Origami & Folding
what one sheet can do
Generator

What the order settles, and what it leaves

A generator in the flat-folding library, called 7 times across 2 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.

object-order 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

What the order settles, and what it leavesFor strips with a given number of creases on a given grid: how many distinct folded profiles there are, how many of those are reached by more than one crease pattern, and how many of those ambiguities survive being handed the complete layer order as well.what the outline and the thickness leave open, and what the order closesprofilesambiguousthe order settlesand does not3 creases on 12ths69166103 creases on 16ths1844712354 creases on 12ths23371692the last column is patterns that fold to the same object, so no better photograph reaches them

view: "census", sizes: [3,12, 3,16, 4,12]

What the order settles, and what it leavesFor strips with a given number of creases on a given grid: how many distinct folded profiles there are, how many of those are reached by more than one crease pattern, and how many of those ambiguities survive being handed the complete layer order as well.what the outline and the thickness leave open, and what the order closesprofilesambiguousthe order settlesand does not3 creases on 12ths69166103 creases on 16ths1844712354 creases on 12ths23371692the last column is patterns that fold to the same object, so no better photograph reaches them

view: "census", sizes: [3,12, 4,12]

What the order settles, and what it leavesFor strips with a given number of creases on a given grid: how many distinct folded profiles there are, how many of those are reached by more than one crease pattern, and how many of those ambiguities survive being handed the complete layer order as well.what the outline and the thickness leave open, and what the order closesprofilesambiguousthe order settlesand does not3 creases on 12ths69166104 creases on 12ths23371692the last column is patterns that fold to the same object, so no better photograph reaches them

view: "pair", k: 3, grid: 12

The same object, from two different patternsAbove, two strips with creases in different places. Below each, the folded object with its layer order drawn out: the horizontal extent of each bar is where that piece of the strip lands, and its height is its place in the stack. The two objects are identical — same bands, same thicknesses, same pairs of layers joined at every fold.two strips, creased in different places3 creases on 12ths of the strip1/122/124/12marked MVVand what it folds to2/124/125/12marked MMVand what it folds tothe two folded objects agree in every band and at every 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.

The order does not name it either

A photograph of a folded model carries an outline and a layer count, and that is not enough to recover the pattern. Hand the observer the layer order as well — everything the object physically is — and most of the ambiguity goes. Most. What is left are pairs of genuinely different crease patterns that fold to the same object, which no better photograph reaches.

Which side is showing

Two earlier rungs asked what a folded object records about the pattern that made it, first from its outline and then from its complete layer order. Neither observation is one anybody can make. A photograph of duo paper carries the outline, the thickness and the colour showing at every point — and over the whole census the colour separates nothing at all.

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