Rigid-foldability — where it appears
Named by 10 essays across one field — each of them below, with the objects they name alongside it.
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.
A state no motion reaches
Flat-foldability asks whether a folded state exists. Rigid-foldability asks whether there is a path to it. The two sets are different, and the difference can be counted on a single vertex.
The only pattern that moves
A rigid motion is not a generic property of a folded pattern. Move one vertex of a Miura by a thousandth of a panel and the sheet has no isometric folded position of that kind at all — and the amount by which it fails is first order in the displacement, so no move is small enough to be free.
The family the Miura belongs to
Move one vertex of a Miura and the sheet has no rigid folded position at all — which leaves the obvious question unanswered. What else moves? A row of paper reflected in each of a fan of lines is flat-foldable for nothing at all, and whether it also folds rigidly turns out to be a condition on a table of cosines: it has to be a column of numbers times a row of numbers.
Where an error goes
A misplaced crease in a folded sheet has to be paid for somewhere, and this subject has two answers already — the error is folded too, and the hinge is where it ends up. There is a third. In a quadrilateral mesh a mistake in one row has no consequence in that row at all: it is felt by the columns, which is to say by every other row on the sheet.
The condition that is not flat-foldability
Take away the assumption that one crease family runs straight through every vertex and ask what makes a quadrilateral mesh move. It is not flat-foldability. There is a one-parameter family of meshes, every one of them developable and flat-foldable at every vertex to machine precision, and exactly one member of it folds — the Miura. Slide a single vertex along the ray that keeps every condition exact and the sheet stops moving, first order in the displacement.
A collision is an order
Paper passing through paper is treated here as a thing that happens during a motion and is caught by watching for it. At the flat state it is not an event at all: it is the absence of an ordering, and it can be proved rather than observed. Four of the six quadrilateral meshes this site solves for rigid folding place perfectly and admit no ordering of their nine panels — so every one of them must pass through itself, and none of them was ever driven to find out.
Fourth of eight, and still not chosen for it
A deployable is sold on compaction: large in use, small in transit. Measured, the pattern that actually gets built converts folding into compaction at 0.67 sheet-widths of crease per layer, which is fourth of the eight printed patterns — nearly three times worse than the Yoshimura, which nobody deploys, and nearly three times better than the hexagon twist, which nobody deploys either. The ranking does not pick out the pattern that flew from anywhere on the shelf, and that is the finding.
Two mechanisms at one point
Two creases drawn across each other cannot fold flat — Maekawa's count refuses them at every angle. They move perfectly well as rigid panels, and they move in two ways: bend along one line while the other stays flat, or the reverse. Every other developable vertex of degree four has two ways too, and in both of them all four creases move together at a fixed ratio. The crossing is the case where the two motions have nothing to do with each other.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Fold angleClosureDegrees of freedomKawasaki's theoremKinematicsMiura-oriSpherical linkageDegree-fourMechanismQuadrilateral meshRigid foldingAnisotropy