Origami & Folding
what one sheet can do
Generator

How many things have to pull

A generator in the folding nobody designed library, called 9 times across 5 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.

dof-census 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

How many things have to pullThe degrees of freedom of four folding geometries and the number of drivers each therefore needs. A wing that opens without a muscle at every crease is not a wing with clever muscles; it is a pattern whose state is determined by one number.geometryfreedomsdrivers neededone degree-four vertexfour assignments, one motion each11Miura, 5 × 412 interior vertices, still one freedom11parallel corrugationno interior vertex to couple1112 vertices, uncoupledwhat a pattern costs when nothing constrains it1212or a sequencerone freedom is one actuator — the count is what makes a passive deployment possible at all

cases: [vertex, miura, corrugation, free], cols: 5, rows: 4, angle: 0.4

How many things have to pullThe degrees of freedom of four folding geometries and the number of drivers each therefore needs. A wing that opens without a muscle at every crease is not a wing with clever muscles; it is a pattern whose state is determined by one number.geometryfreedomsdrivers neededone degree-four vertexfour assignments, one motion each11Miura, 5 × 412 interior vertices, still one freedom11parallel corrugationno interior vertex to couple1112 vertices, uncoupledwhat a pattern costs when nothing constrains it1212or a sequencerone freedom is one actuator — the count is what makes a passive deployment possible at all

cases: [vertex, miura, corrugation, free], cols: 5, rows: 4

How many things have to pullThe degrees of freedom of four folding geometries and the number of drivers each therefore needs. A wing that opens without a muscle at every crease is not a wing with clever muscles; it is a pattern whose state is determined by one number.geometryfreedomsdrivers neededone degree-four vertexfour assignments, one motion each11Miura, 5 × 412 interior vertices, still one freedom11parallel corrugationno interior vertex to couple1112 vertices, uncoupledwhat a pattern costs when nothing constrains it1212or a sequencerone freedom is one actuator — the count is what makes a passive deployment possible at all

cases: [vertex, free], cols: 4, rows: 3

How many things have to pullThe degrees of freedom of four folding geometries and the number of drivers each therefore needs. A wing that opens without a muscle at every crease is not a wing with clever muscles; it is a pattern whose state is determined by one number.geometryfreedomsdrivers neededone degree-four vertexfour assignments, one motion each116 vertices, uncoupledwhat a pattern costs when nothing constrains it66or a sequencerone freedom is one actuator — the count is what makes a passive deployment possible at all

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 corrugation has one resting state

A folded wing held short of shut stores energy in its hinges, and a wing that could stay both open and folded with nothing holding it would need that energy to have two bottoms. A corrugation cannot provide them: every crease in it folds by one angle, so the energy of any set of crease springs is a parabola in that angle and has exactly one resting state, however much the springs disagree. A single degree-four vertex has two branches through the flat state, and the same springs give it two resting states on almost every setting tried.

Four finders, one option

Four unrelated lineages arriving at the same corrugation is read as evidence that the corrugation is good. It is at least as much evidence that there was nothing else to arrive at: how far a folded sheet shrinks is exactly its average layer count, so a lineage choosing a packing ratio is choosing a number of layers and nothing else — and the quantity that is genuinely free turns out to be almost uncorrelated with it.

No motor in the fold

An insect's wing has muscles at its base and nothing out along its length, so the pattern has to carry the deployment by itself. The condition that makes that possible is a count: one degree of freedom means one number determines every panel, which means one thing has to pull.

The census returns one

The rung below this one asked for a census: every pattern reaching a stated packing ratio while opening from a single input, with its crease density. The census is makeable for the corrugations and it comes back with one member. A corrugation piles its panels over a footprint as wide as its longest panel, so its ratio is the total length divided by that longest one — and that equals the panel count only when every panel is the same.

Which crease to push

Deciding one fold angle settles every other one on a quadrilateral mesh, which is what makes a self-folding sheet buildable with a single actuator. It leaves a question that sounds like an afterthought: which crease. Driving each of a mesh's twenty-four in turn gives twenty-four different answers to how far an error in it travels — and on the sheet that repeats one vertex, it gives several answers to what shape the sheet takes.

Every generator · The folding nobody designed field · The patterns a reader can fold