Origami & Folding
what one sheet can do
Field

Curves and material

Curved creases, developable surfaces, and everything the zero-thickness sheet was lying about.
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

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

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

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.

All essays