Origami & Folding
what one sheet can do
Generator

One crossing, and then another: 1/5 of the square

A generator in the axioms and construction library, called 17 times across 5 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

n-division is one function. Everything below was drawn by it when the site was last published, at settings 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.

With nothing specified

One crossing, and then another: 1/5 of the squareThe anti-diagonal is folded once. Crossing it with the line from the corner through the mark at one half gives one third; crossing it with the line through one third gives one quarter, and so on. Every crossing is exact, and the ladder costs one fold per rung after the first.1/25 folds to reach 1/5, and every mark on the way is exactthe solid line is the one fold used at every step; the dashed lines are the stepsnothing here converges — each crossing lands on its fraction and stops

view: "table", upTo: 9, steps: 10

Exact at every rungEach rung of the crossing ladder, the fraction it lands on as an exact ratio of integers, the number of folds it took, and — for comparison — how far the halving method still is from the same fraction after ten folds. One arrives and the other approaches.partswhere the crossing landsfoldshalving, after 10 folds21/22off by 1.00e-131/33off by 6.51e-541/44off by 2.54e-651/55off by 1.91e-761/66off by 2.39e-871/77off by 4.25e-981/88off by 9.74e-1091/99off by 2.69e-10the fractions are computed as pairs of integers, so “exact” is settled by a comparison and not by a tolerancethe same construction in doubles agrees to about three parts in 10¹⁷, so this is not a claim about roundingit is a claim about arriving

view: "by-hand", sigmaMM: 0.5, mm: 150

Exact against accurateHow far each method's final crease lands from the true fraction, for a folder who places every crease 0.5 millimetres out on a 150 millimetre sheet. The exact method's error grows with the number of parts because it makes a fold for each of them; the convergent one's does not.4681012141600.10.20.30.40.50.60.7parts the strip is divided intohow far the crease lands out, mmthe exact ladderFujimotothey cross at 4

view: "decay", n: 9

What each method does to one mistakeA single error injected at the first fold, with nothing else going wrong, and what is left of it at each fold after. The exact ladder shrinks it too — its trouble is that it makes a fold for every part and each of them adds one of its own.12345678910-8-6-4-20folds since the mistakewhat is left of it, powers of tenthe exact ladderFujimoto

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

Cheap where it reaches

Two folds from a bare square put marks at a half, a third, a quarter, a fifth, a sixth, an eighth and a twelfth — and at no seventh, ninth or eleventh at all. A rule that reaches every fraction takes n folds to reach one nth. The systematic route and the short one disagree everywhere, and neither of them knows about the other.

Exact is not accurate

There are two ways of dividing a strip into equal parts: a ladder that lands on the fraction as a rational number, and Fujimoto's method, which never arrives. Read as mathematics that settles it. Read as instructions for somebody with a sheet of paper it settles nothing, and past four parts the method that never arrives is the one whose crease lands nearer the mark.

One crossing, and then another

Folding a strip into thirds by Fujimoto's method halves the error at every fold and never reaches a third. There is a construction that arrives instead: cross the square's diagonal with a line through the mark you already have, and the crossing lands on the next fraction exactly — one fold per step, all the way down.

The grid a division makes

Dividing a square into thirds in both directions is a construction: four creases, each exact, each landing on a rational the ladder can name. The object it leaves behind is a three-by-three map of stamps, and how many ways that folds is the oldest open problem in the subject — 1,368 at three, 300,608 at four, and unknown at five.

The numbers a fold reaches

Folding solves cubics, which is one fact about one fold. The reason the subject has a theory rather than a bag of tricks is a second fact about all of them: the lengths a folder can mark are closed under addition, subtraction, multiplication, division, square roots and cube roots. Constructions can therefore be built out of constructions — and no tower of them ever arrives at a fifth root.

All the figures · The axioms and construction field · The patterns a reader can fold