Origami & Folding
what one sheet can do
Generator

One assignment, every pile it allows

A generator in the flat-folding library, called 17 times across 4 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.

layer-census 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 assignment, every pile it allowsA strip with its creases marked, drawn once for each way the layers may be stacked. Every one of these is the same crease pattern with the same mountains and valleys; they differ only in which layer lies over which, which the pattern never said.creases at 0.25, 0.50, 0.75, marked VVV3 legal stackings of 4 segments, read from the bottom of the pile up12341: 1 · 3 · 4 · 2taco-taco12342: 3 · 1 · 2 · 4taco-taco12343: 3 · 4 · 1 · 2assignmentthe paper lands in the same place every time — only the order through the pile differs

view: "orders", creases: [0.25, 0.5, 0.75], assignment: "MMM"

One assignment, every pile it allowsA strip with its creases marked, drawn once for each way the layers may be stacked. Every one of these is the same crease pattern with the same mountains and valleys; they differ only in which layer lies over which, which the pattern never said.creases at 0.25, 0.50, 0.75, marked MMM3 legal stackings of 4 segments, read from the bottom of the pile up12341: 2 · 1 · 4 · 3assignment12342: 4 · 2 · 1 · 3taco-taco12343: 2 · 4 · 3 · 1taco-tacothe paper lands in the same place every time — only the order through the pile differs

view: "orders", creases: [0.25, 0.5, 0.75]

One assignment, every pile it allowsA strip with its creases marked, drawn once for each way the layers may be stacked. Every one of these is the same crease pattern with the same mountains and valleys; they differ only in which layer lies over which, which the pattern never said.creases at 0.25, 0.50, 0.75, marked VVV3 legal stackings of 4 segments, read from the bottom of the pile up12341: 1 · 3 · 4 · 2taco-taco12342: 3 · 1 · 2 · 4taco-taco12343: 3 · 4 · 1 · 2assignmentthe paper lands in the same place every time — only the order through the pile differs

view: "count", creases: [0.25, 0.5, 0.75]

The same creases, every way of marking themEvery mountain-and-valley assignment of one set of crease positions, with the number of legal layer orderings each one admits. Some admit none, some exactly one, and some several — and the letters look equally definite in all three cases.legal layer orderings, by assignment3VVVseveral2MVVseveral1VMVone2MMVseveral2VVMseveral1MVMone2VMMseveral3MMMseveralcreases at 0.25, 0.50, 0.75 — the assignment only decides which way each U-turn wraps

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.

More than one way to lie flat

A crease pattern with its mountains and valleys marked is spoken of as though it named a folded object. It does not. The legal stackings can be counted exactly in one dimension, the count is routinely more than one, and its size is a property of the pattern that nobody quotes.

No height to swap

A folded strip is a permutation of segments, and the smallest change a hand can make to it is a swap of two heights: 672 stackings, 560 of them isolated. A folded sheet has no height. Its layers are ordered by statements about which panels share ground, and on every printed pattern the search can finish, the answer is one stacking and no way out of it.

Nothing slides past anything

A marked strip has several legal stackings and this site has counted them at length. Nobody asked whether a folder holding one can reach another by lifting a flap over its neighbour: five hundred and sixty of six hundred and seventy-two stackings have no such move at all, and whether any exists depends on the parity of the segment count.

One marking, many objects

A crease pattern with every mountain and valley written on it is spoken of as though it named a folded model. At four creases it does. At six it need not, and at the eight-crease vertex in the middle of the first base anybody folds, a single marking can be folded into four genuinely different objects — same creases, same letters, four answers.

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