Origami & Folding
what one sheet can do
Generator

Different patterns, one folded object

A generator in the flat-folding library, called 8 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.

same-outline 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

Different patterns, one folded objectCrease patterns that a photograph of the folded strip cannot tell apart. The creases are in different places and the folded profiles — the outline, and the number of layers over every point of it — are identical.4 patterns, one folded profile1/122/124/127/1225311/123/126/128/1225312/124/125/127/1225312/125/127/128/122531foldedthe layer counts under each band are the same in every row, and so are the widths

k: 4, gsize: 12, view: "pair"

Different patterns, one folded objectCrease patterns that a photograph of the folded strip cannot tell apart. The creases are in different places and the folded profiles — the outline, and the number of layers over every point of it — are identical.4 patterns, one folded profile1/122/124/127/1225311/123/126/128/1225312/124/125/127/1225312/125/127/128/122531foldedthe layer counts under each band are the same in every row, and so are the widths

k: 4, gsize: 12, view: "pair", pick: 1

Different patterns, one folded objectCrease patterns that a photograph of the folded strip cannot tell apart. The creases are in different places and the folded profiles — the outline, and the number of layers over every point of it — are identical.4 patterns, one folded profile1/122/124/128/122531/124/128/1210/122532/124/125/128/122532/126/129/1210/12253foldedthe layer counts under each band are the same in every row, and so are the widths

k: 4, gsize: 12, view: "census"

How much the folded object remembersEvery folded profile a strip with four creases on a twelfths grid can reach, sorted by how many different crease patterns reach it. The profiles on the left name their pattern; the ones to the right of them are reached by two, three or four patterns that no photograph of the folded strip could tell apart.4 creases on a grid of 12: how many patterns share one folded profile162onepattern602patterns13patterns104patterns30% of the profiles are ambiguous, and the folded strip says nothing about which pattern made it

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.

The shadow does not name the pattern

A photograph of a folded model carries an outline and a thickness at every point of it, and that is the whole of what it carries. It is not enough. Crease patterns in genuinely different places fold to identical outlines with identical layer counts, and nearly a third of the folded objects a short strip can reach are reached by more than one pattern.

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