Origami & Folding
what one sheet can do
Generator

One vertex, folded away two creases at a time

A generator in the flat-folding library, called 9 times across 3 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.

crimp-ladder 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

One vertex, folded away two creases at a timeA vertex of degree six, and the sequence of smaller vertices the crimp reduction takes it through. Each step folds the sector strictly smaller than both its neighbours away between them, which removes two creases and merges three sectors into one. The shaded wedge is the sector about to go.the vertex on the paper6 creases0 crimps, and what is left is one straight crease with one letterthe four conditions do not all hold · a stacking does not exist

view: "refusal", angles: [13.8, 68.7, 71.1, 94.2, 95.1, 17.1], assignment: "MVVVMV"

Where the reduction stopsA vertex of degree six, and the sequence of smaller vertices the crimp reduction takes it through. Each step folds the sector strictly smaller than both its neighbours away between them, which removes two creases and merges three sectors into one. The shaded wedge is the sector about to go.every condition holds here6 creases4 creasesevery condition holds at the vertex on the paper — and one crimp later the smallest sector has the same letter on both sidesthe four conditions all hold · a stacking does not exist

view: "ladder", angles: [34, 96, 146, 84], assignment: "MVMV"

One vertex, folded away two creases at a timeA vertex of degree six, and the sequence of smaller vertices the crimp reduction takes it through. Each step folds the sector strictly smaller than both its neighbours away between them, which removes two creases and merges three sectors into one. The shaded wedge is the sector about to go.the vertex on the paper4 creases0 crimps, and what is left is one straight crease with one letterthe four conditions do not all hold · a stacking does not exist

view: "ladder", angles: [52, 44, 38, 66, 71, 89], assignment: "MVMVVV"

One vertex, folded away two creases at a timeA vertex of degree six, and the sequence of smaller vertices the crimp reduction takes it through. Each step folds the sector strictly smaller than both its neighbours away between them, which removes two creases and merges three sectors into one. The shaded wedge is the sector about to go.the vertex on the paper6 creases0 crimps, and what is left is one straight crease with one letterthe four conditions do not all hold · a stacking does not exist

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 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.

Crimp it away and ask again

Four conditions decide whether a vertex folds flat, and they decide it exactly at a vertex whose sectors are all different sizes. Everywhere else they over-count: two markings of every tied four-crease vertex, twelve of the degree-six vertex this site prints nine of on one sheet. What decides the case is not a fifth condition but a procedure — fold the smallest sector away and ask the smaller vertex.

The cost is in the coincidences

How big an instance is, is what a hardness statement is about, and it is the weaker predictor of what deciding one costs. Hold the degree fixed and vary only how many of a vertex's sectors are equal: the work of deciding it rises by a factor of nearly three, against a factor of two for doubling the number of creases. The expensive instances are the ones a designer draws on a grid.

Every generator · The flat-folding field · The patterns a reader can fold