Origami & Folding
what one sheet can do
Concept

Threshold — where it appears

A value at which behaviour changes qualitatively rather than by degree. Folding is full of them — where a mechanism jams, where a surface stops existing, where a search's answer flips — and they are usually more informative than the gradual parts.

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

cuts at 85% of the pitch0.15 of material between one cut and the nextcut length, as a fraction of the pitchligament6 piecesone piece at every length up to 97%and the count is worked out from the cuts, not measured off a picture

One cut short of falling apart

Everything a cut sheet can do is bought out of the material between the end of one cut and the start of the next. That material shrinks to nothing in a straight line as the cuts grow, and the sheet stays in one piece the whole way down — until the instant it does not, and then it is in six.

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 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 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 the two things a domain has to be long enough forthe bar starts where the demand is first met and runs to the rightunique in the whole design11 bases and upstill paired at 45 degrees17 bases and upwhere a honeycomb crossover may sitfirst one inside both bands: 21 baseswhere a square-lattice crossover may sitfirst one inside both bands: 24 bases4812162024283236a domain runs between two crossovers, so its length is a whole multiple of the period

A domain too short to be unique

A staple holds the scaffold by pairing with a stretch of it, and two arguments decide how long that stretch has to be. One is combinatorics — a stretch of seven bases has about four hundred other places in a 7,249-base strand it would also match. The other is thermodynamics, and it is the one that binds: a duplex that is unique at eleven base pairs still comes apart at the temperature the design is held at, and staying paired takes seventeen. Rounded up to the crossover period, that is three periods on both lattices, and the lattice the helix prefers is the one whose three-period domain leaves some of its staples unattached.

biology · Molecular folding metres of crease a square metre: what each pattern asks for, and what each paper allowsthe pale bars are patterns and the dark ones are paperswhat foil-backed tissue allows641026 µm, a crease 0.16 mm acrosswhat washi allows416740 µm, a crease 0.24 mm acrosswhat kami allows238170 µm, a crease 0.42 mm acrosswhat copier paper allows1667100 µm, a crease 0.60 mm acrossThe waterbomb tessellation89printed at 160 mmThe Yoshimura pattern82printed at 170 mmThe tapered corrugation41printed at 160 mmThe hexagon twist41printed at 150 mmThe Miura fold36printed at 170 mmThe preliminary base32printed at 150 mmThe square twist31printed at 150 mmFold and cut — the triangle11printed at 150 mma crease occupies about 6 sheet thicknesses, so the closest two creases can be laid is that, and the ceiling is its reciprocal

The density a paper allows

Every density these essays measure is a quotient a pattern hands over, and nothing has asked what the paper's own answer is. It has one: a crease occupies a band a few thicknesses across, so two creases closer than that are not two creases, and a sheet of a given thickness carries a largest density. Copier paper allows 1,667 metres of crease a square metre and the densest pattern on the printed shelf asks for 89 — a factor of nineteen below the worst paper's ceiling. The material is not what limits a crease pattern's density at any fineness anybody folds.

material · Crease density 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.

DegeneracyDesign spaceIdealisationStraight skeletonTree methodUniversal moleculeKirigamiAngle deficitThe big-little-big lemmaBoundaryConeConnectivity

All concepts