Origami & Folding
what one sheet can do
Generator

Folding a strip into thirds — the fujimoto generator

A generator in the axioms and construction library, called 13 times across 5 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.

fujimoto 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

Folding a strip into thirdsA guess, and then halving. Each fold moves the mark to the midpoint of one of the two pieces, and each fold halves the distance to the exact division — so the error falls geometrically from whatever the first guess was. The sequence of halvings is read off the binary expansion of the fraction rather than chosen, and the halving of the error is asserted rather than observed.guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

n: 3

Folding a strip into thirdsA guess, and then halving. Each fold moves the mark to the midpoint of one of the two pieces, and each fold halves the distance to the exact division — so the error falls geometrically from whatever the first guess was. The sequence of halvings is read off the binary expansion of the fraction rather than chosen, and the halving of the error is asserted rather than observed.guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

n: 3, start: 0.5, folds: 6

Folding a strip into thirdsA guess, and then halving. Each fold moves the mark to the midpoint of one of the two pieces, and each fold halves the distance to the exact division — so the error falls geometrically from whatever the first guess was. The sequence of halvings is read off the binary expansion of the fraction rather than chosen, and the halving of the error is asserted rather than observed.guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

n: 5, start: 0.3, folds: 8

Folding a strip into fifthsA guess, and then halving. Each fold moves the mark to the midpoint of one of the two pieces, and each fold halves the distance to the exact division — so the error falls geometrically from whatever the first guess was. The sequence of halvings is read off the binary expansion of the fraction rather than chosen, and the halving of the error is asserted rather than observed.guessoff by 0.10000fold 1off by 0.05000fold 2off by 0.02500fold 3off by 0.01250fold 4off by 0.00625fold 5off by 0.00313fold 6off by 0.00156fold 7off by 0.00078fold 8off by 0.00039solid mark — the fold is aiming at 1/5; dashed — at where 1/5 has gonehalving word R R L L — period 4, from 2^4 − 1 = 5 × 3every fold halves the error exactly, so 8 folds divide it by 256

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.

Dividing a loop into n

Fujimoto's method divides a strip into any number of equal parts by folding badly and then folding the error in half, over and over. It converges because each step halves what is left over. On a closed loop there is no edge for the leftover to sit against, and what replaces the edge is the loop's own closure.

Exact is not accurate

This site has 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.

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 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 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.

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