Origami & Folding
what one sheet can do
Generator

One fold, and a third

A generator in the who found it, and when library, called 8 times across 4 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.

haga-construction 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 fold, and a thirdThe top-left corner brought down to the midpoint of the bottom edge. The crease is the perpendicular bisector, so it meets the vertical edges at 3/8 and 7/8; and the folded top edge crosses the right-hand edge at exactly two thirds. Nothing is measured and every value is exact.the midpointwhat comes outcrease on the left edge 3/8crease on the right edge 7/8the folded edge crosses at2/3exact, and a trisectionthe corner is placed by folding, not by measuring — which is why theresult is exact

m: 0.5

One fold, and a thirdThe top-left corner brought down to the midpoint of the bottom edge. The crease is the perpendicular bisector, so it meets the vertical edges at 3/8 and 7/8; and the folded top edge crosses the right-hand edge at exactly two thirds. Nothing is measured and every value is exact.the midpointwhat comes outcrease on the left edge 3/8crease on the right edge 7/8the folded edge crosses at2/3exact, and a trisectionthe corner is placed by folding, not by measuring — which is why theresult is exact

m: 0.4

One fold, and a thirdThe top-left corner brought down to the midpoint of the bottom edge. The crease is the perpendicular bisector, so it meets the vertical edges at 3/8 and 7/8; and the folded top edge crosses the right-hand edge at exactly two thirds. Nothing is measured and every value is exact.the midpointwhat comes outcrease on the left edge 0.4200crease on the right edge 0.8200the folded edge crosses at0.5714not the classical values herethe corner is placed by folding, not by measuring — which is why the result isexact

m: 0.5, sq: 300

One fold, and a thirdThe top-left corner brought down to the midpoint of the bottom edge. The crease is the perpendicular bisector, so it meets the vertical edges at 3/8 and 7/8; and the folded top edge crosses the right-hand edge at exactly two thirds. Nothing is measured and every value is exact.the midpointwhat comes outcrease on the left edge 3/8crease on the right edge 7/8the folded edge crosses at2/3exact, and a trisectionthe corner is placed by folding, not by measuring — which is why theresult is exact

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.

A record is not a proof

Every other claim here can be re-derived from the figure that makes it, and a wrong one shows. A date cannot: it is checked once, by hand, against a record that is itself a survivor. This field is the one most likely to be wrong and least likely to be caught, and saying so is the only defence it has.

A schoolteacher's theorem

Kazuo Haga folded a corner of a square to the midpoint of the far side and found exact thirds. The construction needs one fold, no measurement and no compass, the numbers that come out are exactly 3/8, 7/8 and 2/3, and it was found by a biology teacher looking for something to do with a classroom.

The largest triangle in a square

The biggest equilateral triangle a square sheet holds is tilted by exactly fifteen degrees and uses 46.4% of the paper. Both numbers come out of a quadratic — which means a compass reaches this optimum too, and folding's advantage is not needed here at all.

Two creases at once

The seven axioms describe what one fold can do, and the restriction to one fold is a rule somebody imposed rather than a property of paper. Allow two creases to be made simultaneously and the reachable degree rises — and the hendecagon nobody could fold becomes foldable.

All the figures · The who found it, and when field · The patterns a reader can fold