Origami & Folding
what one sheet can do
Concept

Straight skeleton — where it appears

The figure obtained by shrinking a polygon inward at a uniform rate and recording where its edges meet. It is what the fold-and-cut construction folds along and what the universal molecule fills a polygon with.

Named by 12 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 the shrink5 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.5391 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here

The last free parameter

Once the packing is fixed, one number is left in the whole design: how far a leftover polygon can be shrunk before it stops being a polygon. Everything else about the crease pattern has already been decided.

design · Molecules 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 quadrilateralconvexmolecule builtpentagonconvexmolecule builtL, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkingdart, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkinga convex polygon shrinks inward and stays a polygon; a reflex corner is a wall the shrink runs intoso a non-convex region is split into convex pieces first, and choosing the split is a searchwhich is where a construction that always works hands the difficulty to whatever comes before it

The molecule that does not exist

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. Hand it a reflex corner and it does not produce a worse pattern — it produces nothing, and the difficulty moves backwards to whoever chose the polygons.

design · Molecules 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 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 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 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 shrinking changes every side and not their alternating sumthe interior crease is 0.151, at least half of 0.301shrunk byfour sidess₁ − s₂ + s₃ − s₄nothing0.965 0.628 0.740 0.7770.3010.0820.766 0.471 0.597 0.5920.3010.1650.567 0.314 0.455 0.4070.3010.2470.368 0.157 0.312 0.2230.3010.3290.170 0.001 0.170 0.0390.301

Half the difference is a theorem

A quadrilateral's molecule has one crease that does not touch the outline, and on every quadrilateral measured it was at least half the difference between the sums of opposite sides. That was a measurement and has a proof. Shrinking the shape changes every side but not their alternating sum. The proof gives the crease exactly: the difference over twice the sine of half the two angles at the side that vanishes first. A projection argument extends the bound to every even polygon and explains why no odd polygon has one.

design · Molecules

Named alongside it

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

Tree methodThe fold-and-cut theoremUniversal moleculeAxial polygonCrease patternDegeneracyDesign spaceThresholdCompletenessConstructive proofFlat-foldabilityOptimisation

All concepts