Origami & Folding
what one sheet can do
Concept

Rigid origami — where it appears

Folding in which the panels stay flat and every deformation happens at the creases. It is the version manufactured out of metal, plastic and silicon, and almost nothing folded by hand satisfies it.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

the creaseno panels at all128.3° of turning4 segments8 panels128.3° of turning8 segments16 panels128.3° of turning16 segments32 panels128.3° of turningone crease, cut into flat pieceseach panel keeps its width across the crease and loses length along it as the count risesa finer approximation is a better picture and the same total kink, spread over more joints

A curve has no panels

A rigid folding is a finite list of flat pieces joined along lines. A curved crease has no such list, and refining one does not help: the kink at each joint falls as one over the segment count, and the total of the kinks does not fall at all, because it is a constant of the curve.

material · Curved creases the opposite crease, ×1the odd crease, ×0.26794910how far the vertex has foldedsectors 60° · 90° · 120° · 90°constant to 8.2e-13 over 199 points of the motionand equal to cos((α+β)/2) ÷ cos((α−β)/2), which the solver never forms

The vertex is geared

A rigid four-crease vertex has one degree of freedom, which says that one number decides everything and not how. The how is a fixed ratio: the tangents of the half fold angles at two creases stay in constant proportion for the whole of the motion, and the proportion is a function of the sector angles and nothing else.

rigid · Rigid folding

Named alongside it

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

Curved creaseDegrees of freedomDevelopable surfaceFold angleKinematicsMechanismMiura-oriPanelRuling lineTotal turning

All concepts