Origami & Folding
what one sheet can do
Concept

The fold-and-cut theorem — where it appears

The result that any straight-line drawing can be produced by folding a sheet flat and making one straight cut. It is a universality theorem and it says nothing about how large the resulting pattern is.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

the cut line3 straight edgesthe pattern3 skeleton arcs3 perpendiculars1 interior vertexassignments that fold30 of 646 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once

One straight cut

Any drawing made of straight lines can be folded so that the whole drawing lands on a single line, and one cut releases it. The construction is a shrinking process, and it explains itself the moment the shrinking is drawn.

flat-folding · Flat-foldability 4681012010203040sides of the polygoncreasescreases in the patternperpendicularsskeleton arcsa 12-sided outline needs 36 creases and one skeleton nodea convex outline is the cheap casea reflex corner splits the shrinking front, and this solver refuses those rather than guessing

What universality costs

The fold-and-cut theorem says any straight-line drawing can be flattened onto a single line. It says nothing about how much crease pattern that takes, and the amount is a measurable quantity — computed here by running the construction rather than by estimating it.

complexity · Flat-foldability 10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 54° to the foldwaist 0.309 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed

The star that was cut before it was proved

Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.

history · Folklore of folding each row is an exhaustive count over the patterns that construction producedon the edgedeepest piletimes smallercrease densitythe printed patterns8 patterns62%19.415.5×7.9twist tessellations12 patterns52%10.02.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2

Four ways to draw a pattern

Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.

complexity · Typical instances the cut line10 straight edgesthe pattern10 skeleton arcs0 perpendiculars1 interior vertexassignments that fold420 of 102410 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once

One cut for a star

The fold-and-cut construction here could reach a triangle, a pentagon and a house, and refused everything that turned back on itself, because shrinking an outline with a reflex corner needs an event the shrink did not implement. With split events it reaches a five-pointed star — ten creases through one point, four hundred and twenty letterings that fold, and every edge of the outline landing on one line to a part in 10^16.

flat-folding · Flat-foldability the outline, shrunk — and the one corner that moves outwardwhat the shrink finds5 edges, 1 of them meeting at a corner that turns back3 skeleton nodes7 arcs traced by the cornersthe last of them forms at 0.181 of a sheet

The corner that splits the shrink

The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.

design · Molecules the cut line10 straight edgesthe pattern10 skeleton arcs0 perpendiculars1 interior vertexassignments that fold420 of 102410 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once

A tree cannot argue

A molecule fills a polygon with creases taken from its straight skeleton, and a straight skeleton is a tree. So a molecule's panels have almost no closed chains for its letters to contradict themselves round — one to three, against thirty-six on the smallest tessellation patch. Two hundred and eighty independent letterings across seven outlines, including an L and a five-pointed star, and not one of them disagrees with itself.

design · Molecules the stack under a single cuta clean cut taken as 0.6 mm of stackstarnewsprintcopier paperkamiwashifoil-backed tissueunryu tissue3 points6 layers390 µm600 µm420 µm240 µm156 µm108 µm4 points8 layers520 µm800 µm560 µm320 µm208 µm144 µm5 points10 layers650 µm1.0 mm700 µm400 µm260 µm180 µm6 points12 layers780 µm1.2 mm840 µm480 µm312 µm216 µm8 points16 layers1.0 mm1.6 mm1.1 mm640 µm416 µm288 µm10 points20 layers1.3 mm2.0 mm1.4 mm800 µm520 µm360 µm12 points24 layers1.6 mm2.4 mm1.7 mm960 µm624 µm432 µm16 points32 layers2.1 mm3.2 mm2.2 mm1.3 mm832 µm576 µm20 points40 layers2.6 mm4.0 mm2.8 mm1.6 mm1.0 mm720 µma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.6 mm of stack

How many wedges the paper allows

A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.

history · Folklore of folding

Named alongside it

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

Straight skeletonConstructive proofCrease patternFlat-foldabilityKirigamiSymmetryAngle bisectorAssignmentAxial polygonBoundary vertexCompletenessDesign technique

All concepts