Origami & Folding
what one sheet can do
Generator

Dividing a square into 3, exactly

A generator in the axioms and construction library, called 18 times across 9 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.

dividing-the-square 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

Dividing a square into 3, exactlyHaga's theorem. Folding one corner onto a point part-way along an opposite edge produces exact rational divisions elsewhere on the sheet — so a square can be divided into any whole number of parts by folding alone, with no measurement and no accumulated error.3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

parts: 8

Dividing a square into 8, exactlyHaga's theorem. Folding one corner onto a point part-way along an opposite edge produces exact rational divisions elsewhere on the sheet — so a square can be divided into any whole number of parts by folding alone, with no measurement and no accumulated error.8 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

parts: 3

Dividing a square into 3, exactlyHaga's theorem. Folding one corner onto a point part-way along an opposite edge produces exact rational divisions elsewhere on the sheet — so a square can be divided into any whole number of parts by folding alone, with no measurement and no accumulated error.3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

parts: 5

Dividing a square into 5, exactlyHaga's theorem. Folding one corner onto a point part-way along an opposite edge produces exact rational divisions elsewhere on the sheet — so a square can be divided into any whole number of parts by folding alone, with no measurement and no accumulated error.5 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

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.

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.

Designing on a grid

Box pleating gives up the efficiency of a free packing and buys creases that land where they are supposed to. For a design with hundreds of folds that is not a compromise — it is the only thing that makes it foldable.

Dividing without measuring

A square can be divided into any whole number of equal parts by folding alone — exactly, with no ruler, and with no error to accumulate. The construction is one fold and a theorem nobody expected.

Folding a strip into thirds

A third cannot be constructed by the axioms, so it is not constructed. It is guessed, and then halved into place — an algorithm rather than a construction, with an error that falls by exactly half at every fold.

One fold at a time, and there are exactly seven of them

A fold is specified by bringing points and lines into coincidence. There are seven ways to do that, the list is provably complete, and one of the seven does something no compass can.

The eleven-sided one nobody can fold

Folding reaches the heptagon, which a compass cannot. It does not reach the hendecagon, and the obstruction is a single prime factor: ten has a five in it, a fold solves cubics, and no arrangement of cubics produces a five.

The kindergarten was a geometry class

Froebel put paper folding into mass education in the 1830s, and did it as mathematics rather than as craft. His three categories — the folds of life, of beauty, and of knowledge — are the first systematic treatment of folding anybody wrote down, and the third one is a geometry syllabus.

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.

What the grid settles

Box pleating is usually defended as a trade: give up efficiency, buy creases that land where they should. There is a second thing it buys and nobody quotes it — on a lattice the best possible packing is a finite question with an answer, while off the lattice nobody knows the best packing of six circles in a square and probably never will.

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