Origami & Folding
what one sheet can do
Theme

Paper is not ideal

Zero thickness, no stretch, infinitely sharp creases, perfect memory. Every one of those is false, and the interesting engineering lives in exactly where each fails.
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.

 the patternconcentric arcs, alternatingwhat the sheet doesa shape with no flat state at allthe curve is in the crease; the saddle is the paper refusing to stretch Curves and material

A crease that curves

Bend a crease and the paper either side is forced into a shape nobody creased. The flat-folding theorems say nothing about it, because they are statements about straight creases meeting at a point.

 cylinderreachablecurved one way onlyconereachablecurved one way, from a pointsphereunreachablecurved two ways — impossiblesaddleunreachablecurved two ways — impossible Curves and material

What a flat sheet can become

A sheet that cannot stretch can only take shapes that are flat in one direction at every point. Cylinders and cones are reachable; a sphere is not, and no amount of folding will get one.

 no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded Curves and material

Four things that are not true

Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.

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