining anything. The sheets this collection builds by identifying a rectangle's edges are the same objects a body grows, reached by an operation no organism performs — and the difference shows up in where the boundary is and in what has to close.

biology · Surface in a volume 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 worst strain straight tucks leave, against the number of starts placed for straindots: the placement's own worst strain; dashed: I² ⁄ 8m², with I = ∫ √(sin s ⁄ s) ds over the cap-5-4-3-2-1starts, on a doubling scalelog₁₀ of the worst strain2481632the dialled cap: 90°I = 1.464small dots: caps of 30° and 150°every doubling of the starts divides the worst strain by four, on every cap

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.

material · Developability

Named alongside it

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

Gaussian curvatureAngle deficitConeGrowthKawasaki's theoremDifferential growthMaekawa's theoremMetricAssignmentThe big-little-big lemmaBoundaryBoundary vertex

All concepts