Origami & Folding
what one sheet can do
Concept

Degeneracy — where it appears

A configuration where distinct solutions coincide or a construction stops being well posed. The flat state is one for every rigid folding, which is why a solve is never started from it.

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

102°60°78°120°one smallest sectorthe lemma constrains one pair4 foldable assignmentsof the 16 markings120°60°60°120°two smallest sectors equalthe lemma constrains nothing8 foldable assignmentsof the 16 markings

Where the lemma says nothing

The big-little-big lemma asks for a sector strictly smaller than both its neighbours, and the word doing the work is strictly. At a vertex whose two smallest sectors are equal the lemma has no opinion at all — and those are the vertices origami actually uses. The count of markings the conditions admit doubles, discontinuously, at exactly the angles everybody folds.

flat-folding · Sector angles 0.287267creases in the moleculewhere the corner sits7 creases6 creasesat the marked shape three of the polygon's edges vanish at the same instant, and either side of it they vanish one at a time

The skeleton changes its mind

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. It does not fill it continuously. Slide one corner along its edge and the number of creases in the molecule sits at six, jumps, and sits at seven — so two designs a hairsbreadth apart have crease patterns that are not small variations on one another.

design · Molecules the bar is the shortest crease in the pattern, on a scale of powers of tenthe hexagonal patch at four turns of its polygons, everything else heldturn 0.21.4e-2130 creases · every one carries an arc · 2.25 mm on a 160 mm sheetturn 0.357.9e-6142 creases · 12 of them carry no arc · 1.3 µm on a 160 mm sheetturn 0.57.5e-3154 creases · every one carries an arc · 1.20 mm on a 160 mm sheetturn 0.72.8e-2154 creases · every one carries an arc · 4.53 mm on a 160 mm sheetone turn of one patch drops four orders of magnitude below the others, and it is the turn this collection prints

Twelve creases a micrometre long

A patch this collection has drawn for a long time carries a hundred and forty-two creases and a hundred and thirty arcs, and nobody had asked what the other twelve were. They are fragments left where the clip caught a pleat almost exactly at a corner — between one and nine micrometres long on a printed sheet, at one turn angle out of four, and it is the turn the collection prints.

flat-folding · Boundary the bar is the second-smallest sector at a typical vertexthe triangular patch at seven turns, with the same panels and the same creases at all of themturn 0.261.92°smallest sector 60.00° · next 61.92° · no lettering exists, proved by exhaustionturn 0.2160.71°smallest sector 60.00° · next 60.71° · no lettering exists, proved by exhaustionturn 0.215560.06°smallest sector 60.00° · next 60.06° · no lettering exists, proved by exhaustionturn 0.21660.00°smallest sector 60.00° · next 60.00° · a lettering existsturn 0.2260.00°smallest sector 59.52° · next 60.00° · a lettering existsturn 0.2560.00°smallest sector 56.10° · next 60.00° · a lettering existsturn 0.3560.00°smallest sector 46.15° · next 60.00° · a lettering existsthe verdict changes exactly where that sector passes sixty degrees and stops being the second smallest

Where a sector crosses sixty

Turn the twist polygons of a tessellation patch a hundredth of a radian further and the pattern goes from having no mountain-valley labelling at all to having one immediately. Nothing about its graph changes across the transition — the same eighty-three panels, the same hundred and forty-two creases, the same four labellings at every one of its sixty vertices. What changes is which sector at a vertex is the smallest one.

flat-folding · Genericity 00.20.40.60.8100.10.20.30.40.5where the corner sits along the top edgeshortest crease, as a share of the sideincircle7 creasesa crease being born6 creases, one dyingthe pentagon [0,0] [1,0] [1,1] [t,1] [0,h] · the band is where the shortest crease is under 0.5 mm on a 100 mm polygon, drawn ten times wider

The event is an incircle

The universal molecule changes its crease count at isolated shapes, and near one it carries a crease too short to fold. On the pentagon family where the change was found, the shape is exactly the one with a circle touching all five sides, and the short crease is born two to twenty times more slowly than it dies — so the unfoldable shapes lie mostly past the event, and the cleanest escape is to move onto it.

design · Molecules 00.20.40.60.811.21.400.10.20.30.4where the corner sits along the top edgeshortest crease, as a share of the height7 creasesat every positiondots: the flipsthe pentagon [0,0] [w,0] [w,1] [t,1] [0,0.7] · a zero of the curve is a flip, where the shortest crease has no length

Six creases belong to the square

Every measurement of the pentagon whose molecule changes its mind was made on one family, and four of that family's sides lie on a square. A square has an incircle, so for the whole first stretch of the corner's travel four side lines touch one circle and the molecule carries a node where four creases meet: six creases, held degenerate for an interval. Widen the rectangle by a sixth and the molecule has seven creases at every position. What happens instead is two flips, where the short crease shrinks to nothing and regrows between another pair of sides — and a pentagon near an incircle is rarer than one near a flip by about the square.

design · Molecules

Named alongside it

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

ThresholdDesign spaceStraight skeletonTree methodUniversal moleculeThe big-little-big lemmaCrease patternGenericityPatchSector anglesAssignmentBoundary

All concepts