Origami & Folding
what one sheet can do
Concept

Universal molecule — where it appears

The crease pattern that fills a polygon of a circle packing, built by shrinking the polygon inward at a uniform rate. It always works for a convex polygon and runs into trouble at a reflex corner.

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

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 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 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 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.

Straight skeletonTree methodDegeneracyDesign spaceThresholdAxial polygonCrease patternDesign techniqueThe fold-and-cut theoremIncircleInsetOptimisation

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