Origami & Folding
what one sheet can do
Generator

The letterings that fold, and what joins them

A generator in the flat-folding library, called 14 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.

move-graph 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

The letterings that fold, and what joins themEvery mountain-and-valley lettering a single vertex folds flat in, joined wherever one small change turns one into another. Whether the picture is one piece or several is the question: a change of any two creases joins all of them, and a change of two neighbouring creases does not.8 letterings fold · 2 pieces under two neighbouring creasesflip two creases that are next to one another, which is the change a folder makes by handMMVVVVMMMMVVMVVVMVMVMMMVVMVVVMVMMMVMVVVVMMVVMMMMsectors 43° · 110° · 121° · 57° · 16° · 13°the two pieces are each other turned overevery crease at once: changes piece

view: "pattern"

What a local move cannot reachCrease patterns with more than one vertex, with the letterings their conditions admit sorted into the pieces a local move joins. The count of pieces is exactly two to the power of the number of creases whose two ends are both inside the paper — the creases a folder cannot change without changing two vertices at once.a single vertex is always one piece; a pattern with more is notand the number of pieces is decided by the creases that never reach the edge of the paperThe preliminary base1 vertices inside the paper112 letterings admitted1 piece of 1120 creases buried2^0 = 1The square twist4 vertices inside the paper256 letterings admitted16 pieces of 164 creases buried2^4 = 16The hexagon twist6 vertices inside the paper4096 letterings admitted64 pieces of 646 creases buried2^6 = 64Fold and cut — the triangle1 vertices inside the paper30 letterings admitted1 piece of 300 creases buried2^0 = 1

view: "graph", degree: 6, sampler: "generic", move: "pair"

The letterings that fold, and what joins themEvery mountain-and-valley lettering a single vertex folds flat in, joined wherever one small change turns one into another. Whether the picture is one piece or several is the question: a change of any two creases joins all of them, and a change of two neighbouring creases does not.8 letterings fold · 1 piece under any two creasesflip two creases anywhere round the vertex, which is the smallest change Maekawa allowsMMVVVVMMMMVVMVVVMVMVMMMVVMVVVMVMMMVMVVVVMMVVMMMMsectors 43° · 110° · 121° · 57° · 16° · 13°one piece: every folding is reachableevery crease at once: stays inside its own piece

view: "graph", degree: 6, sampler: "generic"

The letterings that fold, and what joins themEvery mountain-and-valley lettering a single vertex folds flat in, joined wherever one small change turns one into another. Whether the picture is one piece or several is the question: a change of any two creases joins all of them, and a change of two neighbouring creases does not.8 letterings fold · 2 pieces under two neighbouring creasesflip two creases that are next to one another, which is the change a folder makes by handMMVVVVMMMMVVMVVVMVMVMMMVVMVVVMVMMMVMVVVVMMVVMMMMsectors 43° · 110° · 121° · 57° · 16° · 13°the two pieces are each other turned overevery crease at once: changes piece

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.

The creases that cannot move

One vertex's foldings are always joined up. A pattern's are not, and the number of pieces they fall into is exactly two to the power of the number of creases with an interior vertex at each end — four on a square twist, six on a hexagon twist, none at all on a preliminary base. The creases a local change cannot reach are the creases that never reach the edge of the paper.

The pieces without the list

The letterings a pattern folds in fall into pieces no folder can cross, and the count was found by writing every lettering down — which stops at eighteen creases. The Miura has thirty-eight, the Yoshimura eighty-six, and the number of pieces can be read off the drawing without listing anything: four million and two hundred and eighty-one million million.

Walking between two foldings

The letterings a vertex folds in are always counted and never navigated. Counting says a generic degree-six vertex has eight of them; navigating says that changing any two creases turns any one into any other, and that changing two neighbouring creases does not — and that the vertices which come apart are the ones with no coincidences in them, which is the opposite of what every other measurement here would suggest.

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