Origami & Folding
what one sheet can do
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From craft to hardware

The same mathematics that folds a paper crane deploys a solar array, packs an airbag and threads a stent through an artery. The scale changes; the constraints do not.
at every vertexthree of one, one of the other15 interior vertices, all identicalwhat the sheet gainsone degree of freedom, not manyit opens and closes in both directions at oncea negative Poisson's ratio22 mountain and 16 valley creases · 39.3 sheet-widths of foldingmountainvalleyraw edge Tessellations

One vertex, repeated

Take a single flat-foldable vertex and tile the plane with it. The sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson's ratio that the paper never had.

 nearly flatwidth ×0.87 height ×0.50half closedwidth ×0.58 height ×0.82nearly packedwidth ×0.11 height ×0.99both dimensions shrink together — pulling it open in one direction opens it in the other Tessellations

A sheet with one freedom

A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.

 found in crushed drink cans, tree bark and deployable boomsthe pattern is a consequence of thin-wall buckling, not of a designmountainvalley Tessellations

Patterns nobody designed

Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.

 folds flatfolds rigidlyas panelsin the outer setthe bird basemost traditional modelsanything with a squash foldin the inner setthe Miura foldthe Yoshimura patterneverything ever manufacturedpaper cheats by bending very slightly; sheet metal does not Rigid folding

Panels instead of paper

Flat-foldability asks whether a pattern can reach a flat state. Rigid-foldability asks whether it can get there without any face bending on the way. The second is much stronger, and everything that gets manufactured lives inside it.

 zero thicknesspanels meet exactlyfour layers of real materialeach fold has to clear the ones belowhinge offset to the surfacethe panel rotates about the right linea crease pattern describes a surface with no thicknessand everything anybody builds has some Rigid folding

The sheet has a thickness

Every crease pattern describes a surface with no thickness. Everything anybody builds has some, and getting it around a corner is the central problem of turning origami into hardware.

 Miura solar array17×Space Flyer Unit, 1995airbag folding25×stored for years, opens in 30 msheart stent6×threaded through an arterystarshade11×26 m disc, 2.5 m launch tubemap fold9×the original problempackeddeployedthe ratio is what is bought; one degree of freedom is what makes it reliable Rigid folding

Folding that gets built

Solar arrays, airbags, stents and starshades. The requirement is always the same — large in use, small in transit, along a path nobody has to trust to chance — and folding is what answers it.

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