Origami & Folding
what one sheet can do
Generator

A vertex is a one-degree-of-freedom mechanism

A generator in the rigid folding library, called 11 times across 6 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

one-vertex-motion is one function. Everything below came out of it during this build, at arguments 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.

At its defaults

A vertex is a one-degree-of-freedom mechanismOne four-crease vertex at four points in its motion. The four dihedral angles are locked to one another, so setting any one of them sets the rest — which means the vertex has a single degree of freedom and can be built out of rigid panels and hinges.12% folded42% folded72% folded95% foldedno face bends anywhere in the motion — which is what makes it a mechanism rather than a fold

alpha: 60, states: [0.12, 0.42, 0.72, 0.95]

A vertex is a one-degree-of-freedom mechanismOne four-crease vertex at four points in its motion. The four dihedral angles are locked to one another, so setting any one of them sets the rest — which means the vertex has a single degree of freedom and can be built out of rigid panels and hinges.12% folded42% folded72% folded95% foldedno face bends anywhere in the motion — which is what makes it a mechanism rather than a fold

alpha: 60, states: [0.12, 0.42, 0.72, 0.95]

A vertex is a one-degree-of-freedom mechanismOne four-crease vertex at four points in its motion. The four dihedral angles are locked to one another, so setting any one of them sets the rest — which means the vertex has a single degree of freedom and can be built out of rigid panels and hinges.12% folded42% folded72% folded95% foldedno face bends anywhere in the motion — which is what makes it a mechanism rather than a fold

alpha: 45, states: [0.1, 0.4, 0.7, 0.95]

A vertex is a one-degree-of-freedom mechanismOne four-crease vertex at four points in its motion. The four dihedral angles are locked to one another, so setting any one of them sets the rest — which means the vertex has a single degree of freedom and can be built out of rigid panels and hinges.10% folded40% folded70% folded95% foldedno face bends anywhere in the motion — which is what makes it a mechanism rather than a fold

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

A property you can dial

Steel has one Poisson's ratio. A Miura-folded sheet has a surface of them, and where on that surface it sits is set by the panel shape and by how far it happens to be folded — which is why it is a mechanism rather than a material.

A sheet with one freedom

A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.

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.

Paper that folds itself

A self-folding sheet has to supply the fold and then choose what to fold into. The second half is where these things fail, and no amount of torque helps, because the two outcomes are equally downhill.

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 motion has no letters to choose

A flat-folding search picks a letter for every crease and can pick badly. A rigid folding does not pick anything: the fold angles are real numbers, determined by the panels through equations that have a solution or do not. Replacing a discrete choice with a continuous solve removes every ordering question at once, and introduces a failure of its own.

Every generator · The rigid folding field · The patterns a reader can fold