Origami & Folding
what one sheet can do
Generator

A wire made of paper

A generator in the flat-folding library, called 11 times across 3 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

sat-gadget 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

A wire made of paperA strip of four creases assigned V M M V. Every local condition holds, the shape is fixed, and there are exactly two ways to stack it — so the strip carries one bit, and the bit lives in the layer order rather than in the paper. This is the piece the hardness proof is built out of, and it is the piece that can be checked here.state 0state 1V M M V — the same pattern in both2 valid stackings, found by enumerationwhat a junction would addthree wires meeting, with the layer orders forced to disagree —which is a clause, and which is where the reduction gets its powernot drawn and not verified: nothing here decides layer order in two dimensions

assignment: "VMMV", show: "both"

A wire made of paperA strip of four creases assigned V M M V. Every local condition holds, the shape is fixed, and there are exactly two ways to stack it — so the strip carries one bit, and the bit lives in the layer order rather than in the paper. This is the piece the hardness proof is built out of, and it is the piece that can be checked here.state 0state 1V M M V — the same pattern in both2 valid stackings, found by enumerationwhat a junction would addthree wires meeting, with the layer orders forced to disagree —which is a clause, and which is where the reduction gets its powernot drawn and not verified: nothing here decides layer order in two dimensions

assignment: "MVVM", show: "both"

A wire made of paperA strip of four creases assigned V M M V. Every local condition holds, the shape is fixed, and there are exactly two ways to stack it — so the strip carries one bit, and the bit lives in the layer order rather than in the paper. This is the piece the hardness proof is built out of, and it is the piece that can be checked here.state 0state 1M V V M — the same pattern in both2 valid stackings, found by enumerationwhat a junction would addthree wires meeting, with the layer orders forced to disagree —which is a clause, and which is where the reduction gets its powernot drawn and not verified: nothing here decides layer order in two dimensions

assignment: "MVVM", show: "wire"

A wire made of paperA strip of four creases assigned V M M V. Every local condition holds, the shape is fixed, and there are exactly two ways to stack it — so the strip carries one bit, and the bit lives in the layer order rather than in the paper. This is the piece the hardness proof is built out of, and it is the piece that can be checked here.state 0state 1M V V M — the same pattern in both2 valid stackings, found by enumeration

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.

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.

A short reason to say no

When a folding question comes back yes it brings an object anybody can check. When it comes back no it usually brings nothing but the assurance that a search looked everywhere. At one vertex that is false: a refusal comes with a witness one or two steps long, out of a search space of a hundred and twelve, and the witness is a vertex the crease pattern does not contain.

Hardness is about the worst one

Flat-foldability is NP-hard, and every crease pattern on this site is decided in under a second. Both are true, and holding them together is the difference between using the result and repeating it: hardness is a statement about the worst instance a family contains, and nobody folds the worst one.

The gadgets that make it hard

Flat-foldability is NP-hard, and the proof is a construction rather than an obstruction: a machine for turning any satisfiability problem into a sheet of paper that folds exactly when the problem has an answer.

All the figures · The flat-folding field · The patterns a reader can fold