Origami & Folding
what one sheet can do
Generator

A strip folded by the all-layers machine

A generator in the what it costs to know library, called 15 times across 7 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.

fold-sequence 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

A strip folded by the all-layers machineThe pile of paper after each fold, drawn from the simulator's own states rather than from a description of them. The dashed line marks where the next fold happens. Each layer is one run of the strip that has not yet been folded anywhere along its length.creases at 0.20, 0.40, 0.60, 0.80 — assignment MVMVflat1 layerafter fold 12 layersafter fold 23 layersafter fold 34 layersafter fold 45 layers4 folds, and the finished pile satisfies the assignment — checked against the layer-ordering rules

creases: [0.2, 0.4, 0.6, 0.8], assignment: "MVMV", model: "one"

A strip folded by the one-layer machineThe pile of paper after each fold, drawn from the simulator's own states rather than from a description of them. The dashed line marks where the next fold happens. Each layer is one run of the strip that has not yet been folded anywhere along its length.creases at 0.20, 0.40, 0.60, 0.80 — assignment MVMVflat1 layerafter fold 12 layersafter fold 23 layersafter fold 34 layersafter fold 45 layers4 folds, and the finished pile satisfies the assignment — checked against the layer-ordering rules

creases: [0.4, 0.5], assignment: "MV", model: "all"

The all-layers machine cannot finish this stripThe pile of paper after each fold, drawn from the simulator's own states rather than from a description of them. The dashed line marks where the next fold happens. Each layer is one run of the strip that has not yet been folded anywhere along its length.creases at 0.40, 0.50 — assignment MVflat1 layerno sequence of all-layers folds finishes this pattern — the search exhausted 5 states

creases: [0.2, 0.4, 0.6, 0.8], assignment: "MVMV", model: "all"

A strip folded by the all-layers machineThe pile of paper after each fold, drawn from the simulator's own states rather than from a description of them. The dashed line marks where the next fold happens. Each layer is one run of the strip that has not yet been folded anywhere along its length.creases at 0.20, 0.40, 0.60, 0.80 — assignment MVMVflat1 layerafter fold 12 layersafter fold 23 layersafter fold 34 layersafter fold 45 layers4 folds, and the finished pile satisfies the assignment — checked against the layer-ordering rules

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.

Deciding is not making

Four earlier essays here ask which machines can flatten a strip at all, and the answer sorts them into a lattice with one column full and three with holes in it. Asked instead what each machine can produce, the three sort completely differently: the machine that may choose its block reaches every folded state of every strip tried, the machine that takes one layer reaches exactly four whatever the strip is and however long, and the machine that takes the whole pile is the only one whose answer depends on the spacing at all.

The fold a machine can make

A theorem that says a folded state exists says nothing about getting there. A machine that folds every layer at once is stopped by a strip with two creases in it — one that folds flat perfectly well, and that a pair of hands folds in about four seconds.

The half no notation records

Every notation this subject has invented writes down the crease pattern or the sequence of folds, and the crease pattern is the half that does not decide the folded object. The field's interchange format has a place for the other half and nothing fills it in — including the files published here, which carry every vertex, edge and letter of a Yoshimura and none of the three hundred bits that would say which of its layer orders the folded object is.

The patient machine is the weak one

A machine that folds one layer at a time sounds like a machine with more freedom, not less. It has less, and the reason is the most ordinary fact about paper there is: it is joined, so whatever a machine declines to hold it also cannot move.

Two directions that will not separate

A map has rows and columns, and a strip of stamps is a map with one row. The obvious hope is that the two-dimensional count is built from the one-dimensional one — fold the rows, then fold the columns. It is not: a two-by-three map folds 60 ways against a product of 12, and the discrepancy grows from a factor of two to a factor of thirty-eight over the counts anybody has.

What a dashed line can say

Before the Yoshizawa–Randlett symbols a model could not be transmitted, and the subject was not cumulative. The basic notation says exactly one thing — fold this crease, this way, now — which is precisely a simple fold, and the share of flat foldings that simple folds reach collapses from 71% to 13% as a model grows.

Where the machine catches up

The weakest machine in the subject folds every layer at once and is stopped by a strip with two creases in it. On a strip of equal stamps it is stopped by almost nothing: every one of the 288 folded states a six-stamp strip has is reachable by a sequence of all-layers folds, and on every unevenly creased strip tried it reaches none of them. At seven stamps the completeness ends, and finding out where it ended is what checking it past six was for.

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