Origami & Folding
what one sheet can do
Generator

The count of references and the reach they buy

A generator in the axioms and construction library, called 12 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.

reference-reach 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 count of references and the reach they buyFor no folds, one fold and two: how many reference points the sheet carries, and the radius of the largest disc that can be placed on it without covering one. The count grows by two orders of magnitude and the worst gap closes by less than one.the bar is the largest gap between anywhere on the sheet and a referencea folder who needs a crease somewhere in particular is asking about this and not about the countno folds0.7074 references · worst point 0.50, 0.50 · under 0.710one fold0.3549 references · worst point 0.25, 0.25 · under 0.3572 folds0.089565 references · worst point 0.93, 0.93 · under 0.092141 times the references for 8.0 times the reach, which is what a count hides

view: "map", depth: 2

Where the paper has the least to fold againstThe reference points a given number of folds puts on a square sheet, with the largest disc that can be placed on the paper without covering any of them. The disc's centre is the place a folder has the furthest to go to find something to align to.565 references · worst gap 0.089 of a sheet · under 0.092the disc is where a folder has the least to align againstthe faint lines are the folds themselves; a reference is where two of them cross on the paper

view: "ladder"

The count of references and the reach they buyFor no folds, one fold and two: how many reference points the sheet carries, and the radius of the largest disc that can be placed on it without covering one. The count grows by two orders of magnitude and the worst gap closes by less than one.the bar is the largest gap between anywhere on the sheet and a referencea folder who needs a crease somewhere in particular is asking about this and not about the countno folds0.7074 references · worst point 0.50, 0.50 · under 0.710one fold0.3549 references · worst point 0.25, 0.25 · under 0.3572 folds0.089565 references · worst point 0.93, 0.93 · under 0.092141 times the references for 8.0 times the reach, which is what a count hides

view: "third"

The third fold, sampled because it cannot be listedTwo folds from a bare square reach five hundred and sixty-five references and leave a gap of nearly a tenth of a sheet. The third round specifies more than a quarter of a million distinct folds, which is past enumeration; taking a few hundred of them at random closes the gap by a factor of six.the bar is the largest gap between anywhere on the sheet and a referencea third fold specifies more folds than can be listed, so a sample of them is taken insteadnone of them0.089565 references, from two folds50 of them0.0763,498 references · 0.02% of the round100 of them0.0508,056 references · 0.04% of the round200 of them0.04723,480 references · 0.07% of the round400 of them0.02274,694 references · 0.15% of the round800 of them0.014270,882 references · 0.29% of the roundevery row is a lower bound on what the whole round would buy, because leaving folds out can only make the gap larger

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.

How far from the nearest reference

One fold puts nine reference points on a square sheet and two folds put five hundred and sixty-five. That is sixty-three times as many points, and it brings the worst-covered spot on the paper from a third of a sheet away to a twelfth — four times closer. A count of references is not a measure of what a fold buys, because a set of points can be arbitrarily crowded and still leave most of the sheet out of reach.

The third fold cannot be listed

Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.

Every generator · The axioms and construction field · The patterns a reader can fold