Origami & Folding
what one sheet can do
Concept

Mechanism — where it appears

An assembly of rigid parts joined so that it can move. A rigidly folding sheet is one, and the whole engineering interest in this subject is that a flat sheet can be a mechanism at all.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

the pattern as it isevery edge keeps its length6.7e-16the same pattern, moved by 0.01and one of them cannot1.7e-21e-181e-161e-141e-121e-101e-81e-61e-41e-21largest change in any edge length, in panel widthswhat an isometry has to do, and what it manages5 × 4 panels, at 50% folded, every edge of both comparedthe moved pattern is fitted the best single panel its own edge lengths allow before being folded at allso the gap is not a bad choice of panel — it is what is left when the best choice has been made

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.

rigid · Rigid folding cuts at 85% of the pitch0.15 of material between one cut and the nextcut length, as a fraction of the pitchligament6 piecesone piece at every length up to 97%and the count is worked out from the cuts, not measured off a picture

One cut short of falling apart

Everything a cut sheet can do is bought out of the material between the end of one cut and the start of the next. That material shrinks to nothing in a straight line as the cuts grow, and the sheet stays in one piece the whole way down — until the instant it does not, and then it is in six.

material · Kirigami 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 the period cell of the Miuraone period, with its neighbours round it2 interior vertices in the cell7 crease pieces drawnperiod 1.000 × 2.000one column wide and two rows high, because the zigzag returns after twothe cell is a rectangle of ordinary paper until somebody says its edges are one edge

A mechanism that closes on itself

A rigid-foldable pattern is a mechanism: panels as rigid plates, creases as hinges, and a motion counted by degrees of freedom at each vertex. Close the sheet into a tube and the mechanism has to come back to itself after a circuit — a constraint that is not at any vertex and that the degree-of-freedom count does not see.

rigid · Rigid folding

Named alongside it

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

Degrees of freedomKinematicsRigid-foldabilityConnectivityConstraintFold angleGluingHingeIdealisationKirigamiMiuraMiura-ori

All concepts