One family, and the one member of it that moves
quad-family 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
view: "propagate", rho: 0.6
view: "patterns", factors: [0.7, 1, 1.4], rho: 0.6
view: "residual", rho: 0.6
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.
- a mesh with no straight crease family and no two vertices alike folds rigidly, closing to 1e-12 ×2
- developability holds at every interior vertex — its sectors close to 360° — 4 checked ×1
- fixing one crease settles 2 of 12 creases under the flat conditions and 12 of 12 under the rigid ones ×1
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 4 checked ×1
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 4 checked ×1
- sliding one vertex along the ray Kawasaki forces leaves the worst residual at 9e-16 at every setting ×1
- the a flat-foldable quadrilateral mesh is put past all four theorems before it is drawn ×1
- the a general quadrilateral mesh that folds is put past all four theorems before it is drawn ×1
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 4 checked ×1
- the four vertices round one face are solved one at a time, and the walk returns to the crease it started at ×1
- the obstruction leaves zero first order in the displacement, at about 0.82 radians per panel length ×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.
One crease decides the sheet
Fix one crease of a flat-folding problem, propagate every condition the subject has, and three creases out of a hundred and fifty-eight follow. Fix one fold angle of a rigid one and every crease on the sheet follows, with a single consistent answer. The same experiment, two questions, opposite answers — and it is why a self-folding sheet needs one biased vertex rather than one per vertex.
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.
All the figures · The rigid folding field · The patterns a reader can fold