Two different questions
rigid-versus-flat is one function. Everything below was drawn by it when the site was
last published, at settings taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
With nothing specified
outer: [the bird base, most traditional models, anything with a squash fold], inner: [the Miura fold, the Yoshimura pattern, everything ever manufactured]
outer: [the bird base, most traditional models], inner: [the Miura fold, everything ever manufactured]
outer: [a general quadrilateral mesh, any pattern with a squash fold], inner: [the Miura fold, the meshes solved for here]
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- both sets have examples in them and few enough to fit beside the diagram ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
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.
From a shell to a solar array
The Miura fold was published in 1970 and flew on a satellite in 1995. The gap is not ignorance — the pattern was known, understood and available the whole time — and the same twenty-five year lag appears between every folding result and the hardware that uses it.
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.
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 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.
What the vertex does on the way
A four-crease vertex is a linkage on a sphere. Solving its closure gives the fold angles at every moment, and two theorems that are usually proved about the flat state turn up in the answer without being put there.
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.
All the figures · The rigid folding field · The patterns a reader can fold